# Diagnose Like a Radiologist:
Hybrid Neuro-Probabilistic Reasoning for Attribute-Based Medical Image
Diagnosis

Gangming Zhao, Quanlong Feng, Chaoqi Chen, Zhen Zhou, and Yizhou Yu
(Corresponding author: Yizhou Yu) G. Zhao, C. Chen and Y. Yu are with
Department of Computer Science, the University of Hong Kong, Pokfulam, Hong
Kong. E-mail<EMAIL_ADDRESS>Q. Feng is with Department of Geographical
Information Engineering, China Agricultural University, Beijing, China. Z.
Zhou is with the AI Lab, Deepwise Healthcare, Beijing 100080, China.

###### Abstract

During clinical practice, radiologists often use attributes, e.g.
morphological and appearance characteristics of a lesion, to aid disease
diagnosis. Effectively modeling attributes as well as all relationships
involving attributes could boost the generalization ability and verifiability
of medical image diagnosis algorithms. In this paper, we introduce a hybrid
neuro-probabilistic reasoning algorithm for verifiable attribute-based medical
image diagnosis. There are two parallel branches in our hybrid algorithm, a
Bayesian network branch performing probabilistic causal relationship reasoning
and a graph convolutional network branch performing more generic relational
modeling and reasoning using a feature representation. Tight coupling between
these two branches is achieved via a cross-network attention mechanism and the
fusion of their classification results. We have successfully applied our
hybrid reasoning algorithm to two challenging medical image diagnosis tasks.
On the LIDC-IDRI benchmark dataset for benign-malignant classification of
pulmonary nodules in CT images, our method achieves a new state-of-the-art
accuracy of 95.36% and an AUC of 96.54%. Our method also achieves a 3.24%
accuracy improvement on an in-house chest X-ray image dataset for tuberculosis
diagnosis. Our ablation study indicates that our hybrid algorithm achieves a
much better generalization performance than a pure neural network architecture
under very limited training data.

###### Index Terms:

Bayesian Networks, Deep Neural Networks, Medical Image Analysis, Neuro-
Probabilistic Reasoning

##  1 Introduction

Due to their rapid progress in the past ten years, deep neural networks have
achieved tremendous success in boosting the performance of image recognition
[27, 50, 18, 19] and other visual computing tasks [48, 38]. Such progress has
also propelled forward many related research areas including medical image
analysis. Nowadays, deep neural networks have been widely used for medical
image analysis and diagnosis. They have demonstrated unprecedented accuracy on
a variety of tasks, such as the detection of acute intracranial hemorrhage in
head CT images [29], breast cancer diagnosis using mammography [39], the
detection of diabetic retinopathy in retinal fundus photographs [14], and the
classification of skin cancers [9].

Despite the high accuracy deep neural networks have achieved in medical image
diagnosis, they still exhibit two major limitations, insufficient
generalization ability and verifiability. It is well known that deep neural
networks require large amounts of training data. When high quality training
data is scarce, as is typically the case in the medical image domain, the
performance of trained deep neural networks on novel testing data may
deteriorate significantly. In addition, deep neural networks typically produce
a final result without showing how and why the result can be reached. This is
undesired because very often computer-generated diagnostic results only serve
as decision support for human doctors, who need to verify their credibility
before accepting them.

To improve verifiability, a medical image diagnosis algorithm needs to learn
the knowledge reasoning process followed by radiologists during clinical
practice. They start from visual evidences in medical images and reach
diagnostic conclusions by referring to causal relationships between diseases
and the visual evidences. For example, radiologists use intuitive
morphological and appearance characteristics of a pulmonary nodule, such as
lobulation, spiculation, and texture, as visual evidences to assess whether
the nodule is benign or malignant [21] (Fig. 1). In this paper, such
characteristics of a lesion or, in general, radiological abnormalities in a
medical image are defined as the attributes of the lesion or image. Thus,
equipping a machine learning model with the capability of simultaneously
predicting attributes as well as underlying diseases improves the causality
and verifiability of the model now that radiologists can verify the predicted
attributes as well as the knowledge reasoning process that goes from
attributes to diagnostic conclusions.

Figure 1: Sample 2D slices of 3D pulmonary nodules from the LIDC-IDRI dataset,
each with one of the following eight attributes: Subtlety, Internal Structure,
Calcification, Sphericity, Margin, Lobulation, Spiculation, Texture. The
annotated grade of each attribute is shown between the parentheses. For
example, Subtlety (5) means its annotated grade is 5.

In addition to causal relationships between diseases and attributes, there
also exist relationships, such as co-occurrence, among the attributes
themselves. Effectively modeling and exploiting the hidden relationships among
these attributes as well as the relationships between attributes and the
underlying diseases can boost the accuracy of attribute and disease
prediction. Nonetheless, there have been few methods in medical image
diagnosis exploiting such relationships.

To address generalization ability, we need to look at the strengths and
limitations of both neural and probabilistic learning algorithms for
relational modeling and reasoning. Examples of neural algorithms include graph
neural networks [26] and relation networks [51] while examples of
probabilistic learning algorithms include Bayesian networks [47] and factor
graphs [11]. Neural algorithms are good at learning from large amounts of
empirical data. The trained models can interpolate and approximate known
information very well to reach high accuracy, but have relatively poor
generalization ability, causality and explainability. On the other hand,
probabilistic learning algorithms, which can be considered as a type of
symbolic AI methods, can learn from smaller datasets. The trained models have
better generalization ability, causality and explainability, but lower
accuracy when facing real-world empirical data.

In this paper, to exploit the complementary strengths of neural and
probabilistic learning algorithms as well as overcome the limitations of both
ends, we introduce a hybrid neuro-probabilistic reasoning algorithm for
verifiable attribute-based medical image diagnosis. Specifically, the proposed
hybrid algorithm consists of multiple stages. It first employs a deep neural
network backbone to extract features from an input image. Such features build
the foundation of reasoning that happens in later stages. Next, a Bayesian
network (BN) branch and a graph convolutional network (GCN) branch process the
features extracted by the backbone and perform relational reasoning in
parallel. The BN branch first converts features into attributes as evidences,
then performs probabilistic reasoning using the causal relationships between
the disease and attributes, and finally produces their marginal posterior
distributions. On the other hand, the GCN branch performs feature-based
relational modeling. It represents attributes and generic relationships among
them using a graph. The GCN branch exploits the relationships among the graph
nodes to enhance the feature of every attribute. More accurate diagnoses and
attribute classifications can be achieved using the enhanced features from the
GCN. Although the BN and GCN belong to two parallel branches, they are not
completely independent. We tightly couple the BN and GCN branches so that the
BN is utilized to improve the causality and generalization ability of the GCN
while the accuracy of the GCN is retained. Such tight coupling is achieved via
a cross-network attention mechanism and the fusion of their classification
results. By fusing the results from the two branches, causality enforced by
the BN branch may be partially compromised. Therefore, a second Bayesian
network module is appended at the end of the entire network to reinforce
causality.

The entire network except the Bayesian network modules can be trained using
gradient back-propagation as the inference phase of a Bayesian network is
differentiable and gradients can be propagated through the two Bayesian
networks in our algorithm. However, we notice that the compatibility between
the structure of the Bayesian networks and the rest of our hybrid network
significantly affects the performance of the entire network. Thus, once we
learn the initial structure of the Bayesian networks using ground-truth
annotations, the training procedure proceeds by alternating between two
phases. The first phase updates the CNN and GCN weights using stochastic
gradient descent while the second phase updates the structures and parameters
of the two Bayesian networks using the attribute classification results most
recently predicted by their preceding stages in our hybrid network as training
labels.

Note that verifiability is more stringent than explainability. For computer-
generated diagnoses to be verifiable, the internal decision process of an
algorithm not only should be explainable, but also needs to comply with the
domain knowledge of human doctors. As mentioned earlier, in medical image
diagnosis, radiologists start from visual evidences in medical images and
reach diagnostic conclusions by referring to causal relationships between
diseases and the visual evidences. Thus the algorithm needs to collect the
same types of visual evidences as a doctor would do, and also learn the causal
relationships between evidences and diseases as a doctor would do. The
proposed algorithm in this paper models visual evidences as attributes and
collects such visual evidences through attribute prediction. It further models
causal and non-causal relationships using BN and GCN, which quantitatively
encode such medical domain knowledge as “the presence of attribute A in this
CT scan increases/decreases the likelihood that the patient has disease D” or
“the occurrence of attribute A in this CT scan increases/decreases the chance
that attribute B also occurs in the same scan”. Every node in these networks
(BN and GCN) represents a high-level concept (a disease or an attribute), and
a disease diagnosis exploits the relationships between the disease and a set
of other high-level concepts. In comparison to previous medical image
diagnosis methods, radiologists may find the knowledge reasoning process of
the proposed algorithm more consistent with their own practice and,
consequently, the disease diagnoses made by our method more credible.

We have successfully applied our hybrid reasoning algorithm to two challenging
medical image diagnosis tasks. The first task is benign-malignant
classification of pulmonary nodules in chest computed tomography (CT) images
in the LIDC-IDRI benchmark dataset. As mentioned earlier, for pulmonary nodule
classification, morphological and appearance attributes have been commonly
used to assist clinical diagnosis [21]. The second task is tuberculosis (TB)
diagnosis using chest X-ray images. For this task, we adopt an in-house
dataset carefully annotated by senior medical experts. Every image in this TB
dataset is not only annotated with a disease diagnosis, but also multiple
types of radiological abnormalities, such as ‘Pulmonary Consolidation’,
‘Pulmonary Cavitation’, ‘Diffuse Nodules’, and ‘Fibrotic Appearance’,
potentially caused by tuberculosis. According to the experimental results of
pulmonary nodule classification on the LIDC-IDRI benchmark, our method
achieves a new state-of-the-art accuracy of 95.36% and an AUC of 96.54%. On
our in-house dataset for tuberculosis classification, our algorithm also
achieves a 4.98% improvement in accuracy in comparison to previously best-
performing algorithm for attribute-based medical image classification.

Our main contributions in this paper can be summarized as follows.

  * •

We introduce a hybrid neuro-probabilistic reasoning algorithm for attribute-
based medical image diagnosis. To support causal and verifiable relational
modeling and reasoning, this algorithm tightly couples Bayesian networks and a
graph convolutional network. Such coupling is achieved via a cross-network
attention mechanism and a classification result fusion scheme.

  * •

We devise an effective training procedure for the proposed hybrid network. It
alternates between two phases. The first phase updates the CNN backbone and
GCN weights while the second phase updates the structures and parameters of
the Bayesian networks.

  * •

We have successfully applied the proposed hybrid reasoning algorithm to two
representative medical image diagnosis tasks, benign-malignant classification
of pulmonary nodules in chest CT images and tuberculosis diagnosis using chest
X-ray images. The proposed algorithm achieves state-of-the-art performance on
the LIDC-IDRI benchmark dataset for the first task and an in-house dataset for
the second task.

