Bayesian learning of joint distributions of objects

Anjishnu Banerjee Jared Murray David B. Dunson

Statistical Science, Duke University

###### Abstract

There is increasing interest in broad application areas in defining flexible
joint models for data having a variety of measurement scales, while also
allowing data of complex types, such as functions, images and documents. We
consider a general framework for nonparametric Bayes joint modeling through
mixture models that incorporate dependence across data types through a joint
mixing measure. The mixing measure is assigned a novel infinite tensor
factorization (ITF) prior that allows flexible dependence in cluster
allocation across data types. The ITF prior is formulated as a tensor product
of stick-breaking processes. Focusing on a convenient special case
corresponding to a Parafac factorization, we provide basic theory justifying
the flexibility of the proposed prior and resulting asymptotic properties.
Focusing on ITF mixtures of product kernels, we develop a new Gibbs sampling
algorithm for routine implementation relying on slice sampling. The methods
are compared with alternative joint mixture models based on Dirichlet
processes and related approaches through simulations and real data
applications.

##  1 INTRODUCTION

There has been considerable recent interest in joint modeling of data of
widely disparate types, including not only real numbers, counts and
categorical data but also more complex objects, such as functions, shapes, and
images. We refer to this general problem as mixed domain modeling (MDM), and
major objectives include exploring dependence between the data types, co-
clustering, and prediction. Until recently, the emphasis in the literature was
almost entirely on parametric hierarchical models for joint modeling of mixed
discrete and continuous data without considering more complex object data. The
two main strategies are to rely on underlying Gaussian variable models
(Muthen, 1984) or exponential family models, which incorporate shared latent
variables in models for the different outcomes (Sammel et al., 1997; Dunson,
2000, 2003). Recently, there have been a number of articles using these models
as building blocks in discrete mixture models relying on Dirichlet processes
(DPs) or closely-related variants (Cai et al., 2011; Song et al., 2009; Yang &
Dunson, 2010). DP mixtures for mixed domain modeling were also considered by
Hannah et al. (2011); Shahbaba & Neal (2009); Dunson & Bhattacharya (2010)
among others. Related approaches are increasingly widely-used in broad machine
learning applications, such as for joint modeling of images and captions (Li
et al., 2011), and have rapidly become a standard tool for MDM.

Although such joint Dirichlet process mixture models (DPMs) are quite
flexible, and can accommodate joint modeling with complicated objects such as
functions (Bigelow & Dunson, 2009), they suffer from a key disadvantage in
relying on conditional independence given a single latent cluster index. For
example, as motivated in Dunson (2009, 2010), the DP and related approaches
imply that two subjects $i$ and $i^{\prime}$ are either allocated to the same
cluster ($C_{i}=C_{i^{\prime}}$) globally for all their parameters or are not
clustered. The soft probabilistic clustering of the DP is appealing in leading
to substantial dimensionality reduction, but a single global cluster index
conveys several substantial practical disadvantages. Firstly, to realistically
characterize joint distributions across many variables, it may be necessarily
to introduce many clusters, degrading the performance in the absence of large
sample sizes. Secondly, as the DP and the intrinsic Bayes penalty for model
complexity both favor allocation to few clusters, one may over cluster and
hence obscure important differences across individuals, leading to misleading
inferences and poor predictions. Often, the posterior for the clusters may be
largely driven by certain components of the data, particularly when more data
are available for those components, at the expense of poorly characterizing
components for which less, or more variable, data are available.

To overcome these problems we propose Infinite Tensor Factorization (ITF)
models, which can be viewed as next generation extensions of the DP to
accommodate dependent object type-specific clustering. Instead of relying on a
single unknown cluster index, we propose separate but dependent cluster
indices for each of the data types whose joint distribution is given by a
random probability tensor. We use this to build a general framework for
hierarchical modeling. The other main contribution in this article is to
develop a general extension of blocked sliced sampling, which allows for an
efficient and straightforward algorithm for sampling from the posterior
distributions arising with the ITF; with potential application in other
multivariate settings with infinite tensors, without resorting to finite
truncation of the infinitely many possible levels.

##  2 PRELIMINARIES

We start by considering a simple bivariate setting $p=2$ in which data for
subject $i$ consist of $y_{i}=(y_{i1},y_{i2})^{\prime}\in\mathcal{Y}$, with
$\mathcal{Y}=\mathcal{Y}_{1}\otimes\mathcal{Y}_{2}$,
$y_{i1}\in\mathcal{Y}_{1}$, and $y_{i2}\in\mathcal{Y}_{2}$ for $i=1,\ldots,n$.
We desire a joint model in which $y_{i}\sim f$, with $f$ a probability measure
characterizing the joint distribution. In particular, letting
$\mathcal{B}(\mathcal{Y})$ denote an appropriate sigma-algebra of subsets of
$\mathcal{Y}$, $f$ assigns probability $f(B)$ to each
$B\in\mathcal{B}(\mathcal{Y})$. We assume $\mathcal{Y}$ is a measurable Polish
space, as we would like to keep the domains $\mathcal{Y}_{1}$ and
$\mathcal{Y}_{2}$ as general as possible to encompass not only subsets of
Euclidean space and the set of natural numbers but also function spaces that
may arise in modeling curves, surfaces, shapes and images. In many cases, it
is not at all straightforward to define a parametric joint measure, but there
is typically a substantial literature suggesting various choices for the
marginals $y_{i1}\sim f_{1}$ and $y_{i2}\sim f_{2}$ separately.

