This work proposes the use of  BCART models for insurance pricing, and in particular, claims frequency prediction. These tree-based models  can automatically perform variable selection and detect non-linear effects and possible interactions among explanatory variables. The obtained optimal trees are relatively accurate, stable and are straightforward to interpret
by a visualization of the tree structure. These are desirable aspects for insurance pricing.
We have introduced the  framework of the BCART models and presented  MCMC algorithms for general non-Gaussian distributed data where data augmentation may be needed in its implementation. We have included BCART models for Poisson, NB and ZIP distributions, which are the commonly used distributions for claims frequency. For the NB and ZIP models, we explored two different ways to deal with exposures.
Remarkably, we conclude from the simulation examples and real data analysis that the non-standard ways of embedding exposures can provide us with better tree models, which is in line with the conclusions of <cit.>.

Furthermore, we introduced  a  tree model selection approach based on DIC, which has been seen to be an effective approach using both simulation examples and real insurance data. In particular,  we conclude from the real insurance data analysis that the ZIP-BCART with exposure embedded in the zero mass component is the best candidate for claims frequency modelling. It is worth remarking that
a general zero-inflated NB BCART can be implemented and may further improve the accuracy, but this will require more latent variables to be introduced and will make the convergence of MCMC algorithm harder/slower; see <cit.> for some insights.

Below we
comment on potential further improvements of the BCART models for claims frequency modelling.

* In the MCMC algorithms we have only used four common proposals, namely, Grow, Prune, Change, and Swap, which have made the algorithm to quickly converge to a local optimal region. In order to make it better explore the tree space, other proposals such as those in <cit.> can be suggested to improve the mixing of simulated trees. However, we suspect this will significantly increase the computational time, particularly, for high-dimensional large data set and for models requiring data augmentation. To mitigate this effect, we might consider to use a non-uniform choice of splitting variables in the tree prior so as to achieve a better variable selection, e.g., the Dirichlet prior proposed in <cit.>.

* The proposed models have imposed several assumptions in order to simplify calculations. For example, we used conjugate prior for the terminal node distributions, and additional independence assumption as in (<ref>) and (<ref>). To further improve the
analysis, it might be beneficial to incorporate different specifications
of the prior for the same distribution scenario without using conjugate priors or independence, while this may require other techniques such as Laplace
approximation (see <cit.>). We refer also to <cit.> for an  interesting incorporation of some hierarchical priors.

* We have proposed to use a single (optimal) tree induced from the BCART models for claims frequency prediction. The main reason for this choice is, as we discussed in the introduction, for ease of interpretation. Since stakeholders and regulators may not be statisticians who are able to understand very complex statistical models, a single tree offers  intuitive and visual results to them.
Although we have proposed an approach to find one single optimal tree, some sub-optimal trees (in the convergence region of the MCMC) which possess similar/different tree structures, may also be as informative as the single optimal tree and should not be simply ignored. Further research can be done in this direction to make better use of the posterior trees by clustering or merging them; see, e.g.,  <cit.>.

* To further improve the accuracy of these Bayesian tree-based models we could explore BART for claims frequency modelling. The BART models are tree ensembles; each tree in BART only accounts for a small part of the overall fit, potentially improving the performance, but  model interpretability needs to be explored before it can be used for insurance pricing. To this end, we believe  some insights  from  <cit.> would be helpful.

In this paper, we have focused on insurance claims frequency. A natural next step is to construct a full insurance pricing BCART model, including both claims frequency and severity.

We are  thankful to the anonymous referee for their constructive suggestions which have led to a significant improvement of the manuscript.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleNon-life Insurance Pricing with Generalized Linear Models

John Wiley & Sons

titleActuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-malus Systems

Taylor & Francis

journaltitleScandinavian Actuarial Journal
titleA data driven binning strategy for the construction of insurance tariff classes

Taylor & Francis

journaltitleNorth American Actuarial Journal
titleBoosting insights in insurance tariff plans with tree-based machine learning methods

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series A (General)
titleGeneralized linear models