##  2 Related Work

###  2.1 Attribute Learning

Attributes, such as texture, color, and shape, are of great importance to
describe objects. Attribute learning has been studied in computer vision for
many years [10, 28, 1, 30, 34, 33, 42]. Ferrari et al. [10] proposed to use
low-level semantic features for attribute representation and they presented a
probabilistic generative model for visual attributes, together with an image
likelihood learning algorithm. Human faces have many attributes, and remain a
challenge for attribute learning. Kumar et al. [28] trained binary classifiers
to recognize the presence or absence of describable aspects of facial visual
appearance using traditional hand-crafted features. Liu et al. [37] proposed a
CNN framework for face localization and attribute prediction, respectively.
Attributes have also been exploited in tasks such as zero-shot learning [30,
23]. Effectively modeling the hidden relationships among attributes is useful
for improving the accuracy of attribute prediction and causal association.
Nonetheless, most of the early works in attribute learning did not model
relationships among attributes and explore such relationships for attribute
reasoning. The development of graph neural networks (GNN) [26] made it
possible to learn relationships among attributes. For example, Meng et al.
[41] used message passing to perform end-to-end learning of image
representations, their relationships as well as the interplay among different
attributes. They observed that relative attribute learning naturally benefits
from exploiting the graph of dependencies among different image attributes. In
this paper, we not only utilize a graph neural network to model the
correlations among attributes, but also embed Bayesian networks into the whole
framework to better model causality.

Figure 2: Our proposed framework consists of three stages: 1) Deep feature
extraction using an CNN-FPN backbone; 2) hybrid attribute relational modeling
and reasoning using a pair of coupled Bayesian network and deep graph network;
3) causality reinforcement using a second Bayesian network. Note that BN-1 and
BN-2 are distinct in both parameters and structures.

###  2.2 Bayesian Networks

Bayesian networks (BN), introduced by Judea Pearl [46], represent a natural
approach to model causality and perform logical reasoning. Bayesian network
learning includes two phases, structure learning and parameter learning. The
most intuitive method for structure learning is that of ‘search and score,’
where one searches the space of directed acyclic graphs (DAGs) using dynamic
programming and identifies the one that minimizes the objective function [58].
For parameter learning, the most frequently used method is maximum likelihood
estimation (MLE) [46]. However, when given a dataset, BN cannot learn a
feature representation, which limits its further development. Recently,
researchers have started to focus on the integration between BN and deep
learning. For example, Rohekar et al. [49] proposed to utilize BN models for
learning better deep neural networks. Meanwhile, a few improved versions of BN
have been proposed for new applications in computer vision. For example, Barik
[3] improved BN using low-rank conditional probability tables. Elidan et al.
[8] used the Copula Bayesian Network model for representing multivariate
continuous distributions. The method in this paper differs from the above work
in that it exploits the representation power of deep neural networks and the
causality modeling capability of BN by embedding them into a unified
framework.

###  2.3 Bayesian Neural Networks

In addition to poor interpretability, deep neural networks also suffer from
overfitting on limited labeled data and the incapability of uncertainty
analysis. To tackle these issues, researchers combined Bayesian methods with
deep learning to construct Bayesian neural networks (BNN) for a probabilistic
representation of uncertainty of the latter [25, 12, 56]. Specifically, a BNN
estimates posterior distributions for the weights in a neural network. There
exist a few estimation methods, such as variational inference [25] and dropout
variational inference [12]. However, the proposed method in this paper differs
significantly from Bayesian neural networks. Unlike BNNs, our proposed method
aims to improve the causality and interpretability of deep learning via the
fusion of deep learning and traditional graph-based probabilistic reasoning.
It cannot perform uncertainty analysis. Nonetheless, due to the few trainable
parameters in Bayesian networks, the proposed method can clearly mitigate
overfitting on small training datasets, as demonstrated in one of our ablation
studies.

###  2.4 Neuro-symbolic Learning

The integration of connectionist (neural networks) and symbolic AI has long
been a key issue in machine learning [54, 53, 65, 66, 32]. Early works [54,
53] primarily focused on the injection of symbolic information as prior
knowledge into a neural network. More recently, with the wide spread of deep
learning, researchers [65, 66, 32] started to combine the merits of deep
feature representation with symbolic reasoning to build strong, reliable and
explainable neural-symbolic AI systems. Relevant studies have been conducted
on visual question answering (VQA) [65], semantic parsing [66] and handwritten
formula recognition [32]. One of the key issues in neural-symbolic learning is
that most symbolic approaches are not differentiable, making end-to-end
training difficult. In this study, we partially solved this issue by
performing gradient back-propagation through both Bayesian networks, which
serve as the symbolic reasoning module in our hybrid network, for a better
feature representation and a more efficient training procedure.

##  3 Algorithm

###  3.1 Overview

The proposed hybrid neuro-probabilistic reasoning algorithm takes an image
patch as the input and outputs a classification result, as shown in Fig. 2.
The proposed algorithm consists of the following three stages.

The Backbone Stage:

a backbone based on ResNet [18] or EfficientNet [57] is utilized for deep
feature representation, which builds the foundation for reasoning in later
stages. Moreover, a feature pyramid network (FPN) [36] is integrated into the
backbone for multi-scale feature aggregation (Fig. 2). A distinct global
average pooling (GAP) operator is applied to the feature map at each scale of
the FPN to reduce the number of parameters. And for reducing information loss
in low-level feature maps, we ensure that the dimension of the resulting
feature from GAP is kept the same at different scales of the FPN. For
instance, when the size of a feature map in the FPN is $64\times 64\times
256$, we first reshape it to $32\times 16\times 2048$ by concatenating the
third dimensions of every $2\times 4$ neighbors in the first two dimensions
together, then use GAP to produce a 2048-dimensional feature vector. All GAP-
reduced features go through parallel FC layers before they are fused via a sum
operator and a subsequent FC layer to obtain the final feature, $F_{0}$, at
the backbone stage.

The Reasoning Stage:

both a Bayesian network and a graph convolutional network are adopted for
relational reasoning in parallel. The BN branch first converts deep features
into attributes through attribute classification, then performs probabilistic
causal reasoning using these attributes, and finally outputs marginal
posterior distributions. The GCN branch first transforms the deep features
from the backbone into a feature representation for individual attributes and
represents the relationships among the attributes using an undirected graph,
then enhances the feature of each attribute via graph convolutions. The BN and
GCN are tightly coupled together through a cross-network attention mechanism
and the fusion of their classification results, which will be described in
Section 3.4.

The Second BN Module:

to reinforce causality of the proposed hybrid algorithm, a second BN is
appended at the end of the entire network.

###  3.2 Integration between BN and Backbone

A Bayesian network is incorporated in the reasoning stage to model
probabilistic dependencies among various attributes for disease diagnosis.
Specifically, a Bayesian network,
$\mathcal{B}=<\mathcal{V_{B}},\mathcal{E_{B}},\Theta>$, is a directed acyclic
graph (DAG) $<\mathcal{V_{B}},\mathcal{E_{B}}>$ with a conditional probability
table (CPT) for each node, and $\Theta$ represents all parameters (CPTs),
which encodes the joint probability distribution of the BN. Each node
$v_{i}\in\mathcal{V_{B}}$ stands for a random variable, and a directed edge
$e\in\mathcal{E_{B}}$ between two nodes $(v_{i},v_{j})$ indicates $v_{i}$
probabilistically depends on $v_{j}$. The training of a Bayesian network
undergoes two phases, structure learning and parameter learning [46, 47, 7].
We adopt dynamic programming with the Bayesian information criterion (BIC)
[58] for structure learning to determine the topology of the DAG and maximum
likelihood estimation for parameter learning to determine $\Theta$. Our BN
module first learns its structure and then updates all parameters, i.e. the
conditional probability tables at all nodes in the network. In this paper,
each attribute has a corresponding node in the Bayesian network. Without loss
of generality, assume there is only one disease we wish to diagnose, and that
disease is an extra node in the network.

To integrate the Bayesian network with our backbone, the final feature from
the backbone, $F_{0}$, is fed to another FC-layer with weight matrix
$\mathcal{W}_{\mathcal{B}}$ to obtain a feature map $F_{\mathcal{B}}$. Thus,
$F_{\mathcal{B}}=\mathcal{W}_{\mathcal{B}}F_{0}(\in\mathbb{R}^{(N+1)\times
C})$, where $N$ is the number of attribute categories and $C$ is the number of
grades for each attribute or disease. Each attribute must have at least two
grades to indicate whether that attribute is present or not. In practice,
there is often a need to have more than two grades for each attribute or
disease to indicate intermediate levels of certainty or severity. Note that
different grades of the same attribute are mutually exclusive while different
attributes are not because more than one attribute could be present at the
same time. Thus, $F_{\mathcal{B}}$ can be interpreted as the concatenation of
$N+1$ $C$-dimensional vectors, one for each node in the Bayesian network. We
further define $P^{0}_{B}=\sigma(F_{\mathcal{B}})(\in[0,1]^{(N+1)\times C})$,
where the softmax operator $\sigma$ is applied to each $C$-dimensional vector
in $F_{B}$, and the result represents a discrete distribution over the grades
of an attribute or disease. This distribution becomes the input evidence at
the corresponding node in the Bayesian network.

Once we perform inference in the Bayesian network using the conditional
probability tables as well as the input evidences at all nodes, we obtain
marginal posterior distributions at all nodes, including the extra node for
disease diagnosis, as the output of the Bayesian network. Let the nodes in the
Bayesian network be $v_{i},i\in\\{0,1,...,n\\}$. Without loss of generality,
according to [46], the marginal posterior distribution $P_{B}(v_{0})$ at node
$v_{0}$ is formulated as

$P_{B}(v_{0})=\int...\int_{V}P(v_{0},v_{1},...,v_{n})\mathrm{d}v_{1}...\mathrm{d}v_{n},$
(1)

where

$P(v_{0},v_{1},...,v_{n})=\Pi_{i=0}^{n}P(v_{i}|Parents(v_{i})),$ (2)

where $Parents(v_{i})$ is NULL if $v_{i}$ does not have any parent nodes. The
equation in (2) is derived using the local Markov property of Bayesian
networks. In practice, to evaluate (1), we use the belief propagation
algorithm in [45].

###  3.3 GCN Module

In addition to the Bayesian network, we also adopt a graph convolutional
network in the reasoning stage to perform feature-based relational modeling
and reasoning among attributes to facilitate medical image diagnosis.
Specifically, a GCN is based on an undirected graph
$\mathcal{G}=(\mathcal{V_{G}},\mathcal{E_{G}})$, where $\mathcal{V_{G}}$ and
$\mathcal{E_{G}}$ are the set of nodes and edges respectively. Here, there is
a node corresponding to each attribute or disease, and the node holds a
feature vector associated with that attribute or disease. Again, assume there
is only one disease we wish to diagnose, and the total number of nodes is
$N+1$. If vertices $v_{i}$ and $v_{j}$ are connected in the graph, there might
exist certain type of relations between them. We associate a learnable weight
with every edge in the graph to indicate the strength of the connection. The
GCN in our algorithm consists of $L$ layers, all of which share the same
number of nodes. But the edge weights in these layers may be different.