If we only had data for the $j$th variable, $y_{ij}$, then one possible
strategy is to use a mixture model in which

$\displaystyle
f_{j}(B)=\int_{\Theta_{j}}\mathcal{K}_{j}(B;\theta_{j})dP_{j}(\theta_{j}),\quad
B\in\mathcal{B}(\mathcal{Y}_{j}),$ (1)

where $\mathcal{K}_{j}(\cdot;\theta_{j})$ is a probability measure on
$\\{\mathcal{Y}_{1},\mathcal{B}(\mathcal{Y}_{1})\\}$ indexed by parameters
$\theta_{j}\in\Theta_{j}$, $\mathcal{K}_{j}$ obeys a parametric law (e.g.,
Gaussian), and $P_{j}$ is a probability measure over
$\\{\Theta_{j},\mathcal{B}(\Theta_{j})\\}$. A nonparametric Bayesian approach
is obtained by treating $P_{j}$ as a random probability measure and choosing
an appropriate prior. By far the most common choice is the Dirichlet process
(Ferguson, 1973), which lets $P_{j}\sim DP(\alpha P_{0j})$. Under the
Sethuraman (1994) stick-breaking representation, one then obtains,

$\displaystyle
f_{j}(B)=\sum_{h=1}^{\infty}\pi_{h}\mathcal{K}_{j}(B;\theta_{h}^{*}),\mbox{with}$
$\displaystyle\pi_{h}=V_{h}\prod_{l<h}(1-V_{l}),\quad\theta_{h}^{*}\sim
P_{0j},$ (2)

and $V_{h}\sim\mbox{Be}(1,\alpha)$, so that $f_{j}$ can be expressed as a
discrete mixture. This discrete mixture structure implies the following simple
hierarchical representation, which is crucially used for efficient
computation:

$\displaystyle
y_{ij}\sim\mathcal{K}_{j}(\theta_{C_{i}}^{*}),\quad\theta_{h}^{*}\sim
P_{0j},\quad\mbox{pr}(C_{i}=h)=\pi_{h},$ (3)

where $C_{i}$ is a cluster index for subject $i$. The great success of this
model is largely attributable to the divide and conquer structure in which one
allocates subjects to clusters probabilistically, and then can treat the
observations within each cluster as separate instantiations of a parametric
model. In addition, there is a literature showing appealing properties, such
as minimax optimal adaptive rates of convergence for DPMs of Gaussians (Shen &
Ghosal, 2011; Tokdar, 2011).

The standard approach to adapt expression (1) to accommodate mixed domain data
is to simply let $f(B)=\int_{\Theta}\mathcal{K}(B;\theta)dP(\theta)$, for all
$B\in\mathcal{B}(\mathcal{Y})$, where $\mathcal{K}(\cdot;\theta)$ is an
appropriate joint probability measure over
$\\{\mathcal{Y},\mathcal{B}(\mathcal{Y})\\}$ obeying a parametric law.
Choosing such a joint law is straightforward in simple cases. For example,
Hannah et al. (2011) rely on a joint exponential family distribution
formulated via a sequence of generalized linear models. However, in general
settings, explicitly characterizing dependence within
$\mathcal{K}(\cdot;\theta)$ is not at all straightforward and it becomes
convenient to rely on a product measure (Dunson & Bhattacharya, 2010):

$\displaystyle\mathcal{K}(B;\theta)=\prod_{j}\mathcal{K}(B_{j};\theta_{j}),\quad
B=\bigotimes_{j=1}^{p}B_{j},\quad B_{j}\in\mathcal{B}(\mathcal{Y}_{j}).$ (4)

If we then choose $P\sim DP(\alpha P_{0})$ with
$P_{0}=\bigotimes_{j=1}^{p}P_{0j}$, we obtain an identical hierarchical
specification to (3), but with the elements of $y_{i}=\\{y_{ij}\\}$
conditionally independent given the cluster allocation index $C_{i}$.

As mentioned in §$1$, this conditional independence assumption given a single
latent class variable is the nemesis of the joint DPM approach. We consider
more generally a multivariate
$C_{i}=(C_{i1},\ldots,C_{ip})^{T}\in\\{1,\ldots,\infty\\}^{p}$, with separate
but dependant indices across the disparate data types. We let,

$\displaystyle\mbox{pr}(C_{i1}=h_{1},\ldots,C_{ip}=h_{p})=\pi_{h_{1}\cdots
h_{p}},$ $\displaystyle\mbox{with}\quad h_{j}=1,\ldots,\infty,j=1,\ldots,p,$
(5)

where $\pi=\\{\pi_{h_{1}\cdots h_{p}}\\}\in\Pi_{p}^{\infty}$ is an infinite
$p$-way probability tensor characterizing the joint probability mass function
of the multivariate cluster indices. It remains to specify the prior for the
probability tensor $\pi$, which is considered next in §3.