Available at SSRN 2319328

titleNon-life Insurance: Mathematics & Statistics

titleA Course in Credibility Theory and its Applications

titleMachine learning in P&C insurance: A review for pricing and reserving

titleEffective Statistical Learning Methods for Actuaries

Springer Actuarial, Open Access, https://link.springer.com/book/10.1007/978-3-031-12409-9, Available at SSRN: https://ssrn.com/abstract=3822407

titleStatistical Foundations of Actuarial Learning and its Applications

Available at SSRN: https://ssrn.com/abstract=2870308

titleData Analytics for Non-Life Insurance Pricing (January 9, 2023).

journaltitleInsurance: Mathematics and Economics
titleAutocalibration and Tweedie-dominance for insurance pricing with machine learning

journaltitleEuropean Actuarial Journal
titleBias regularization in neural network models for general insurance pricing

PhD thesis, University of Connecticut

titleInsurance Analytics with Tree-Based Models

journaltitleInsurance: Mathematics and Economics
titleImbalanced learning for insurance using modified loss functions in tree-based models

journaltitleInsurance: Mathematics and Economics
titleActuarial intelligence in auto insurance: Claim frequency modeling with driving behavior features and improved boosted trees

journaltitleDuration-Weighted Probabilities, and Automatic Construction of Tariff Cells (October 24, 2022)
titleLocal bias adjustment, duration-weighted probabilities, and automatic construction of tariff cells

CRC press

titleClassification and Regression Trees

Taylor & Francis

journaltitleJournal of the American Statistical Association
titleBayesian CART model search

Oxford University Press

titleA Bayesian CART algorithm

The Korean Statistical Society

journaltitleCommunications for Statistical Applications and Methods
titleA review of tree-based Bayesian methods

Institute of Mathematical Statistics

journaltitleThe Annals of Applied Statistics
titleBART: Bayesian additive regression trees

journaltitlearXiv preprint arXiv:2108.07636
titleAccounting for shared covariates in semi-parametric Bayesian additive regression trees

Taylor & Francis

journaltitleJournal of the American Statistical Association
titleLog-linear Bayesian additive regression trees for multinomial Logistic and count regression models

Annual Reviews

journaltitleAnnual Review of Statistics and its Application
titleBayesian additive regression trees: A review and look forward

journaltitleJournal of the Royal Statistical Society. Series B (Statistical Methodology)
titleBayesian regression tree ensembles that adapt to smoothness and sparsity

journaltitleAnnals of Statistics
titlePosterior concentration for Bayesian regression trees and forests

Wiley Online Library

titleSemiparametric mixed-scale models using shared Bayesian forests

titleDelta boosting implementation of negative binomial regression in actuarial pricing

Cambridge University Press

journaltitleASTIN Bulletin: The Journal of the IAA
titleAddressing imbalanced insurance data through zero-inflated Poisson regression with boosting

Oxford University Press

titleSeeking efficient data augmentation schemes via conditional and marginal augmentation

Taylor & Francis

journaltitleJournal of Computational and Graphical Statistics
titleThe art of data augmentation

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series B (Statistical Methodology)
titleBayesian measures of model complexity and fit

journaltitleMachine Learning
titleBayesian treed models

Clarendon Press Oxford

journaltitleBayesian Statistics
titleBayesian treed generalized linear models

Wiley, New York

journaltitleEncyclopedia of Statistical Sciences Update
titleBayesian model selection

Courier Corporation

titleBranching Processes

Taylor & Francis

journaltitleJournal of the American Statistical Association
titleBayesian regression trees for high-dimensional prediction and variable selection

journaltitlearXiv preprint arXiv:2304.12505
titleTheory of Posterior Concentration for Generalized Bayesian Additive Regression Trees

Taylor & Francis

journaltitleJournal of Computational and Graphical Statistics
titleBayesian CART: Prior specification and posterior simulation

journaltitleThe Annals of Applied Statistics
titleVariable selection for BART: an application to gene regulation

Oxford University Press

journaltitleJournal of the Royal Statistical Society: Series B (Statistical Methodology)
titleVariable selection with ABC Bayesian forests

Wiley Online Library

titleMultinomial probit Bayesian additive regression trees

titleReversible jump MCMC computation and Bayesian model determination

International Society for Bayesian Analysis

journaltitleBayesian Analysis
titleEfficient Metropolis–Hastings proposal mechanisms for Bayesian regression tree models

journaltitlearXiv preprint arXiv:1312.2171
titlebartMachine: Machine learning with Bayesian additive regression trees

Taylor & Francis

journaltitleJournal of the American Statistical Association
titleThe calculation of posterior distributions by data augmentation

International Society for Bayesian Analysis

journaltitleBayesian Aanalysis
titleDeviance information criteria for missing data models

journaltitleStatistics and Computing
titleUnderstanding predictive information criteria for Bayesian models

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series B (Statistical Methodology)
titleThe deviance information criterion: 12 years on

journaltitleJournal of Machine Learning Research
titleAsymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory.