A general graph convolution operation $\mathcal{F}$ at the $l$-th layer can be
formulated as follows

$\mathcal{F}(\mathcal{G}_{l},\mathcal{W}_{l})=\mbox{Update}(\mbox{Aggregate}(\mathcal{G}_{l},\mathcal{W}^{agg}_{l}),\mathcal{W}^{update}_{l}),$
(3)

where $\mathcal{G}_{l}=(\mathcal{V_{G}}_{l},\mathcal{E}_{l})$ is the graph at
the $l$-th layer; $\mathcal{W}^{agg}_{l}$ and $\mathcal{W}^{update}_{l}$ are
the learnable weights of the aggregation and update functions of the $l$-th
layer, respectively. We use a max-pooling node feature aggregator to pool the
pairwise differences of features at node $v_{i}$ and all of its neighbors. But
the feature difference between $v_{i}$ and one of its neighbors is modulated
by the edge weight between them before the pooling operation. The node feature
updater is a multi-layer perceptron (MLP) with batch normalization and ReLU as
the activation function. The above MLP concatenates the original node feature
with the aggregated feature as its input.

We adopt the residual graph convolution introduced in [31]. Let
$H_{\mathcal{G}_{l}}\in\mathcal{R}^{N\times D_{l}}$ be the feature map holding
the features at all nodes of $\mathcal{G}_{l}$, where $N$ is the number of
attributes and $D_{l}$ is the feature dimension in the $l$-th layer. The
residual graph convolution between the $l$-th and ($l+1$)-th layers of the GCN
is formulated as follows,

$H_{\mathcal{G}_{l+1}}=\mathcal{F}(\mathcal{G}_{l},\mathcal{W}_{l})+H_{\mathcal{G}_{l}}.$
(4)

To integrate the above GCN with our backbone, we need to obtain $N+1$ features
from the final feature in the backbone, one for each node in the GCN. For this
purpose, $F_{0}$ is fed to another FC-layer with weight matrix
$\mathcal{W}_{\mathcal{G}}$ to obtain feature map $H_{\mathcal{G_{0}}}$. Thus,
$H_{\mathcal{G_{0}}}=\mathcal{W}_{\mathcal{G}}F_{0}(\in\mathbb{R}^{(N+1)\times
D_{0}})$, where $N$ is the number of attribute categories and $D_{0}$ is the
input feature dimension for every node in the GCN. $H_{\mathcal{G_{0}}}$ thus
can be interpreted as the concatenation of $N+1$ $D_{0}$-dimensional feature
vectors, one for each node in the first layer of the GCN as input.

The final feature map of the GCN is
$H_{\mathcal{G}_{L}}\in\mathbb{R}^{(N+1)\times D_{L}}$. We further use a FC-
layer followed by a softmax operator to obtain attribute and disease
classification results in the following equation,

$P_{G}=\sigma(\mathcal{W}_{P}H_{\mathcal{G}_{L}})$ (5)

where $P_{G}\in[0,1]^{(N+1)\times C}$ denotes the attribute and disease
predictions.

###  3.4 Coupling between BN and GCN

Although both the BN and GCN are utilized for relational reasoning, they are
not completely independent. In this work, we tightly couple BN and GCN via a
cross-network attention mechanism and the fusion of classification results. As
for the former, BN provides attention for the node features in each layer of
the GCN. Specifically, the marginal posterior distributions at all nodes of
the BN are taken as the input of $L$ node attention modules, one for each
layer of the GCN to figure out the relative importance of attributes in that
layer. Node attention values prescribed by the $l$-th attention module are
defined as follows,

$M_{attn}^{l}=\sigma(\mathcal{W}_{l_{1}}\mbox{ReLU}(\mathcal{W}_{l_{0}}P_{B}(\mathcal{V}))),l\in{1,2,...,L}$
(6)

where $M_{attn}^{l}\in\mathcal{R}^{N+1}$ is the node attention vector for the
$l$-th layer of the GCN, $\mathcal{V}$ includes all attribute and disease
nodes of the BN, and $P_{B}(\mathcal{V})\in\mathcal{R}^{(N+1)\times C}$
represents the concatenated marginal distributions at all nodes of the BN.
Note that there is one attention value in $M_{attn}^{l}$ for each node in the
$l$-th layer of the GCN. We name $M_{attn}^{l}$ the spatial attention map for
GCN nodes since nodes are spatial entities of a graph, similar to pixels in a
CNN.

We also extend squeeze-and-excitation based channel-wise attention in SENet
[20] to graph convolutional networks as follows to further enhance the
features of every layer in the GCN,

$\displaystyle C_{attn}^{l}=$
$\displaystyle\sigma(\mathcal{W}_{l_{ex}}\mbox{ReLU}(\mathcal{W}_{l_{sq}}GAP(H_{\mathcal{G}_{l}})))$
(7)

where ${C_{attn}^{l}}\in\mathcal{R}^{D_{l}},l\in{1,2,...,L}$ is the channel-
wise attention vector for the $l$-th layer of the GCN, $\mathcal{W}_{l_{sq}}$
and $\mathcal{W}_{l_{ex}}$ are the weights for squeezing and excitation,
respectively.

Once both spatial and channel-wise attention vectors have been calculated,
firstly the same channel of all features in $\mathcal{G}_{l}$ is multiplied by
its corresponding channel-wise attention value, then the entire feature at
each node in $\mathcal{G}_{l}$ is multiplied by its corresponding spatial
attention value. Such spatially and channel-wise modulated features at the
$l$-th layer of the GCN serve as the input to the next graph convolution to
produce the feature map $H_{\mathcal{G}_{l+1}}$ at the $(l+1)$-th layer.

As for the fusion of the disease classification results from BN and GCN, a
residual fusion scheme is formulated as follows,

$P^{d}_{fusion}=w_{B}\cdot
P^{d}_{B}+(1-w_{B})\cdot\sigma(W_{0}\mbox{Concat}(P^{d}_{G},P^{d}_{B})).$ (8)

For attribute classification results, the same residual fusion scheme is
adopted as follows,

$P^{a}_{fusion}=w^{\prime}_{B}\cdot
P^{a}_{B}+(1-w^{\prime}_{B})\cdot\sigma(W^{\prime}_{0}\mbox{Concat}(P^{a}_{G},P^{a}_{B})).$
(9)

In these equations, $P^{d}_{B}$ and $P^{a}_{B}$ are the marginal posterior
distributions at the disease and attribute nodes in the BN, $P^{d}_{G}$ and
$P^{a}_{G}$ are the probabilistic disease and attribute classification results
from the GCN, $w_{B}$ and $w^{\prime}_{B}$ are two learnable trade-off
coefficients, $W_{0}$ and $W^{\prime}_{0}$ are the weight matrices of two
fully connected layers, and Concat() is the concatenation operator.

###  3.5 The Second BN Module

It should be noted that by fusing the results of both BN and GCN branches,
causality enforced by the BN branch may be compromised to some extent. To
tackle this issue, we append a second Bayesian network at the end of the
entire network to reinforce causality. The fused attribute and disease soft
classification results in (9) and (8) are used as the input evidences at the
nodes in this second Bayesian network. The marginal posterior distribution at
the disease node, denoted as $P^{d}_{final}$, can also be defined using
equations similar to those in (1) and (2). In practice, it is also evaluated
using the belief propagation algorithm. This marginal posterior distribution
at the disease node of the second BN becomes the final disease prediction of
the entire network. Note that BN-1 and BN-2 are distinct in both parameters
and structures.

###  3.6 Training Scheme

Our entire network except the BN modules can be trained using stochastic
gradient descent. Meanwhile, we observe that the inference phase of a BN is
differentiable with respect to its inputs and parameters, but not its
structure. That implies gradients can be backpropagated through a BN from its
outputs (i.e. marginal posterior distributions) to its inputs (i.e.
evidences). Parameter learning in a BN is tightly coupled with structure
learning, and is driven by global statistics of the training set. Performing
parameter learning separately from structure learning using stochastic
gradient descent is unlikely to be very effective. Moreover, we discovered
that the compatibility between the Bayesian network structures and the neural
parts of our hybrid network significantly affects the overall performance.
Therefore, we propose the following two alternating phases to train the entire
hybrid network.

  * •

The first phase updates the weights of the CNN backbone and the GCN branch
using gradient backpropagation by fixing the structures and parameters of the
two BNs. Gradients are simply backpropagated through these two BNs.

  * •

The second phase updates the structures and conditional probability tables of
the BN modules using the attribute and disease classification results most
recently predicted by their preceding stages in the hybrid network as training
labels. As for structure learning, dynamic programming is used with the same
score function as in [58]. Once the structure of BN modules are determined,
their conditional probability tables are updated immediately using maximum
likelihood estimation.

It should be noted that we actually set a maximum value on the number of times
the structures and parameters of the two BNs are updated (20 times in
practice). Once the two phases have been alternated for such a number of
times, we fix the structures and parameters of the two BNs and only update the
weights in the neural parts of our hybrid network. In this way, we can
guarantee the convergence of our training process.

The ground-truth annotations of the training samples are only used to train
the initial structure and parameters of the BN modules. In subsequent
iterations, for the first BN, the ground-truth attribute and disease
annotations of a training sample are replaced with its attribute and disease
classification results most recently produced by the backbone, i.e.
$P^{0}_{B}$; for the second BN, they are replaced with its most recent fused
attribute and disease classification results, $P^{d}_{fusion}$ and
$P^{a}_{fusion}$ in (8) and (9). The effectiveness of this training strategy
has been empirically verified. Potential reasons for the effectiveness are
twofold: first, attribute classification results produced at intermediate
stages of our hybrid network are soft labels, which are actually
distributions, and have more complete information than the ground-truth
attribute labels represented as one-hot vectors; second, using soft attribute
labels generated within the hybrid network makes the trained Bayesian network
structures more compatible with the neural parts of the hybrid network.

We rely on the belief propagation algorithm to perform inference in a Bayesian
network and compute the marginal posterior distributions for its variables.
Since we train the weights in the neural networks by propagating gradients
through the Bayesian networks, we need to obtain the gradient of the belief
propagation algorithm. A method for computing this gradient is described in
the Appendix.

####  3.6.1 Training Loss

During the first phase of the above alternating training procedure, we adopt
the deep supervision strategy by distributing multiple supervision signals to
various stages of our hybrid network. Most of the time, each supervision
signal is primarily responsible for training a subset of the modules in the
network. The overall loss function during the first phase of training is a
weighted combination of five loss functions,

$\displaystyle\mathcal{L}_{all}=w_{1}\mathcal{L}^{0}_{G}+$ $\displaystyle
w_{2}\mathcal{L}^{0}_{B}+w_{3}\mathcal{L}_{a}+w_{4}\mathcal{L}_{d}+w_{5}\mathcal{L}_{final}$
(10)

where $\sum^{5}_{i=1}w_{i}=1$. In our experiments, we always set
$w_{i}=0.2,i\in{1,2,..,5}$.