##  3 PROBABILISTIC TENSOR FACTORIZATIONS

###  3.1 PARAFAC Extension

Suppose that $C_{ij}\in\\{1,\ldots,d_{j}\\}$, with $d_{j}$ the number of
possible levels of the $j$th cluster index. Then, assuming that $C_{i}$ are
observed unordered categorical variables, Dunson & Xing (2009) proposed a
probabilistic Parafac factorization of the tensor $\pi$:

$\displaystyle\pi=\sum_{h=1}^{k}\lambda_{h}\psi_{h}^{(1)}\otimes\cdots\otimes\psi_{h}^{(p)},$
(6)

where $\lambda=\\{\lambda_{h}\\}$ follows a stick-breaking process,
$\psi_{h}^{(j)}=(\psi_{h1}^{(j)},\ldots,\psi_{hd_{j}}^{(j)})^{T}$ is a
probability vector specific to component $h$ and outcome $j$, $\otimes$
denotes the outer product.

We focus primarily on generalizations of the Parafac factorization to the case
in which $C_{i}$ is unobserved and can take infinitely-many different levels.
We let,

$\displaystyle\pi_{c_{1}\cdots c_{p}}$ $\displaystyle=$
$\displaystyle\mbox{pr}(C_{1}=c_{1},\ldots,C_{p}=c_{p})=\sum_{h=1}^{\infty}\lambda_{h}\prod_{j=1}^{p}\psi_{hc_{j}}^{(j)}$
$\displaystyle\lambda_{h}$ $\displaystyle=$ $\displaystyle
V_{h}\prod_{l<h}(1-V_{l}),\quad V_{h}\sim\mbox{Be}(1,\alpha)$
$\displaystyle\psi_{hr}^{(j)}$ $\displaystyle=$ $\displaystyle
U_{hr}^{(j)}\prod_{s<r}(1-U_{hs}^{(j)}),\quad
U_{hr}^{(j)}\sim\mbox{Be}(1,\beta_{j}),$ (7)

A more compact notation for this factorization of the infinite probability
tensor $\pi$ is,

$\displaystyle\pi=\sum_{h=1}^{\infty}\lambda_{h}\bigotimes_{j=1}^{p}\psi_{h}^{(j)},$
(8) $\displaystyle\lambda\sim\mbox{Stick}(\alpha),\
\psi_{h}^{(j)}\sim\mbox{Stick}(\beta_{j}),$ (9)

which takes the form of a stick-breaking mixture of outer products of stick-
breaking processes. This form is carefully chosen so that the elements of
$\pi$ are stochastically larger in those cells having the smallest indices,
with rapid decreases towards zero as one moves away from the upper right
corner of the tensor.

It can be shown that tensors realizations from the ITF distribution are valid
in the sense that they sum to $1$ with probability $1$. We can be flexible in
terms where exactly these cluster indices occur in a hierarchical Bayesian
model. Next in §3.2, we formulate a generic mixture model for MDM, where the
ITF is used characterize the cluster indices of the parameters governing the
distributions of the disparate data-types.

###  3.2 Infinite Tensor Factorization Mixture

Assume that for each individual $i$ we have a data ensemble
$(y_{i1},\ldots,y_{ip})\in\mathcal{Y}$ where
$\mathcal{Y}=\bigotimes_{j=1}^{p}\mathcal{Y}_{j}$. Let
$\mathcal{B}(\mathcal{Y})$ be the sigma algebra generated by the product sigma
algebra
$\mathcal{B}(\mathcal{Y}_{1})\times\cdots\times\mathcal{B}(\mathcal{Y}_{p})$.
Consider any Borel set
$B=\bigotimes_{j=1}^{p}B_{j}\in\mathcal{B}(\mathcal{Y})$. Given cluster
indices $(C_{i1}=c_{i1},\ldots,C_{ip}=c_{ip})$, we assume that the ensemble
components are independent with

$\displaystyle f(y_{i1}\in B_{1},\ldots,y_{ip}\in
B_{p}\,|\,C_{i1}=h_{1},\ldots,C_{ip}=h_{p})$
$\displaystyle=\prod_{j=1}^{p}\mathcal{K}_{j}(B_{j};\theta_{j,h_{j}}).$ (10)

$\mathcal{K}_{j}(\cdot;\theta_{j,h})$ is an appropriate probability measure on
$\\{\mathcal{Y}_{j},\mathcal{B}(\mathcal{Y}_{j})\\}$ as in equation (1).
Marginalizing out the cluster indices, we obtain

$\displaystyle f(y_{i1}\in B_{1},\ldots,y_{ip}\in B_{p})$
$\displaystyle=\sum_{h_{1}=1}^{\infty}\cdots\sum_{h_{p}=1}^{\infty}\pi_{h_{1},\ldots,h_{p}}\prod_{j=1}^{p}\mathcal{K}_{j}(B_{j};\theta_{j,h_{j}}),$
(11)

where $\pi_{h_{1},\ldots,h_{p}}=\mbox{pr}(C_{i1}=h_{1},\ldots,C_{ip}=h_{p})$.
We let $\pi\sim\mbox{{ITF}}(\alpha,\beta)$ and we call the resulting mixture
model an infinite tensor factorization mixture,
$f\sim\mbox{{ITM}}(\alpha,\beta)$. To complete the model specification, we let
$\theta_{j,h_{j}}\sim P_{0j}$ independently as in (2).