NIH Public Access

booktitleProceedings of the International Conference on Machine Learning. International Conference on Machine Learning
titleLognormal and gamma mixed negative binomial regression

Taylor & Francis

journaltitleCommunications in Statistics-Theory and Methods
titleBayesian analysis of zero-inflated distributions

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series B (Methodological)
titleEstimation of finite mixture distributions through Bayesian sampling

Taylor & Francis

journaltitleStatistical Theory and Related Fields
titleThe balance property in neural network modelling

journaltitleGenetics Selection Evolution
titleA comparison between Poisson and zero-inflated Poisson regression models with an application to number of black spots in Corriedale sheep

noteR package version4.1-15
titlerpart: Recursive Partitioning and Rregression Trees

noteR package version 1.0
titleinsuranceData: A Collection of Insurance Datasets Useful in Risk Classification in Non-life insurance

John Wiley & Sons, Ltd Chichester, UK

journaltitleWiley StatsRef: Statistics Reference Online
titleBayesian Additive Regression Trees, Computational Approaches

booktitleThe 22nd International Conference on Artificial Intelligence and Statistics
titleXBART: Accelerated Bayesian additive regression trees

journaltitleJournal of Statistical Software
titleNonparametric machine learning and efficient computation with Bayesian additive regression trees: the BART R package

journaltitlePreprint, https://arxiv.org/pdf/1903.00816.pdf
titleStability of decision trees and logistic regression

journaltitleStatistics and Computing
titleHierarchical priors for Bayesian CART shrinkage

booktitleInternational Workshop on Artificial Intelligence and Statistics
titleManaging multiple models

Wiley Online Library

journaltitleStatistics in Medicine
titleIdentifying representative trees from ensembles

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# Feeding the zombies: Synthesizing brain volumes
using a 3D progressive growing GAN

###### Abstract

Deep learning requires large datasets for training (convolutional) networks
with millions of parameters. In neuroimaging, there are few open datasets with
more than 100 subjects, which makes it difficult to, for example, train a
classifier to discriminate controls from diseased persons. Generative
adversarial networks (GANs) can be used to synthesize data, but virtually all
research is focused on 2D images. In medical imaging, and especially in
neuroimaging, most datasets are 3D or 4D. Here we therefore present
preliminary results showing that a 3D progressive growing GAN can be used to
synthesize MR brain volumes.

##  1 Introduction

Generative adversarial networks (GANs) [1] can today produce very realistic
synthetic images [2]. GANs use adversarial training, where a generator creates
an image from noise and a discriminator classifies each image as synthetic or
real. During the training, the generator becomes better at generating
realistic images, and the discriminator becomes better at discriminating
images as synthetic or real. GANs can broadly be divided into noise-to-image
GANs [1, 2], which produce an image from a noise vector, and image-to-image
GANs, which produce an image from another image (image to image translation)
[3, 4].

GANs can be particularly useful in the medical imaging field [5], where large
datasets (e.g. data from more than 100, 1,000 or 10,000 subjects) are uncommon
[6]. However, medical imaging data are normally not 2D, but 3D or 4D, and
therefore a 3D GAN should be used to synthesize realistic volumes. Compared to
2D GANs, the current progress on 3D noise-to-image GANs is rather limited,
with a few exceptions [7, 8].

3D CycleGAN [4] has been used for image-to-image translation of medical
volumes [9, 10, 11, 12] and 2D noise-to-image GANs have been used to
synthesize medical images [13, 6, 14, 15, 16], but the work on 3D noise-to-
image GANs in the medical imaging domain is very limited. [17] used a 3D GAN
to synthesize MR patches of 32 x 32 x 32 voxels, but did not generate full
size volumes.

In this work, we evaluate if a 3D progressive growing GAN [2] can be used to
synthesize realistic T1-weighted MR volumes.