The first two supervision signals corresponding to losses
$\mathcal{L}^{0}_{B}$ and $\mathcal{L}^{0}_{G}$ are respectively located at
the beginning of the BN and GCN branches. Two classifiers are involved. One of
them is simply $P^{0}_{B}=\sigma(F_{\mathcal{B}})(\in[0,1]^{(N+1)\times C})$,
the input evidences at the nodes of the first Bayesian network. The cross-
entropy loss for this classifier is written as follows,

$\displaystyle\mathcal{L}^{0}_{B}=$
$\displaystyle-\sum_{i=1}^{N+1}\sum_{j=1}^{C}(y_{ij}\log(p^{B,0}_{ij})+(1-y_{ij})\log(1-p^{B,0}_{ij}))$
(11)

The second classifier is defined as
$P^{0}_{G}=\sigma(\mathcal{W}_{\mathcal{T}}H_{\mathcal{G_{0}}})(\in[0,1]^{(N+1)\times
C})$, which takes $H_{\mathcal{G_{0}}}$ as the input and performs both
attribute and disease classification. The cross-entropy loss for this
classifier is written as follows,

$\displaystyle\mathcal{L}^{0}_{G}=$
$\displaystyle-\sum_{i=1}^{N+1}\sum_{j=1}^{C}(y_{ij}\log(p^{G,0}_{ij})+(1-y_{ij})\log(1-p^{G,0}_{ij}))$
(12)

In the above two loss functions, $y_{ij}$’s denote one-hot encoding of the
ground-truth attribute or disease grades of training samples, $p^{B,0}_{ij}$
and $p^{G,0}_{ij}$ are components of $P^{0}_{B}$ and $P^{0}_{G}$ respectively.
These two classification losses disentangle attributes from the deep features
of the backbone under the supervision of ground-truth labels. These two
supervision signals are primarily responsible for training the backbone as
well as the connection layers between the backbone and the BN or GCN.

The second group of two supervision signals are applied to the fused results
of the BN and GCN in (8) and (9). Two cross-entropy losses $\mathcal{L}_{a}$
and $\mathcal{L}_{d}$ are imposed again on the classification results of
attributes and diseases, respectively. The first loss is imposed on the fused
disease classification result, $P^{d}_{fusion}$ in (8), and is formulated as
follows,

$\displaystyle\mathcal{L}_{d}=$
$\displaystyle-\sum_{j=1}^{C}(y^{d}_{j}\log(p^{d}_{j})+(1-y^{d}_{j})\log(1-p^{d}_{j})),$
(13)

where $y^{d}$ denotes one-hot encoding of the ground-truth disease grades of
training samples, and $p^{d}_{j}$ represents a component of $P^{d}_{fusion}$.
The second loss $\mathcal{L}_{a}$ is imposed on the fused attribute
classification results, $P^{a}_{fusion}$ in (9), and is formulated as follows,

$\displaystyle\mathcal{L}_{a}=$
$\displaystyle-\sum_{i=1}^{N}\sum_{j=1}^{C}(y^{a}_{ij}\log(p^{a}_{ij})+(1-y^{a}_{ij})\log(1-p^{a}_{ij})),$
(14)

where $y^{a}$ denotes one-hot encoding of the ground-truth attribute grades of
training samples, and $p^{a}_{ij}$ represents a component of $P^{a}_{fusion}$.
Since we do not update parameters in the BNs during the first phase of the
training scheme, these two supervision signals are primarily responsible for
training the backbone and the GCN.

The last supervision signal is located at the end of the entire hybrid
network, and is responsible for training all modules in the entire network. A
cross-entropy loss is imposed on the final disease classification result,
$P^{d}_{final}$, defined in Section 3.5 and produced by the second Bayesian
network. It is written as follows,

$\displaystyle\mathcal{L}_{final}=$
$\displaystyle-\sum_{j=1}^{C}(y^{d}_{j}\log(p^{final}_{j})+(1-y^{d}_{j})\log(1-p^{final}_{j})),$
(15)

where $p^{final}_{j}$ represents a component of $P^{d}_{final}$.

TABLE I: Performance Comparison of Lung Nodule Classification Models on the
LIDC-IDRI Dataset. ‘B’ And ‘M’ are the Number of Benign and Malignant Lung
Nodules Used for Model Training.

| Methods | Number | Results(%) (mean$\pm$standard deviation)
---|---|---|---
| B | M | Accuracy | Sensitivity / Recall | Specificity | AUC | Precision | F-score
A | Shen et al., 2017 [55] (Multi-crop CNN) | 528 | 297 | 87.14 | 77.00 | 93.00 | 93.00 | Not given | Not given
B | Hussein et al., 2017 [21] (3D CNN) | 635 | 509 | 91.26 | Not given | Not given | Not given | Not given | Not given
C | Han et al., 2015 [15] (3D GLCM feature+SVM) | 1301 | 644 | 85.38$\pm$0.10 | 70.20$\pm$0.15 | 92.80$\pm$0.20 | 88.19$\pm$0.16 | 82.85$\pm$0.38 | 75.99$\pm$0.10
D | Dhara et al., 2016 [6] (Multi-visual features) | 1301 | 644 | 87.90$\pm$0.17 | 84.50$\pm$0.19 | 89.09$\pm$0.25 | 93.77$\pm$0.15 | 79.31$\pm$0.37 | 81.82$\pm$0.21
E | Xie et al., 2018 [63] (Deep + visual features) | 1301 | 644 | 88.73$\pm$0.15 | 84.40$\pm$0.20 | 90.88$\pm$0.13 | 94.02$\pm$0.20 | 82.09$\pm$0.24 | 83.23$\pm$0.21
F | Xie et al., 2017 [60] (TMME with Resnet-50) | 1301 | 644 | 91.01$\pm$0.10 | 83.83$\pm$0.15 | 94.56$\pm$0.13 | 95.35$\pm$0.15 | 88.40$\pm$0.24 | 86.07$\pm$0.15
G | Xie et al., 2019a [61] (Knowledge-based) | 1301 | 644 | 91.60$\pm$0.15 | 86.52$\pm$0.25 | 94.00$\pm$0.30 | 95.70$\pm$0.24 | 87.75$\pm$0.52 | 87.13$\pm$0.16
H | Xie et al., 2019b [62] (Semi-Supervised) | 1301 | 644 | 92.53$\pm$0.05 | 84.94$\pm$0.17 | 96.28$\pm$0.08 | 95.81$\pm$ 0.19 | Not given | Not given
I | Xu et al., 2020 [64] (Multi-Scale Cost-Sensitive) | 1156 | 556 | 92.64$\pm$ 0.12 | 85.58 $\pm$ 0.44 | 95.87$\pm$1.26 | 94.00 $\pm$0.26 | 90.39 $\pm$0.48 | 87.91 $\pm$0.11
T1 | Low-Level-Feature [10] | 1301 | 644 | 88.73$\pm$0.15 | 84.40$\pm$0.20 | 90.88$\pm$0.13 | 94.02$\pm$0.20 | 82.09$\pm$0.24 | 83.23$\pm$0.21
T2 | Basic-visual-Feature [28] | 1301 | 644 | 91.01$\pm$0.10 | 83.83$\pm$0.15 | 94.56$\pm$0.13 | 95.35$\pm$0.15 | 88.40$\pm$0.24 | 84.07$\pm$0.15
M1 | ResNet-50 | 1301 | 644 | 88.14$\pm$0.23 | 82.17$\pm$0.14 | 90.77$\pm$0.15 | 91.21$\pm$0.14 | 82.18$\pm$0.11 | 82.36$\pm$0.14
M2 | Efficient-B4 | 1301 | 644 | 89.21$\pm$0.12 | 83.83$\pm$0.24 | 91.18$\pm$0.22 | 92.05$\pm$0.27 | 86.88$\pm$0.19 | 83.04$\pm$0.23
M3 | ResNet-50-FPN | 1301 | 644 | 90.01$\pm$0.13 | 84.23$\pm$0.21 | 91.54$\pm$0.21 | 92.81$\pm$0.31 | 84.26$\pm$0.12 | 84.71$\pm$0.21
M4 | Efficient-B4-FPN | 1301 | 644 | 90.91$\pm$0.22 | 85.74$\pm$0.13 | 92.27$\pm$0.15 | 93.23$\pm$0.10 | 87.10$\pm$0.24 | 86.97$\pm$0.14
M31 | ResNet-50-FPN-GCN-Relation [51] | 1301 | 644 | 91.60$\pm$0.15 | 86.52$\pm$0.25 | 92.32$\pm$0.15 | 93.70$\pm$0.24 | 86.75$\pm$0.52 | 85.13$\pm$0.16
M41 | Efficient-B4-FPN-GCN-Relation [51] | 1301 | 644 | 92.15$\pm$0.12 | 86.97$\pm$0.23 | 93.13$\pm$0.11 | 93.89$\pm$0.23 | 87.14$\pm$0.21 | 85.57$\pm$0.24
M32 | ResNet-50-FPN-GRU [41] | 1301 | 644 | 91.41$\pm$0.11 | 86.12$\pm$0.14 | 92.92$\pm$0.19 | 93.61$\pm$0.23 | 87.44$\pm$0.21 | 85.22$\pm$0.15
M42 | Efficient-B4-FPN-GRU [41] | 1301 | 644 | 92.21$\pm$0.10 | 86.94$\pm$0.11 | 93.89$\pm$0.24 | 93.72$\pm$0.12 | 88.24$\pm$0.13 | 86.63$\pm$0.21
O1 | Our-ResNet-50-FPN | 1301 | 644 | 93.74$\pm$0.17 | 89.23$\pm$0.21 | 95.76$\pm$0.24 | 96.12$\pm$0.12 | 94.21$\pm$0.14 | 88.27$\pm$0.31
O2 | Our-Efficient-B4-FPN | 1301 | 644 | 95.31$\pm$0.15 | 90.51$\pm$0.15 | 96.15$\pm$0.22 | 96.47$\pm$0.31 | 95.95$\pm$0.24 | 88.83$\pm$0.45
O2∗ | Our-Efficient-B4-FPN∗ | 1301 | 644 | 95.36$\pm$0.10 | 91.01$\pm$0.16 | 96.47$\pm$0.12 | 96.54$\pm$0.32 | 95.96$\pm$0.21 | 89.13$\pm$0.15

TABLE II: Comparison with Existing Lung Nodule Classification Models under a
Second Training and Testing Protocol.

Methods | Accuracy % | # of dataset
---|---|---
TumorNet [22] | 82.47 | 1145
TumorNet-attribute [22] | 92.31 | 1145
SHC-DCNN [5] | 82.40 | 1432
MCNN [55] | 86.84 | 1100
CNN-MTL [21] | 91.26 | 1340
PN-SAMP-S1 [59] | 92.03 | 1404
PN-SAMP-S2 [59] | 95.30 | 1404
PN-SAMP-M [59] | 97.58 | 1404
Ours | 98.12 | 1404

##  4 Medical Image Diagnosis

In this section, we apply our attribute-based hybrid reasoning algorithm to
two challenging medical image diagnosis tasks. The first task is benign-
malignant classification of pulmonary nodules in chest computed tomography
(CT) images. The second task is tuberculosis diagnosis using chest X-ray
images. We define attributes and perform attribute relationship learning and
reasoning for both tasks.

Following the common practice of previous attribute-based classification
methods, not all attributes are used for training, i.e., we select attributes
that have high causal relations with the diseases, and those unrelated
attributes have been discarded.