The model $y_{i}\sim f$, $f\sim\mbox{{ITM}}(\alpha,\beta)$, can be
equivalently expressed in hierarchical form as

$\displaystyle y_{ij}$ $\displaystyle\sim$
$\displaystyle\mathcal{K}_{j}(\theta_{ij}^{*}),\
\theta_{i}^{*}=P\sum_{h_{1}=1}^{\infty}\cdots\sum_{h_{p}=1}^{\infty}\pi_{h_{1},\ldots,h_{p}}\prod_{j=1}^{p}\delta_{\theta_{j,h_{j}}},$
$\displaystyle\pi$ $\displaystyle\sim$
$\displaystyle\mbox{{ITF}}(\alpha,\beta),\quad\theta_{j,h_{j}}\sim P_{0j},$
(12)

Here, $P$ is a joint mixing measure across the different data types and is
given a infinite tensor process prior,
$P\sim\mbox{{ITP}}(\alpha,\beta,\bigotimes_{j=1}^{p}P_{0j})$. Marginalizing
out the random measure $P$, we obtain the same form as in (3.2). The proposed
infinite tensor process prior provide a much more flexible generalization of
existing priors for discrete random measures, such as the Dirichlet process or
Pitman Yor process.

##  4 POSTERIOR INFERENCE

###  4.1 Markov Chain Monte Carlo Sampling

We propose a novel algorithm for efficient exact MCMC posterior inference in
the ITM model, utilizing blocked and partially collapsed steps. We adapt ideas
from Walker (2007); Papaspiliopoulos & Roberts (2008) to derive slice sampling
steps with label switching moves, entirely avoiding truncation approximations.
Begin by defining the augmented joint likelihood for an observation $y_{i}$,
cluster labels $c_{i}=(c_{i0},c_{i1},\dots,c_{ip})$ and slice variables
$u_{i}=(u_{i0},u_{i1},\dots,u_{ip})$ as

$\displaystyle p(y_{i},c_{i},u_{i}\mid\lambda,\Psi,\Theta)$
$\displaystyle=\bm{1}\left({u_{i0}<\lambda_{c_{i0}}}\right)\prod_{j=1}^{p}{\mathcal{K}_{j}}(y_{ij};\theta^{(j)}_{c_{ij}})\bm{1}\left({u_{ij}<\psi_{c_{i0}c_{ij}}^{(j)}}\right)$
(13)

It is straightforward to verify that on marginalizing $u_{i}$ the model is
unchanged, but including $u_{i}$ induces full conditional distributions for
the cluster indices with finite support. Let
$m_{0h}=\sum_{i=1}^{n}\bm{1}\left({c_{i0}=h}\right)$ and
$\mathcal{D}_{0}=\\{h:m_{0h}>0\\}$. Similarly define
$m_{jhk}=\sum_{i=1}^{n}\bm{1}\left({c_{i0}=h}\right)\bm{1}\left({c_{ij}=k}\right)$
and $\mathcal{D}_{j}=\\{k:\sum_{h=1}^{\infty}m_{jhk}>0\\}$, and let
$k^{*}_{j}=\max(\mathcal{D}_{j})$ for $0\leq j\leq p$. Define
$\mathcal{U}_{0}=\\{u_{i0}:1\leq i\leq n\\}$, $\mathcal{C}_{0}=\\{c_{i0}:1\leq
i\leq n\\}$, $\mathcal{U}_{1}=\\{u_{ij}:1\leq i\leq n,1\leq j\leq p\\}$ and
$\mathcal{C}_{1}=\\{c_{ij}:1\leq i\leq n,1\leq j\leq p\\}$. The superscript
$(-i)$ denotes that the quantity is computed excluding observation $i$.

  1. 1.

Block update $(\mathcal{U}_{0},\lambda,\alpha)$

    1. (a)

Sample $(\alpha\mid\mathcal{C}_{0})$. Standard results (Antoniak, 1974) give

$p(\alpha\mid\mathcal{C}_{0})\propto
p(\alpha)\alpha^{\tilde{c}}\frac{\Gamma(\alpha)}{\Gamma(\alpha+n)}$

for $\tilde{c}=|\mathcal{D}_{0}|$ which can be sampled via Metropolis-Hastings
or using auxiliary variables when $p(\alpha)$ is a mixture of Gamma
distributions (Escobar & West, 1995).

    2. (b)

Sample $(\lambda\mid\alpha,\mathcal{C}_{0})$ by drawing $V_{h}\sim
Beta(1+m_{0h},\alpha+\sum_{l=h+1}^{k_{0}^{*}}m_{0l})$ for $1\leq h\leq
k_{0}^{*}$ and setting $\lambda_{h}=V_{h}\prod_{l<h}(1-V_{l})$

    3. (c)

Label switching moves:

      1. i.

From $\mathcal{D}_{0}$ choose two elements $h_{1},h_{2}$ uniformly at random
and change their labels with probability
$\min(1,(\lambda_{h_{1}}/\lambda_{h_{2}})^{m_{0h_{2}}-m_{0h_{1}}})$

      2. ii.