##  2 Data

We used T1-weighted MR volumes from the Human Connectome Project (HCP) for
training our 3D GAN111Data collection and sharing for this project was
provided by the Human Connectome Project (U01-MH93765) (HCP; Principal
Investigators: Bruce Rosen, M.D., Ph.D., Arthur W. Toga, Ph.D., Van J.Weeden,
MD). HCP fund- ing was provided by the National Institute of Dental and
Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH),
and the National Institute of Neurological Disorders and Stroke (NINDS). HCP
data are disseminated by the Laboratory of Neuro Imaging at the University of
Southern California. [18, 19]. Out of the 1,113 subjects we used the first 900
subjects for training, and reserved the remaining 213 subjects for
evaluations. The T1-weighted volumes were acquired with a $0.7\times 0.7\times
0.7$ mm isotropic voxel size, giving volumes of 260 x 311 x 260 voxels. We
used preprocessed [19] structural images, which have been distortion corrected
and registered to MNI space. Each volume was downsampled a factor 2 and then
cropped to obtain volumes of 128 x 128 x 128 voxels.

##  3 Methods

###  3.1 Data augmentation

To increase the number of training volumes we performed data augmentation by
applying 10 random 3D rotations to each of the 900 volumes, to achieve a total
of 9000 training volumes. The random rotations were generated from a normal
distribution with a mean of 0 and a standard deviation of 10 degrees.

###  3.2 3D PGAN

Noise-to-image GANs are rather difficult to train, especially for high
resolutions, and a common pitfall is the mode collapse problem, where the GAN
synthesizes a single image. A progressive growing GAN (PGAN) [2] is trained
progressively, i.e. by first generating images of 4 x 4 pixels, then images of
8 x 8 pixels, all the way to the desired resolution. Here we investigate if
the same idea can be used to synthesize high resolution volumes, by starting
the training on 4 x 4 x 4 volumes, and the training then continues on 8 x 8 x
8 volumes etc.

We based our 3D PGAN on the 2D PGAN Tensorflow implementation provided in [2].
We replaced all 2D convolutions with 3D convolutions, and added an extra
dimension to all relevant Tensorflow calls. We also added reading and writing
of nifti files using the nibabel Python package [20]. Our 3D PGAN is available
on Github222https://github.com/wanderine/ProgressiveGAN3D to facilitate
replication of our results [21]. Compared to the original 2D PGAN
implementation, we lowered the number of filters from 512 to 128, and also
lowered the learning rate to 0.0003 for 4 x 4 x 4 and 8 x 8 x 8 volumes, and
0.0006 for 16 x 16 x 16 and 32 x 32 x 32 volumes. For the final training steps
of the 32 x 32 x 32 resolution the learning rate was lowered to 0.0001. The
number of real volumes to show before doubling the resolution was increased
from 600,000 to 1,000,000.

Training the 3D PGAN to generate 32 x 32 x 32 volumes took 23 hours using an
Nvidia DGX station (containing 4 Nvidia Tesla V100 graphics cards with 32 GB
of memory each).

##  4 Results

Figure 1 shows some examples of synthetic brains generated by our 3D PGAN.

Fig. 1: Synthetic T1-weighted MR volumes (32 x 32 x 32 voxels, upsampled to
128 x 128 x 128 voxels) generated by our 3D progressive growing GAN.

##  5 Conclusion

We have demonstrated that a 3D progressive growing GAN can be used to
synthesize T1-weighted volumes of 32 x 32 x 32 voxels. The synthetic volumes
can be used for training (convolutional) networks that perform classification
or segmentation. Furthermore, synthetic volumes can be shared freely, as they
do not belong to a specific person, and can therefore facilitate data sharing
[22].

Producing volumes of higher resolution is in theory possible, but our results
indicate that the training is very sensitive to the used hyper parameters.
Additionally, each training step for the 64 x 64 x 64 resolution takes about 3
hours on the Nvidia DGX station. In future work we will continue to tune the
hyper parameters, to synthesize volumes of higher resolution.

## Acknowledgements

This study was supported by Swedish research council grant 2017-04889. Funding
was also provided by VINNOVA Analytic Imaging Diagnostics Arena (AIDA) and the
ITEA3 / VINNOVA funded project ”Intelligence based iMprovement of Personalized
treatment And Clinical workflow supporT” (IMPACT).

## References

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