###  4.1 Pulmonary Nodule Classification

Lung cancer gives rise to the most cancer-related deaths around the world [4].
Early diagnosis and treatment are of great importance to long-term survival of
lung cancer patients. Benign-malignant classification of pulmonary nodules in
chest CTs is vital for timely diagnosis of lung cancer [43]. In practice,
radiologists often gain many clinical insights from the connections between
domain knowledge and clinical symptoms as well as from the dependencies
between diverse attributes and the underlying disease. This is applicable to
pulmonary nodule diagnosis. For instance, when a nodule has an irregular shape
and presents obvious spiculation, it has a high probability of being
malignant.

TABLE III: Distribution of Median Malignancy Levels (MML) in the LIDC-IDRI Dataset for Lung Nodule Classification. Dataset | Benign | Uncertain | Malignant
---|---|---|---
MML | 1 | 2 | 3 | 4 | 5
# of Nodule | 358 | 943 | 612 | 474 | 170

####  4.1.1 LIDC-IDRI Dataset

The LIDC-IDRI dataset [2] from the Cancer Imaging Archive (TCIA) is one of the
largest publicly available lung cancer datasets. It has 1018 clinical chest CT
scans obtained from seven institutions. Each scan is associated with an XML
file that details the locations of nodules on each 512$\times$512 slice. The
diameters of the nodules range from 3mm to 30mm. Each suspicious lesion is
categorized as a non-nodule, a nodule $<$ 3mm, or a nodule $\geq$ 3mm in
diameter. Adopting the same setting as in [15, 6, 61, 62, 64], we only
consider nodules $\geq$ 3mm in diameter since nodules $<$ 3mm are not
considered to be clinically relevant by current screening protocols [15, 6,
22, 55, 52, 16].

The malignancy of each nodule was evaluated with a 5-point scale, from benign
to malignant, by up to four experienced thoracic radiologists. Following the
same procedure used in previous studies [15, 6, 61, 62, 64], we select those
nodules annotated by at least one radiologist and calculate the median
malignancy level (MML) for each of them. Nodules with an MML less than or
higher than 3 are respectively labeled as benign or malignant. To reduce the
uncertainty of nodule malignancy evaluation, nodules with an MML of 3 (noted
as ‘uncertain’) are excluded from our experiments as previous studies did.
Thus there are a total of 1301 benign, 612 uncertain and 644 malignant
nodules. The distribution of nodules over their MML is shown in Table III.
Apart from malignancy, eight attributes are also graded for each nodule, i.e.,
subtlety, internal structure, calcification, sphericity, margin, lobulation,
spiculation, and radiographic solidity. The grade of each attribute is between
1 and 5 except for internal structure (1-4) and calcification (1-6). We scale
the grades of all attributes to 1-5 to maintain consistency, where 1 means the
least obvious and 5 means the most obvious. Fig. 1 shows sample 2D slices with
distinct attributes.

We also normalize all chest CTs to a unified voxel size of $1.0\times
1.0\times 1.0mm^{3}$ using spline interpolation. In addition, following the
same setting as in [61, 62], we assume the location of a nodule is the mean of
the annotated centers of the nodule by radiologists. For each nodule, we crop
a $64\times 64\times 64$ patch centered on its location.

####  4.1.2 Experimental Setup

In all experiments, we perform 10-fold cross validation, and each model is
independently trained and tested 5 times with randomly initialized weights on
each fold. The overall performance of a model is assessed with several
commonly used metrics, including the mean and standard deviation of accuracy,
sensitivity/recall, specificity, precision with the cut-off value of 0.5,
F-score and AUC (area under the receiver operator curve).

All models are trained for 160 epoches from scratch using PyTorch [44] on
NVIDIA Titan X pascal GPUs while Adam [24] being the optimizer with the
initial learning rate set to 1e-3, which is reduced by a factor of 10 after
every 30 epochs. The weight decay is set to 1e-4. Both ResNet and EfficientNet
are adopted as the backbone models. We have implemented their 3D versions for
the LIDC dataset by revising the original 2D versions. The size of the input
image patches is $64\times 64\times 64$, and the batch size is 6 on a single
GPU. Separate models are trained with and without data augmentation, which
includes standard operations including flipping, rotation, and random
cropping.

####  4.1.3 Comparison with the State of the Art

We have compared our hybrid algorithm with several state-of-the-art models for
pulmonary nodule classification on the LIDC-IDRI dataset and the results are
shown in Table I, where O2 and O2∗ represent our models trained without and
with standard data augmentation. The proposed method achieves the best
performance under every evaluation metric when data augmentation is applied
while it is still ranked first under every evaluation metric except for
specificity when data augmentation is not used. Such a performance
demonstrates the effectiveness of hybrid neural and probabilistic reasoning
proposed in this paper, as well as the important role of attribute learning in
medical image diagnosis. Specifically, method A adopts a multi-crop
convolutional neural network trained a subset of 825 nodules from LIDC and
achieves an accuracy of 87.14%. Method B trains a 3D CNN with a subset of 1144
nodules and achieves a higher accuracy (91.26%). All methods from C to H train
models using the same number of nodules, which is 1945. However, method H
introduced additional 1839 unlabeled nodules for semi-supervised learning and
achieves the highest accuracy (92.53%) among methods from C to H. Method I
proposes a multi-scale 3D ResNet with a cost-sensitive loss and delivers a
higher performance (92.64%) than method H using a subset of 1712 nodules. In
comparison, when adopting the same backbone (3D ResNet) as method I, our
method (O1) achieves a better performance (93.74%), indicating the importance
of attribute relationship modeling in enhancing the performance of pulmonary
nodule classification. Most importantly, Table I shows that when using an
EfficientB4-FPN backbone, our model with data augmentation (O2*) achieves the
highest accuracy of 95.36%, and our model without data augmentation (O2) gains
4.41% over the baseline using the same backbone (M4).

Figure 3: Sample results of our method (O2) and four baseline methods (T1, T2,
M41, M42) on the LIDC-IDRI dataset. Short names are defined as follows: SUB
(Subtlety), IS (Internal Structure), CAL (Calcification), SP (Sphericity), MG
(Margin), LB (Lobulation), SL (Spiculation), TE (Texture), and MA
(Malignancy). The radar charts visualize the actual grades of attributes and
disease, and there exist five possible grades for each attribute or disease.
GT stands for the ground truth. The Histograms denote the numerical importance
of different attributes for diagnosing malignancy (MA).  Figure 4: The
Bayesian network structure learned from the LIDC-IDRI dataset. Notations are
defined as follows. SPI: spiculation; MAL: malignancy; SUB: subtlety; CAL:
calcification; MAR: margin; LOB: lobulation. TABLE IV: Ablation Study of Our
Classification Algorithm on the LIDC-IDRI Dataset. ‘BN-1’ And ‘BN-2’ Denote
the First and Second BN Modules, ‘CNA-RES’ Denotes Cross-Network Attention and
Residual Fusion, ‘SEatt’ Means Adopting SE Channel-Wise Attention in the GCN,
‘GradBN’ Means Gradient Backpropagation through the BN Modules, ‘AlterTrain’
Denotes the Proposed Alternating Training Strategy. ‘Relation’ Means Removing
Both BN Modules and Appending the Relational Network from [51] after the GCN
Module.

GCN | BN-2 | CNA-RES | SEatt | BN-1 | GradBN | AlterTrain | Relation | Accuracy | Sensitivity / Recall | Specificity | AUC | Precision | F-score
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 95.31$\pm$0.15 | 90.51$\pm$0.15 | 96.15$\pm$0.22 | 96.47$\pm$0.31 | 95.95$\pm$0.24 | 88.83$\pm$0.45
$\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 94.74$\pm$0.09${}^{\textbf{$\downarrow$}0.57}$ | 88.92$\pm$0.13 | 95.86$\pm$0.41 | 95.83$\pm$0.09 | 92.54$\pm$0.06 | 88.04$\pm$0.04
$\checkmark$ | $\boxtimes$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 92.11$\pm$0.14${}^{\textbf{$\downarrow$}3.20}$ | 88.14$\pm$0.22 | 93.74$\pm$0.15 | 94.91$\pm$0.26 | 88.11$\pm$0.09 | 86.88$\pm$0.12
$\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 91.21$\pm$0.22${}^{\textbf{$\downarrow$}4.10}$ | 87.26$\pm$0.14 | 93.21$\pm$0.13 | 93.47$\pm$0.54 | 88.01$\pm$0.09 | 86.07$\pm$0.56
$\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 93.64$\pm$0.11${}^{\textbf{$\downarrow$}1.67}$ | 88.01$\pm$0.15 | 95.03$\pm$0.13 | 95.02$\pm$0.13 | 91.42$\pm$0.13 | 87.92$\pm$0.04
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 94.01$\pm$0.02${}^{\textbf{$\downarrow$}1.30}$ | 88.64$\pm$0.08 | 95.23$\pm$0.07 | 95.32$\pm$0.11 | 91.11$\pm$0.04 | 87.13$\pm$0.09
$\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 92.41$\pm$0.14${}^{\textbf{$\downarrow$}2.90}$ | 88.14$\pm$0.22 | 93.74$\pm$0.15 | 94.91$\pm$0.26 | 92.11$\pm$0.09 | 86.88$\pm$0.12
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\boxtimes$ | 93.81$\pm$0.10${}^{\textbf{$\downarrow$}1.50}$ | 88.04$\pm$0.32 | 93.74$\pm$0.15 | 94.91$\pm$0.26 | 90.21$\pm$0.07 | 86.96$\pm$0.10
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\boxtimes$ | 94.31$\pm$0.10${}^{\textbf{$\downarrow$}1.00}$ | 88.28$\pm$0.41 | 94.23$\pm$0.12 | 94.87$\pm$0.76 | 89.11$\pm$0.22 | 87.03$\pm$0.20
$\checkmark$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\checkmark$ | 92.15$\pm$0.12${}^{\textbf{$\downarrow$}3.16}$ | 86.97$\pm$0.23 | 93.13$\pm$0.11 | 93.89$\pm$0.23 | 87.14$\pm$0.21 | 85.57$\pm$0.24
$\checkmark$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | 91.31$\pm$0.12${}^{\textbf{$\downarrow$}4.00}$ | 87.83$\pm$0.24 | 93.15$\pm$0.13 | 94.07$\pm$0.12 | 86.65$\pm$0.12 | 87.04$\pm$0.13

In addition, Table II shows that under the training setting and testing
protocol adopted in [59], our algorithm also achieves the highest accuracy,
further demonstrating the effectiveness of our method. In this comparison, we
adopt the same backbone (DenseNet) as in [59], and do not apply any data
augmentation during training. During testing, we adopt the same “off-by-one”
accuracy as in [59], which considers attribute/malignancy grading within
$\pm$1 of the ground truth as correct results. Note that although we do not
exploit the additional supervision signal provided by the nodule segmentation
masks, that are used in [59], our method still achieves the highest accuracy
of 98.12%.