Sample a label $h$ uniformly from $1,2,\dots,k_{0}^{*}$ and propose to swap
the labels $h,h+1$ and corresponding stick breaking weights $V_{h},V_{h+1}$.
Accept with probability $\min(1,a)$ where

$a=\left(\frac{k_{0}^{*}}{k_{0}^{*}+1}\right)^{\bm{1}\left({h=k^{*}_{0}}\right)}\frac{\left(1-V_{h}\right)^{m_{0(h+1)}}}{\left(1-V_{h+1}\right)^{m_{0h}}}$

    4. (d)

Sample $(u_{i0}|c_{i0},\lambda)\sim U(0,\lambda_{c_{i0}})$ independently for
$1\leq i\leq n$

  2. 2.

Update $\mathcal{C}_{0}$. From (13) the relevant probabilities are

$\displaystyle Pr(c_{i0}=h|u_{i},c_{i},\Psi,\lambda)$
$\displaystyle\propto\bm{1}\left({u_{i0}<\lambda_{h}}\right)\prod_{j=1}^{p}\bm{1}\left({u_{ij}<\psi_{hc_{ij}}^{(j)}}\right)$
(14)

However, it is possible to obtain more efficient updates through _partial
collapsing_ , which allows us to integrate over the lower level slice
variables and $\Psi$ instead of conditioning on them. Then we have

$\displaystyle Pr$ $\displaystyle(c_{i0}=k\mid
u_{i0},\mathcal{C}_{1},\mathcal{C}_{0}^{(-i)},\lambda)\propto\bm{1}\left({u_{i0}<\lambda_{h}}\right)$
$\displaystyle\times\prod_{j=1}^{p}\frac{\left(1+m^{(-i)}_{jkc_{ij}}\right)\prod_{l<c_{ij}}\left(\beta_{k}^{(j)}+\sum_{s>l}+m^{(-i)}_{jks}\right)}{\prod_{l\leq
c_{ij}}\left(1+\beta_{k}^{(j)}+\sum_{s\geq c_{ij}}+m^{(-i)}_{jks}\right)}$
(15)

To determine the support of (15) we need to ensure that
$u^{*}_{0}=\min\\{u_{i0}:1\leq i\leq n\\}$ satisfies
$u_{0}^{*}>1-\sum_{l=1}^{k_{0}^{*}}\lambda_{l}.$ If
$\sum_{l=1}^{k_{0}^{*}}\lambda_{l}<1-u^{*}_{0}$ then draw additional stick
breaking weights $V_{k_{0}^{*}+1},\dots,V_{k_{0}^{*}+d}$ independently from
$Beta(1,\alpha)$ until $\sum_{l=1}^{k_{0}^{*}+d}\lambda_{l}>1-u^{*}_{0}$,
ensuring that
$\sum_{l=k_{0}^{*}+d+1}^{\infty}\bm{1}\left({u_{i0}<\lambda_{h}}\right)=0$ for
all $1\leq i\leq n$. Then the support of (15) is contained within
$1,2,\dots,k_{j}^{*}+d$ and we can compute the normalizing constant exactly.

  3. 3.

Block update $(\mathcal{U}_{1},\Psi,\beta)$:

    1. (a)

Update $(\beta_{r}^{(j)}\mid\\{c_{ij}:c_{i0}=r\\},\mathcal{C}_{0})$ for $1\leq
j\leq p$, $1\leq r\leq k_{0}^{*}$. If the concentration parameter is shared
across global clusters (that is, $\beta_{r}^{(j)}\equiv\beta^{(j)}$) then a
straightforward conditional independence argument gives

$\displaystyle p(\beta^{(j)}\mid\\{c_{ij}:c_{i0}=r\\},\mathcal{C}_{0})$
$\displaystyle\propto
p(\beta^{(j)})\prod_{r\in\mathcal{D}_{0}}\left(\beta^{(j)}\right)^{\tilde{c}_{jr}}\frac{\Gamma(\beta^{(j)})}{\Gamma(\beta^{(j)}+n_{r})}$
(16)

where $n_{r}=|\\{i:c_{i0}=r\\}|$ and $\tilde{c}_{jr}=|\\{h:m_{jrh}>0\\}|$.
Note that terms with $n_{r}=1$ (corresponding to top-level singleton
components) do not contribute, since
$\beta^{(j)}{\Gamma(\beta^{(j)})}={\Gamma(\beta^{(j)}+1)}$. The updating
scheme of Escobar & West (1995) is simple to adapt here using
$|\mathcal{D}_{0}|$ independent auxiliary variables.

    2. (b)

For $r\in\mathcal{D}_{0}$ update
$(\psi^{(j)}_{r}\mid\mathcal{C}_{0},\mathcal{C}_{1},\beta_{r}^{(j)})$ by
drawing $U^{(j)}_{rh}\sim
Beta(1+m_{jrh},\beta_{r}^{(j)}+\sum_{l=h+1}^{k_{0}^{*}}m_{jlh})$ for $1\leq
h\leq k^{*}_{j}$

    3. (c)

Label switching moves: For $1\leq j\leq p$,

      1. i.

From $\mathcal{D}_{j}$ choose two elements $h_{1},h_{2}$ uniformly at random
and change their labels with probability $\min(1,a)$ where

$a=\prod_{h_{0}\in\mathcal{D}_{0}}\left(\frac{\psi^{(j)}_{h_{0}h_{1}}}{\psi^{(j)}_{h_{0}h_{2}}}\right)^{m_{jh_{0}h_{2}}-m_{jh_{0}h_{1}}}$

      2. ii.

Sample a label $h$ uniformly from $1,2,\dots,k^{*}_{j}$ and propose to swap
the labels $h,h+1$ and corresponding stick breaking weights. Accept with
probability $min(1,a)$ where

$\displaystyle
a=\left(\frac{k_{j}^{*}}{k_{j}^{*}+1}\right)^{\bm{1}\left({h=k^{*}_{j}}\right)}$
$\displaystyle\times\prod_{h_{0}\in\mathcal{D}_{0}}\frac{\left(1-U^{(j)}_{rh}\right)^{m_{jr(h+1)}}}{\left(1-U^{(j)}_{r(h+1)}\right)^{m_{jrh}}}$
(17)

    4. (d)

Sample $(u_{ij}|c_{i},\Psi)\sim U(0,\psi^{(m)}_{c_{i0}c_{ij}})$ independently
for $1\leq j\leq p$, $1\leq i\leq n$.

  4. 4.

Update $\mathcal{C}_{j}$ for $1\leq j\leq p$ independently. We have

$\displaystyle Pr(c_{ij}=k\mid y,\Theta,u_{ij},c_{i0},\Psi)$
$\displaystyle\propto{\mathcal{K}_{j}}(y_{ij};\theta^{(j)}_{k})\bm{1}\left({u_{ij}<\psi_{c_{i0}k}^{(j)}}\right)$
(18)

As in step 2 we determine the support of the full conditional distribution as
follows: Let $u_{j}^{*}=\min\\{u_{ij}:1\leq i\leq n\\}$. For all
$r\in\mathcal{D}_{0}$, if $\sum_{h=1}^{k_{j}^{*}}\psi_{rh}^{(j)}<1-u_{j}^{*}$
then extend the stick breaking measure $\psi^{(j)}_{r}$ by drawing $d_{r}$ new
stick breaking weights from the prior so that
$\sum_{h=1}^{k_{j}^{*}+d_{r}}\psi_{rh}^{(j)}>1-u_{j}^{*}$. Draw
$\theta^{(j)}_{k_{j}^{*}+1},\dots,\theta^{(j)}_{k_{j}^{*}+d}\sim
p(\theta^{(j)})$ independently (where
$d=\max\\{d_{r}:r\in\mathcal{D}_{j}\\}$). Then update $c_{ij}$ from

$\displaystyle Pr(c_{ij}=k\mid y,\Theta,u_{ij},c_{i0},\Psi)$
$\displaystyle=\frac{{\mathcal{K}_{j}}(y_{ij};\theta^{(j)}_{k})\bm{1}\left({u_{ij}<\psi_{c_{i0}k}^{(j)}}\right)}{\sum_{h=1}^{k_{j}^{*}+d}{\mathcal{K}_{j}}(y_{ij};\theta^{(j)}_{k})\bm{1}\left({u_{ij}<\psi_{c_{i0}k}^{(j)}}\right)}$
(19)

  5. 5.

Update $(\Theta|-)$ by drawing from

$p(\theta_{h}^{(j)}\mid y,\mathcal{C}_{j})\propto
p(\theta_{h}^{(m)})\prod_{\\{i:c_{ij}=h\\}}{\mathcal{K}_{j}}(y_{ij};\theta^{(j)}_{h})$

for each $1\leq j\leq p$ and $1\leq h\leq k^{*}_{j}$

###  4.2 Inference

Given samples from the MCMC scheme above we can estimate the predictive
distribution as

$\displaystyle\hat{f}(y_{n+1}\mid
y_{n})=\frac{1}{T}\sum_{t=1}^{T}\sum_{h_{0}=1}^{k^{*}_{0}}\sum_{h_{1}=1}^{k^{*}_{1}}\cdots\sum_{h_{p}=1}^{k^{*}_{p}}\lambda^{(t)}_{h_{0}}$
$\displaystyle\times\prod_{j=1}^{p}\psi^{(t)}_{h_{0}h_{j}}{\mathcal{K}_{j}}\left(y_{(n+1)j};\theta^{(j)(t)}_{h_{j}}\right)$
(20)

Each of the inner sums in (20) is a truncation approximation, but it can be
made arbitrarily precise by extending the stick breaking measures with draws
from the prior and drawing corresponding atoms from $p(\theta^{(j)})$. In
practice this usually isn’t necessary as any error in the approximation is
small relative to Monte Carlo error.