We have further compared with attribute relationship modeling methods
including M31, M41, M32, and M42. For a fair comparison, methods in the same
comparison always adopt the same backbone, which is either 3D ResNet-50 or
EfficientNet-B4. The relational network module proposed in [51] and the GRU
module proposed in [41] respectively achieve an improvement of 1.59% and 1.30%
in accuracy when compared with the 3D ResNet-50 baseline while our method
improves the accuracy of the same baseline by 3.63%. The same conclusion can
be drawn for the EfficientNet-B4 backbone. We have also compared with two
classic attribute learning methods T1 and T2, and they deliver a lower
accuracy of 88.73% and 91.01%, respectively. In summary, all the above well-
designed experiments demonstrate that the proposed hybrid neuro-probabilistic
reasoning algorithm achieves clearly better performance than existing methods
on the LIDC-IDRI dataset.

Sample input images and their associated attribute and disease classification
results from our method and a few other baseline methods are shown in Fig. 3.
In addition, we provide the importance of individual attributes for pulmonary
nodule classification, _i.e.,_ quantitatively evaluating how important
individual attributes are in making the prediction. To be specific, we measure
how much the final disease classification probability is affected by
deactivating one attribute at a time in the input signals to BN-1 and GCN. For
the input signals to GCN, an attribute can be deactivated by setting its
corresponding feature vector to an average feature for that attribute over the
samples where that attribute attains the lowest possible grade. For the input
signals to BN-1, since each attribute in the LIDC-IDRI dataset is associated
with 5 grades, deactivation means assigning the lowest grade (1) to the
attribute.

Figure 5: Sample chest X-ray images used for building the TB-Xatt dataset,
each with one of the following seven attributes except the last one: Fibrotic
Streaks, Pulmonary Consolidation, Diffuse Nodules, Pulmonary Cavitation,
Atelectasis, Multiple Nodules, and Pleural Effusion. The last image shows a
case of the disease, Pulmonary Tuberculosis.

####  4.1.4 Ablation Study

We have performed a systematic ablation study on the LIDC dataset to verify
the effectiveness of the components in our framework. The results are shown in
Table IV. Specifically, we have explored the effects of both the first and
second BN (BN-1 and BN-2), cross-network attention and residual fusion (CNA-
RES), SE channel-wise attention (SEatt) [20], gradient backpropagation through
BN (GradBN), BN and DNN alternating training (AlterTrain) and the Relational
Network Module (Relation) [51] on the EfficientNet-B4 backbone. Table IV shows
that all proposed components or strategies contribute to the final performance
of our algorithm. More specifically, we have found out that ‘CNA-RES’ is the
most important component, the performance would drop by 1.67% if it was
removed, which verifies the effectiveness of our proposed network coupling
strategy. When both ‘BN-2’ and ‘CNA-RES’ are removed and the final result of
the network is computed by a simple average of the results from BN-1 and GCN,
the network becomes an ensemble model integrating the BN-1 and GCN branches.
The performance drops significantly by 3.2% under this ensemble model setting.
This result indicates that various components in our hybrid framework are
highly coupled, and altogether they deliver a performance much better than a
simple ensemble model. ‘GradBN’ and ‘AlterTrain’ also have a significant
impact, and the performance drops by 1.5% and 1.0% respectively if either of
them is removed. In addition, completely removing BN-1 and its coupling with
GCN would decrease the performance by 2.90% while removing BN-2 alone would
decrease the performance by 0.57%. Moreover, the performance would drop by
3.16% when both BN modules are removed and a relational network is appended
after the GCN module. This result demonstrates the strong dependency modeling
ability of BN modules. Fig. 4 shows the structure of the first BN module
(BN-1) learned from the LIDC-IDRI dataset. This learned BN structure indicates
that the node for malignancy has one parent node (i.e., spiculation) and four
children nodes (i.e., subtlety, calcification, margin and lobulation).

We can draw the following conclusions from the results given in Table IV,

  * •

‘BN-1’, ‘CNA-RES’, ‘GradBN’, ‘AlterTrain’ and ‘SENet’ are important to the
classification performance on the LIDC-IDRI dataset. Note that removing
‘AlterTrain’ actually means using ground-truth attribute labels instead of
soft attribute labels for training the BN modules because the BN structures
will not change if we always use ground-truth attribute labels during
alternating training.

  * •

BN modules are vital for attribute relational modeling. Their performance
surpasses the performance of the advanced relational network.

  * •

Although a simple ensemble model consisting of BN and GCN can already improve
the performance, our proposed cross-network attention mechanism takes full
advantage of BN and GCN, thus performs better.

TABLE V: Distribution of Disease and Radiological Abnormalities in the TB-Xatt
Dataset for Tuberculosis Diagnosis.

Sign/Disease | Fibrotic Streaks | Pulmonary Consolidation | Diffuse Nodules | Pulmonary Cavitation | Atelectasis | Multiple Nodules | Pleural Effusion | Pulmonary Tuberculosis
---|---|---|---|---|---|---|---|---
#Images | 1,640 | 1,200 | 1,500 | 1,400 | 800 | 1,200 | 900 | 1,972

###  4.2 Tuberculosis Diagnosis

Chest radiographs are a type of medical images that can be conveniently
acquired for disease diagnosis. Radiologists can find out many diseases
quickly via observing chest X-ray images, which are useful for early diagnosis
and intervention. Tuberculosis [35] has the highest mortality around the globe
among infectious diseases. However, if it was detected at an early stage from
chest radiographs, the death rate could decrease by 70% [17]. Meanwhile,
tuberculosis is a type of bacterial infection that can give rise to multiple
types of radiological abnormalities in chest radiographs such as diffuse
nodules and fibrotic appearance. Just like morphological characteristics of
lung nodules, these radiological abnormalities can be treated as attributes to
facilitate the diagnosis of tuberculosis by medical experts. Therefore, we
choose tuberculosis diagnosis as an additional task to further evaluate the
proposed hybrid neuro-probabilistic reasoning algorithm, which is able to
model the dependencies between tuberculosis and radiological abnormalities to
improve the reasoning capability of deep learning methods.

Figure 6: Construction of image samples in the TB-Xatt dataset. First, lung
segmentation in chest X-ray images using a trained segmentation model. Second,
resizing the left and right lung patches to the same resolution and placing
them side by side.  TABLE VI: Performance Comparison of Tuberculosis Diagnosis
Models on the TB-Xatt Dataset.

| Methods | Results(%) (mean$\pm$standard deviation)
---|---|---
| Accuracy | Sensitivity / Recall | Specificity | AUC | Precision | F-score
A1 | Low-Level-Feature [10] | 86.14$\pm$0.14 | 83.11$\pm$0.26 | 89.17$\pm$0.10 | 90.11$\pm$0.76 | 81.10$\pm$0.12 | 81.71$\pm$0.11
A2 | Basic-visual-Feature [28] | 87.23$\pm$0.12 | 84.16$\pm$0.27 | 92.00$\pm$0.09 | 91.27$\pm$0.63 | 83.26$\pm$0.71 | 83.44$\pm$0.17
S1 | Attention-Guide [13] | 93.12$\pm$0.10 | 86.17$\pm$0.16 | 91.22$\pm$0.34 | 93.67$\pm$0.43 | 87.21$\pm$0.53 | 85.64$\pm$0.29
S2 | ADINet [40] | 93.43$\pm$0.19 | 87.24$\pm$0.25 | 91.87$\pm$0.11 | 94.11$\pm$0.29 | 87.44$\pm$0.54 | 84.21$\pm$0.13
M1 | ResNet-50 | 90.17$\pm$0.11 | 90.44$\pm$0.12 | 89.16$\pm$0.23 | 90.03$\pm$0.21 | 82.09$\pm$0.12 | 84.15$\pm$0.23
M2 | Efficient-B4 | 93.21$\pm$0.25 | 91.54$\pm$0.22 | 92.55$\pm$0.11 | 92.67$\pm$0.19 | 84.27$\pm$0.98 | 87.23$\pm$0.09
M3 | ResNet-50-FPN | 91.23$\pm$0.28 | 91.96$\pm$0.42 | 91.22$\pm$0.33 | 91.54$\pm$0.07 | 83.27$\pm$0.08 | 85.26$\pm$0.44
M4 | Efficient-B4-FPN | 94.19$\pm$0.19 | 92.78$\pm$0.09 | 92.97$\pm$0.01 | 94.01$\pm$0.10 | 86.11$\pm$0.02 | 89.17$\pm$0.08
M31 | ResNet-50-FPN-GCN-Relation [51] | 91.74$\pm$0.29 | 92.16$\pm$0.32 | 93.23$\pm$0.25 | 92.26$\pm$0.12 | 86.14$\pm$0.11 | 86.54$\pm$0.13
M41 | Efficient-B4-FPN-GCN-Relation [51] | 95.17$\pm$0.11 | 92.29$\pm$0.18 | 94.01$\pm$0.12 | 94.83$\pm$0.21 | 86.77$\pm$0.14 | 90.21$\pm$0.19
M32 | ResNet-50-FPN-GRU [41] | 91.66$\pm$0.13 | 92.06$\pm$0.34 | 94.26$\pm$0.26 | 93.13$\pm$0.12 | 86.01$\pm$0.21 | 87.11$\pm$0.98
M42 | Efficient-B4-FPN-GRU [41] | 94.65$\pm$0.32 | 92.94$\pm$0.18 | 95.19$\pm$0.13 | 94.88$\pm$0.23 | 86.98$\pm$0.12 | 91.03$\pm$0.24
O1 | Our-ResNet-50-FPN | 94.67$\pm$0.11 | 94.22$\pm$0.33 | 96.25$\pm$0.23 | 97.22$\pm$0.43 | 90.29$\pm$0.22 | 90.01$\pm$0.10
O2 | Our-Efficient-B4-FPN | 97.13$\pm$0.19 | 95.51$\pm$0.10 | 97.11$\pm$0.16 | 98.10$\pm$0.34 | 92.44$\pm$0.31 | 91.78$\pm$0.27
O2∗ | Our-Efficient-B4-FPN∗ | 98.41$\pm$0.10 | 96.32$\pm$0.11 | 97.91$\pm$0.21 | 98.91$\pm$0.12 | 93.24$\pm$0.10 | 92.77$\pm$0.14

####  4.2.1 TB-Xatt dataset

As for the task of tuberculosis diagnosis, there are a total of 14200 frontal
chest X-ray images in an in-house dataset named TB-Xatt. Every image in this
dataset has been carefully annotated by certificated and experienced
radiologists. Image annotations include 7 types of radiological abnormalities,
‘Pulmonary Consolidation’, ‘Pulmonary Cavitation’, ‘Diffuse Nodules’,
‘Fibrotic Streaks’, ‘Atelectasis’, ‘Multiple Nodules’, ‘Pleural Effusion’ and
one disease, ‘Pulmonary Tuberculosis’. On average, there are 3 occurrences of
disease or radiological abnormalities per image and 1326 annotated images for
each type of disease or radiological abnormalities. The distribution of images
over disease and radiological abnormalities is presented in Table V. In this
study, we aim to improve the classification performance of ‘Pulmonary
Tuberculosis’ with the help of the seven types of radiological abnormalities,
which are treated as the attributes of a chest X-ray image in our hybrid
algorithm since these seven types of radiological abnormalities are the main
expressions of active tuberculosis in the lung. Fig. 5 shows sample images
with these attributes and disease from the TB-Xatt dataset. Note that the
image samples in the dataset are pre-processed versions of the original X-ray
images. We first perform lung segmentation in the original X-ray images and
extract bounding boxes of the left and right lungs as shown in Fig. 6, then
resize these bounding boxes to 512 $\times$ 256 and place each pair of left
and right bounding boxes side by side as a single image sample. In all
experiments, we perform 10-fold cross validation, and each model is
independently trained and tested 5 times with randomly initialized weights on
each fold, the same as we did on LIDC. The performance of a model is also
assessed with the same set of metrics as LIDC, including accuracy,
sensitivity/recall, specificity, precision, F-score and AUC.