The other common inferential question of interest in the MDM settings is the
dependence between components, for example testing whether component $j1$ and
$j2$ are independent of each other. As already noted, the dependence between
the components comes in through the dependence between the cluster allocations
and therefore, tests for independence between $j1$ and $j2$ is equivalent to
testing for independence between their latent cluster indicators $C_{j1}$ and
$C_{j2}$. Such a test can be constructed in terms of the divergence between
the joint and marginal posterior distributions of $C_{j1}$ and $C_{j2}$. The
Monte Carlo estimate of the Kulback Leibler divergence between the joint and
marginal posterior distributions is given as,

$\displaystyle I$
$\displaystyle(j1,j2)=\frac{1}{T}\sum_{t=1}^{T}\sum_{h_{j1}=1}^{k^{*}_{j1}}\sum_{h_{j2}=1}^{k^{*}_{j2}}\left(\sum_{h_{0}=1}^{k^{*}_{0}}\lambda^{(t)}_{h_{0}}\psi^{(t)}_{h_{0}h_{j1}}\psi^{(t)}_{h_{0}h_{j2}}\right)$
$\displaystyle\times\log\left(\frac{\sum_{h_{0}=1}^{k^{*}_{0}}\lambda^{(t)}_{h_{0}}\psi^{(t)}_{h_{0}h_{j1}}\psi^{(t)}_{h_{0}h_{j2}}}{\left[\sum_{h_{0}=1}^{k^{*}_{0}}\lambda^{(t)}_{h_{0}}\psi^{(t)}_{h_{0}h_{j1}}\right]\left[\sum_{h_{0}=1}^{k^{*}_{0}}\lambda^{(t)}_{h_{0}}\psi^{(t)}_{h_{0}h_{j1}}\right]}\right)$
(21)

Under independence, the divergence should be $0$. Analogous divergences can be
considered for testing other general dependancies, like $3$-way, $4$-way
independences.

##  5 EXPERIMENTS

Our approach can be used for two different objectives in the context of mixed
domain data - for prediction and for inference on the dependence structure
between different data types. We outline results of experiments with both
simulated and real data that show the performance of our approach with respect
to both the objectives.

###  5.1 Simulated Data Examples

To the best of our knowledge, there is no standard model to jointly predict
for mixed domain data as well as evaluate the dependence structure, so as a
competitor, we use a joint DPM. To keep the evaluations fair, we use two
scenarios. In the first the ground truth is close to that of the joint DPM, in
the sense that all the components of the mixed data have the same cluster
structure. The other simulated experiment considers the case when the ground
truth is close to the ITF, where different components of the mixed data
ensemble have their own cluster structure but clustering is dependent. The
goal here in each of the scenarios is to compare joint DPM vs ITF in terms of
recovery of dependence structure and predictive accuracy.

For scenario 1, we consider a set of 1,000 individuals from whom an ensemble
comprising of T, a time series R, a multivariate real-valued response
($\in\Re^{4}$) and C1,C2,C3, 3 different categorical variables have been
collected, to emulate the type of data collected from patients in cancer
studies and other medical evaluations. For the purposes of scenario 1, we
simulate T, R, C1, C2, C3 each from a mixture of 3 clusters. For example, R is
simulated from a two-component mixture of multivariate normals with different
means, R is simulated from a mixture of two autoregressive kernels and each of
the categorical variables from a mixture of two multinomial distributions. If
we label the clusters as $1$ and $2$, for each simulation, either all of the
ensemble (T,R,C1,C2,C3) comes from $1$ or all of it comes from $2$. After
simulation we randomly hold out R in 50 individuals, C1, C2 in 10 each, for
the purposes of measuring prediction accuracy. For the categorical variables
prediction accuracy is considered with a $0-1$ loss function and is expressed
as a percent missclassification rate. For the multivariate real variable R, we
consider squared error loss and accuracy is expressed as relative predictive
error. We also evaluate for some of the pairs their dependence via estimated
mutual information.

For scenario 2, the same set-up as in scenario 1 is used, except for the
cluster structure of the ensemble. Now simulations are done such that T falls
into three clusters and this is dependent on R and C1. C2 and C3 depend on
each other and are simulated from two clusters each but their clustering is
independent of the other variables in the ensemble. We measure prediction
accuracy using a hold out set of the same size as in scenario 1 and also
evaluate the dependence structure from the ITF model.

In each case, we take 100,000 iterations of the MCMC scheme with the first few
1,000 discarded as a burn-in. These are reported in table 1 (left). We also
summarize the recovered dependence structure in table 1 and in table 2. In
scenario 1, the prediction accuracy of ITF and DPM are comparable, with DPM
performing marginally better in a couple of cases. Note that the recovered
dependence structure with the ITF is exactly accurate which shows that the ITF
can reduce to joint co-clustering when that is the truth. In scenario 2,
however there is significant improvement in using the ITF over the DPM with
predictive accuracy. In fact the predictions from the DPM for the categorical
variable are close to noise. The dependence structure recovered the ITF almost
reflects the truth as compared to that from the DPM which predicts every pair
is dependent, by virtue of its construction.

###  5.2 Real Data Examples

For generic real mixed domain data the dependence structure is wholly unknown.
To evaluate how well the ITF does in capturing pairwise dependencies, we first
consider a network example in which recovering dependencies is of principal
interest and prediction is not relevant. We consider data comprising of 105
political blogs (Adamic & Glance, 2005) where the edges in the graph are
composed of the links between websites. Each blog is labeled with its
ideology, and we also have the source(s) which were used to determine this
label. Our model includes the network, ideology label, and binary indicators
for 7 labeling sources (including “manually labeled”, which are thought to be
the most subject to errors in labelings). We assume that ideology impacts
links through cluster assignment only, which is a reasonable assumption here.
We collect 100,000 MCMC iterations after a short burn-in and save the iterate
with the largest complete-data likelihood for exploratory purposes.