####  4.2.2 Experimental Setup

Different from LIDC, the ResNet and EfficientNet backbones are pre-trained on
ImageNet. Adam is also the optimizer with the initial learning rate set to
1e-3, and the batch size is 32. All models are trained 60 epochs and the
learning rate is reduced by a factor of 10 after 20 and 40 epochs.

Figure 7: Sample results of our method (O2) and four baseline methods (T1, T2,
M41, M42) on the in-house TB-Xatt dataset. Short names are defined as follows:
FIB (Fibrotic Streaks), CON (Pulmonary Consolidation), DIF (Diffuse Nodules),
CAV (Pulmonary Cavitation), ATE (Atelectasis), MUL (Multiple Nodules), and PLE
(Pleural Effusion). The colored bars visualize attribute classification
results from individual methods as well as the ground truth. GT stands for the
ground truth. $\checkmark$ and $\times$ visualize the diagnoses of pulmonary
tuberculosis from individual methods as well as the ground truth. The
histograms under the X-ray images visualize the importance of individual
attributes for diagnosing pulmonary tuberculosis.  TABLE VII: Ablation Study
of Our Classification Algorithm on the TB-Xatt Dataset. ‘BN-1’ And ‘BN-2’
Denote the First and Second BN Modules, ‘CNA-RES’ Denotes Cross-Network
Attention and Residual Fusion, ‘SEatt’ Means Adopting SE Channel-Wise
Attention in the GCN, ‘GradBn’ Means Gradient Backpropagation through the BN
Modules, ‘AlterTrain’ Denotes the Proposed Alternating Training Strategy.
‘Relation’ Means Removing Both BN Modules and Appending the Relational Network
from [51] After the GCN Module.

GCN | BN-2 | CNA-RES | SEatt | BN-1 | GradBN | AlterTrain | Relation | Accuracy | Sensitivity / Recall | Specificity | AUC | Precision | F-score
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 97.13$\pm$0.19 | 95.51$\pm$0.10 | 97.11$\pm$0.16 | 98.10$\pm$0.34 | 92.44$\pm$0.31 | 91.78$\pm$0.27
$\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 96.78$\pm$0.11${}^{\textbf{$\downarrow$}0.35}$ | 94.26$\pm$0.13 | 96.78$\pm$0.23 | 97.12$\pm$0.11 | 90.93$\pm$0.17 | 90.03$\pm$0.07
$\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 96.21$\pm$0.44${}^{\textbf{$\downarrow$}0.57}$ | 94.12$\pm$0.31 | 96.22$\pm$0.41 | 97.17$\pm$0.41 | 90.61$\pm$0.20 | 90.01$\pm$0.20
$\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 95.69$\pm$0.12${}^{\textbf{$\downarrow$}1.09}$ | 93.58$\pm$0.17 | 95.44$\pm$0.32 | 96.71$\pm$0.21 | 90.01$\pm$0.10 | 89.96$\pm$0.24
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 96.89$\pm$0.32${}^{\textbf{$\downarrow$}0.24}$ | 94.47$\pm$0.55 | 96.79$\pm$0.42 | 97.87$\pm$0.31 | 91.02$\pm$0.33 | 90.64$\pm$0.25
$\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | 95.69$\pm$0.14${}^{\textbf{$\downarrow$}1.44}$ | 94.55$\pm$0.43 | 94.78$\pm$0.11 | 95.94$\pm$0.12 | 87.11$\pm$0.23 | 91.21$\pm$0.54
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\checkmark$ | $\boxtimes$ | 96.12$\pm$0.24${}^{\textbf{$\downarrow$}0.66}$ | 94.66$\pm$0.31 | 94.51$\pm$0.23 | 95.17$\pm$0.21 | 90.14$\pm$0.32 | 90.11$\pm$0.23
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\boxtimes$ | $\boxtimes$ | 96.53$\pm$0.13${}^{\textbf{$\downarrow$}0.60}$ | 94.71$\pm$0.24 | 94.62$\pm$0.33 | 95.19$\pm$0.47 | 90.28$\pm$0.34 | 90.51$\pm$0.33
$\checkmark$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\boxtimes$ | $\checkmark$ | 94.97$\pm$0.21${}^{\textbf{$\downarrow$}2.81}$ | 93.09$\pm$0.08 | 93.46$\pm$0.29 | 94.93$\pm$0.21 | 86.77$\pm$0.14 | 90.21$\pm$0.19

####  4.2.3 Comparison with the State of the Art

On our in-house TB-Xatt dataset for tuberculosis diagnosis, we have also
compared our hybrid algorithm with existing methods, including two state-of-
the-art models (S1 and S2), two attribute learning methods (A1 and A2) and two
relationship modeling methods (M31, M41 and M32, M42). The results are shown
in Table VI, where O2 and O2∗ represent our models trained without and with
standard data augmentation, respectively. No matter whether data augmentation
is used, the proposed algorithm achieves the best performance under all
evaluation metrics, including accuracy, sensitivity/recall, specificity,
precision with the cut-off value of 0.5, F-score and AUC. Specifically, both
A1 and A2 are classic attribute learning methods that use hand-crafted
features, and achieve an accuracy of 86.14% and 87.23%, respectively. When
compared with state-of-the-art disease classification models S1 and S2, the
proposed hybrid algorithm demonstrates better performance with 4.01% and 3.70%
improvement in accuracy, respectively. Furthermore, as in the pulmonary nodule
classification task, we implemented two attribute relationship modeling
methods (M31, M41 and M32, M42) based on the ResNet-50 and EfficientNet-B4
backbones. Results indicate that our method achieves better performance due to
its strong causality modeling and relationship reasoning capabilities. The
classification performance of our method with ResNet-50-FPN and
EfficientNet-B4-FPN backbones are 94.67% and 97.13%, which respectively gain
3.44% and 2.94% over the two baselines using the same backbones. The results
in Table VI demonstrate the effectiveness of our proposed algorithm on the
tuberculosis diagnosis task. Sample input images and their associated
attribute and disease classification results from our method and a few other
baseline methods are shown in Fig. 7, where we have binary attribute and
disease classifications. We also evaluate the importance of individual
attributes for pulmonary tuberculosis diagnosis. Similarly, we measure how
much the final disease classification probability is affected by deactivating
one attribute at a time in the input signals to BN-1 and GCN. In this case,
for the input signals to BN-1, deactivation means setting the classification
probability of a specific attribute to zero. The deactivation scheme for the
input signals to GCN is the same as that for the LIDC-IDRI dataset.

TABLE VIII: Performance Comparison of Our Tuberculosis Diagnosis Models
Trained with Increasingly Smaller Datasets. The Amount of Training Data in
P1-P4 Has the Following Ratio, 1: 0.75 : 0.5: 0.25.

| Methods | Results(%) (mean$\pm$standard deviation)
---|---|---
| Accuracy | Sensitivity / Recall | Specificity | AUC | Precision | F score
P1 | Low-Level-Feature [10] | 86.14$\pm$0.14 | 83.11$\pm$0.26 | 89.17$\pm$0.10 | 90.11$\pm$0.76 | 81.10$\pm$0.12 | 81.71$\pm$0.11
Basic-visual-Feature [28] | 87.23$\pm$0.12 | 84.16$\pm$0.27 | 92.00$\pm$0.09 | 91.27$\pm$0.63 | 83.26$\pm$0.71 | 83.44$\pm$0.17
Efficient-B4-FPN-GCN-Relation [51] | 94.97$\pm$0.21 | 93.09$\pm$0.08 | 93.46$\pm$0.29 | 94.93$\pm$0.21 | 86.77$\pm$0.14 | 90.21$\pm$0.19
Efficient-B4-FPN | 94.19$\pm$0.19 | 92.78$\pm$0.09 | 92.97$\pm$0.01 | 94.01$\pm$0.10 | 86.11$\pm$0.02 | 89.17$\pm$0.08
| Our-Efficient-B4-FPN | 97.13$\pm$0.19${}^{\textbf{$\uparrow$}2.94}$ | 95.51$\pm$0.10 | 97.11$\pm$0.16 | 98.10$\pm$0.34 | 92.44$\pm$0.31 | 91.78$\pm$0.27
P2 | Low-Level-Feature [10] | 84.23$\pm$0.12 | 80.07$\pm$0.04 | 86.22$\pm$0.20 | 86.78$\pm$0.15 | 78.24$\pm$0.09 | 78.16$\pm$0.27
Basic-visual-Feature [28] | 83.45$\pm$0.22 | 81.07$\pm$0.54 | 89.21$\pm$0.12 | 87.13$\pm$0.01 | 80.55$\pm$0.44 | 79.21$\pm$0.03
Efficient-B4-FPN-GCN-Relation [51] | 91.26$\pm$0.34 | 91.11$\pm$0.25 | 90.24$\pm$0.27 | 91.45$\pm$0.67 | 84.12$\pm$0.03 | 87.64$\pm$0.11
Efficient-B4-FPN | 91.21$\pm$0.14 | 90.16$\pm$0.22 | 89.98$\pm$0.34 | 92.12$\pm$0.22 | 82.56$\pm$0.13 | 86.26$\pm$0.17
| Our-Efficient-B4-FPN | 93.21$\pm$0.44${}^{\textbf{$\uparrow$}2.11}$ | 91.02$\pm$0.33 | 91.20$\pm$0.21 | 93.56$\pm$0.17 | 87.65$\pm$0.12 | 89.43$\pm$0.14
P3 | Low-Level-Feature [10] | 80.11$\pm$0.09 | 75.67$\pm$0.12 | 80.32$\pm$0.55 | 81.26$\pm$0.64 | 72.13$\pm$0.07 | 72.51$\pm$0.62
Basic-visual-Feature [28] | 79.14$\pm$0.24 | 78.27$\pm$0.63 | 82.11$\pm$0.10 | 80.65$\pm$0.44 | 76.17$\pm$0.23 | 74.56$\pm$0.35
Efficient-B4-FPN-GCN-Relation [51] | 87.24$\pm$0.12 | 86.10$\pm$0.65 | 87.21$\pm$0.46 | 87.13$\pm$0.55 | 81.21$\pm$0.46 | 83.27$\pm$0.35
Efficient-B4-FPN | 86.25$\pm$0.22 | 84.23$\pm$0.27 | 84.22$\pm$0.54 | 82.45$\pm$0.32 | 79.15$\pm$0.21 | 80.24$\pm$0.55
| Our-Efficient-B4-FPN | 92.44$\pm$0.65${}^{\textbf{$\uparrow$}6.19}$ | 90.01$\pm$0.21 | 89.26$\pm$0.17 | 89.22$\pm$0.10 | 86.14$\pm$0.25 | 85.23$\pm$0.17
P4 | Low-Level-Feature [10] | 76.21$\pm$0.09 | 79.27$\pm$0.31 | 74.32$\pm$0.45 | 77.26$\pm$0.13 | 66.45$\pm$0.16 | 67.84$\pm$0.17
Basic-visual-Feature [28] | 75.67$\pm$0.54 | 77.15$\pm$0.13 | 76.55$\pm$0.43 | 76.21$\pm$0.22 | 72.56$\pm$0.34 | 71.27$\pm$0.21
Efficient-B4-FPN-GCN-Relation [51] | 80.11$\pm$0.01 | 78.20$\pm$0.15 | 81.31$\pm$0.11 | 80.13$\pm$0.24 | 75.56$\pm$0.31 | 76.27$\pm$0.45
Efficient-B4-FPN | 78.11$\pm$0.12 | 79.26$\pm$0.36 | 78.23$\pm$0.34 | 80.15$\pm$0.24 | 76.27$\pm$0.15 | 73.62$\pm$0.14
| Our-Efficient-B-FPN | 86.98$\pm$0.27${}^{\textbf{$\uparrow$8.87}}$ | 87.23$\pm$0.19 | 87.63$\pm$0.24 | 88.11$\pm$0.12 | 85.23$\pm$0.11 | 80.16$\pm$0.47

TABLE IX: Specification of Datasets Used for the Performance Comparison in Table VIII. Dataset | P1 | P2 | P3 | P4
---|---|---|---|---
#Training Samples | 11360 | 8530 | 5680 | 2840
#Validation Samples | 1420 | 1420 | 1420 | 1420
#Testing Samples | 1420 | 1420 | 1420 | 1420
P/N ratio in Train | 1/8 | 1/8 | 1/8 | 1/8
P/N ratio in Val | 1/3 | 1/3 | 1/3 | 1/3
P/N ratio in Test | 1/3 | 1/3 | 1/3 | 1/3
TABLE X: Classification accuracy of input and output signals of BN-1 and GCN on TB-Xatt dataset. Method | P1 | P4
---|---|---
BN-1 input | 91.04 | 78.22
BN-1 output | 93.10 | 85.44
GCN input | 92.12 | 78.23
GCN output | 96.36 | 80.28
Residual Fusion (GCN and BN-1) | 97.07 | 86.10

####  4.2.4 Ablation Study

We have also conducted an ablation study on the TB-Xatt dataset to verify the
effectiveness of the components in our proposed framework. The results are
shown in Table VII. As in the pulmonary nodule classification task,
EfficientNet-B4 is the chosen backbone, and performance is measured when
individual components are removed or replaced. Similar patterns in performance
have been found here as well, and all the considered components contribute to
the final performance of our hybrid algorithm.

Moreover, we have also tested the performance of our algorithm under the
condition of a limited training set by sampling an increasingly smaller subset
of the TB-Xatt dataset as the training set while keeping the size of the
validation and testing sets unchanged. The sampling results are shown in Table
IX, where P1 means all original training samples, and P2, P3 and P4 mean 75%,
50% and 25% of the original training samples, respectively. For a fair
comparison, we perform stratified sampling and ensure the same ratio between
the number of positive and negative samples under all settings. This ratio is
always 1/8 for the training sets, and 1/3 for the validation and testing sets.
All results are shown in Table VIII, which indicates that the proposed
algorithm can still perform better than the baselines when given a decreasing
number of training samples. Note that all baselines adopt EfficientNet-B4 as
the backbone since we empirically find out it is the best backbone under
different settings. Furthermore, the performance gap between our algorithm and
its corresponding baseline widens when the number of training samples
decreases. When only 25% of the original training samples are used, this
performance gap reaches 8.87% for the EfficientNet-B4 backbone. More
surprisingly, although the number of training samples used in P4 is only half
of that in P3, the performance of our models trained in the P4 setting is
comparable and sometimes even better than the performance of corresponding
baseline models in the P3 setting. These results demonstrate the strong
generalization ability of our models trained using a small dataset.

We further investigate the classification performance of the individual output
signals from the BN-1 and GCN modules to show how the two mechanisms work
collaboratively when the size of training data varies. As shown in Tab. X, (1)
when the training dataset is small, BN-1 achieves better performance than GCN;
(2) when the training dataset is large, GCN achieves better performance than
BN-1. The justification is that (1) when the training dataset is small, BN-1
probabilistically models the causal relationships between attributes and
diseases to mitigate the problem of insufficient data; (2) when the training
dataset is large, GCN has the capability to learn expressive and powerful
feature representations from the large-scale annotated dataset. Moreover, we
notice that our residual fusion scheme always achieves higher accuracy than
both individual networks (BN-1 and GCN). It has become clear that the strong
generalization ability of our trained models is made possible by the
complementarity between our neural and probabilistic reasoning algorithms. The
neural branch in our network realizes its full power when the training set is
large while the probabilistic branch prevents overfitting from happening when
the training set is small as probabilistic models are better at learning from
smaller datasets.

##  5 Conclusions and Discussions

In this paper, we have introduced a hybrid neuro-probabilistic reasoning
algorithm for verifiable attribute-based medical image diagnosis. There are
two parallel branches in our hybrid algorithm, a Bayesian network branch
performing probabilistic causal relationship reasoning and a graph
convolutional network branch performing more generic relational modeling and
reasoning using a feature representation. Tight coupling between these two
branches is achieved via a cross-network attention mechanism and residual
fusion of classification results. We train the hybrid network by alternatively
updating neural network weights and the structure/parameters of the Bayesian
networks. We have successfully applied our hybrid reasoning algorithm to two
challenging medical image diagnosis tasks, benign-malignant classification of
pulmonary nodules in chest computed tomography images and tuberculosis
diagnosis using chest X-ray images. On the LIDC-IDRI benchmark dataset for
pulmonary nodule classification, our method achieves a new state-of-the-art
accuracy of 95.36% and an AUC of 96.54%. Our method also achieves a 2.94%
accuracy improvement on the in-house tuberculosis diagnosis dataset. Our
ablation study indicates that our hybrid algorithm achieves a much more robust
performance than a pure neural network architecture under very limited
training data.

##  Appendix A Gradient of Belief Propagation

Belief propagation is an iterative algorithm, where nodes exchange messages to
update their marginal distributions until convergence. For a predefined
Bayesian network without any loops, there exists a maximum number of
iterations that guarantees convergence. The messages exchanged during belief
propagation are computed using sums of products, which only involve additions
and multiplications. Therefore, the entire belief propagation process is
differentiable.

To be able to propagate gradients through Bayesian networks, we need to
compute the gradient of all converged marginal distributions with respect to
the network inputs, which are evidences at a subset of the nodes. Since both
network inputs and outputs are multi-dimensional, the gradient we wish to
compute is a Jacobian matrix. In the rest of this section, we outline an
approach to compute this gradient. As there exist multiple variants of the
belief propagation algorithm, without loss of generality, we formulate the
gradient for the original belief propagation algorithm proposed by Pearl [45,
47] for polytrees.

Let $x$ be the variable at a typical node $X$, which has $M$ children nodes,
$\\{Y_{1},...,Y_{M}\\}$, and $N$ parents, $\\{U_{1},...,U_{N}\\}$. The current
marginal distribution at $X$ is denoted as $P_{B}(x)$. There are two types of
incoming messages at $X$. The first type, $\lambda_{Y_{j}}(x)$, is received
from one of its children $Y_{j}$. It represents the current strength of the
diagnostic support contributed by the outgoing link $X\rightarrow Y_{j}$. And
the second type, $\pi_{X}(u_{i})$, is received from one of its parents
$U_{i}$. It represents the current strength of the causal support contributed
by the incoming link $U_{i}\rightarrow X$. $P_{B}(x)$, $\lambda_{Y_{j}}(x)$
and $\pi_{X}(u_{i})$ are all vectors of probabilities, representing discrete
marginal or conditional distributions. Note that the evidence at a node can be
modeled as an incoming message from an auxiliary child node. To facilitate
gradient formulation, we synchronize all messages using time steps and every
message has a time stamp. At each time step, every node updates its marginal
distribution and outgoing messages as follows.

$P_{B}^{t}(x)=\alpha\prod_{j}\lambda^{t-1}_{Y_{j}}(x)\sum_{u_{1},...,u_{n}}P(x|u_{1},...,u_{n})\prod_{i}\pi^{t-1}_{X}(u_{i})$
(16) $\displaystyle\lambda^{t}_{X}(u_{i})=$
$\displaystyle\beta\left[\sum_{x}\prod_{j}\lambda^{t-1}_{Y_{j}}(x)\right]$
(17) $\displaystyle\ast\sum_{u_{k}:k\neq i}P(x|u_{1},...,u_{n})\prod_{k\neq
i}\pi^{t-1}_{X}(u_{k})$ $\pi^{t}_{Y_{j}}(x)=\alpha\prod_{k\neq
j}\lambda^{t-1}_{Y_{k}}(x)\sum_{u_{1},...,u_{n}}P(x|u_{1},...,u_{n})\prod_{i}\pi^{t-1}_{X}(u_{i})$
(18)

where $\alpha$ and $\beta$ are normalizing constants, $\lambda^{t}_{X}(u_{i})$
is an outgoing message from $X$ to one of its parents, $\pi^{t}_{Y_{j}}(x)$ is
an outgoing message from $X$ to one of its children, and
$P(x|u_{1},...,u_{n})$ is the predetermined conditional probability table at
$X$.

The marginal distribution and outgoing messages are defined as the variables
associated with a node. Suppose the Bayesian network has $N_{B}$ nodes and
$N_{E}$ edges, and each marginal or conditional distribution is defined on $L$
discrete grades. Since a polytree with $N_{B}$ nodes has $N_{B}-1$ edges and
each edge is associated with two messages propagated along opposite
directions, the total number of variables during belief propagation is
$(N_{B}+2N_{E})L=(3N_{B}-2)L$. According to (16)-(18), at node $X$, the
marginal distribution and outgoing messages at time step $t$ are all functions
of the messages received from its parents and children at the previous time
step. Thus, we can define a $(3N_{B}-2)L\times(3N_{B}-2)L$ Jacobian matrix,
$J_{t}$, between every two consecutive time steps. Note that $J_{t}$ is a
sparse matrix. Suppose it takes $T$ time steps for belief propagation to reach
convergence. The Jacobian matrix for all the time steps together, $J_{all}$,
is thus a product of the Jacobian matrices for individual time steps,
$J_{all}=\prod_{t=1}^{T}J_{t}$. The gradient of the entire belief propagation
algorithm, which consists of the partial derivatives of the final marginal
distributions with respect to the input evidences, forms a submatrix of
$J_{all}$, and can be extracted from $J_{all}$.

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