Fig. 2 shows the network structure, with nodes colored by ideology. It is
immediately clear that there is significant clustering, apparently driven
largely by ideology, but that ideology alone does not account for all the
structure present in the graph. Joint DPM approach would allow for only one
type of clustering and prevent us from exploring this additional structure.
The recovered clustering in fig. 2 reveals a number of interesting structural
properties of the graph; for example, we see a tight cluster of conservative
blogs which have high in- and out- degrees but do not link to one another
(green) and a partitioning of the liberal blogs into a tightly connected
component (purple) and a periphery component with low degree (blue). The
conservative blogs do not exhibit the same level of assortative mixing
(propensity to link within a cluster) as the liberal blogs do, especially
within the purple component.

To get a sense for how stable the clustering is, we estimate the posterior
probability that nodes $i$ and $j$ are assigned to the same cluster by
recording the number of times this event occurs in the MCMC. We observe that
the clusters are generally quite stable, with two notable exceptions. First,
there is significant posterior probability that points 90 and 92 are assigned
to the red cluster rather than the blue cluster. This is significant because
these two points are the conservative blogs which are connected only to
liberal blogs (see fig. 2). While the graph topology strongly suggests that
these belong to the blue cluster, the labels are able to exert some influence
as well. Note that we do not observe the same phenomenon for points 7, 15, and
25, which are better connected. We also observe some ambiguity between the
purple and blue clusters. These are nodes 6, 14, 22, 33, 35 and 36, which
appear at the intersection of the purple/blue clusters in the graph projection
because they are not quite as connected as the purple “core” but better
connected than most of the blue clusters.

Finally, we examine the posterior probability of being labeled ”conservative”
(fig. 3). Most data points are assigned very high or low probability. The five
labeled points stand out as having uncharacteristic labels for their link
structure (see fig 2). Since the observed label doesn’t agree with the graph
topology, the probability is pulled away from 0/1 toward a more conservative
value. This effect is most pronounced in the three better-connected liberal
blogs (lower left) versus the weakly connected conservative blogs (upper
right).

For the second example, we use data obtained from the Osteoarthritis
Initiative (OAI) database, which is available for public access at
http://www.oai.ucsf.edu/. The question of interest for this data is
investigate relationships between physical activity and knee disease symptoms.
For this example we use a subset of the baseline clinical data, version 0.2.2.
The data ensemble comprises of variables including biomarkers, knee joint
symptoms, medical history, nutrition, physical exam and subject
characteristics. In our subset we take an ensemble of size $120$ for $4750$
individuals. We hold out some of the biomarkers and knee joint symptoms and
consider prediction accuracy of the ITF versus the joint DPM model. For the
real variables, mixtures of normal kernels are considered, for the
categorical, mixtures of multinomials and for the time series, mixtures of
fixed finite wavelet basis expansion.

Results for this experiment are summarized in table 3 for 4 held-out
variables. ITF outperforms the DPM in 3 of these 4 cases and marginally worse
prediction accuracy in case of the other variable. It is also interesting to
note that ITF helps to uncover useful relationships between medical history,
physical activity and knee disease symptoms, which has a potential application
for clinical action and treatments for the subsequent patient visits.

##  6 CONCLUSIONS

We have developed a general model to accommodate complex ensembles of data,
along with a novel algorithm to sample from the posterior distributions
arising from the model. Theoretically, extension to any number of levels of
stick breaking processes should be possible, the utility and computational
feasibility of such extensions is being studied. Also under investigation is
connections with random graph/network models and theoretical rates of
posterior convergence.

Table 1: Simulation Example, Scenario 1: Prediction error (top), tests of
independence (bottom)

| ITF | DPM
---|---|---
T | 1.79 | 1.43
C2 | 31$\%$ | 23 $\%$
C3 | 37$\%$ | 36 $\%$

| ITF | DPM | “Truth”
---|---|---|---
C1 vs T | Yes | Yes | Yes
C2 vs T | Yes | Yes | Yes
C3 vs T | Yes | Yes | Yes
C2 vs R | Yes | Yes | Yes

Table 2: Simulation Example, Scenario 2: Prediction error (top), tests of
independence (bottom)

| ITF | DPM
---|---|---
T | 4.61 | 10.82
C2 | 27$\%$ | 55 $\%$
C3 | 34$\%$ | 57 $\%$

| ITF | DPM | “Truth”
---|---|---|---
C1 vs T | Yes | Yes | Yes
C2 vs T | No | Yes | No
C3 vs T | No | Yes | No
C2 vs R | No | Yes | No

Figure 1: Network Example: True Clustering

Figure 2: Network Example: Recovered Clustering

Figure 3: Network Example: Pairwise cluster assignment probability. Left bars correspond to clustering in Fig. 2, top bars correspond to clustering on the ideology label. Table 3: OAI Data example: Relative Predictive Accuracy. The variables are respectively, left knee baseline pain, isometric strength left knee extension,left knee paired X ray reading, left knee baseline radiographic OA. | ITF | DPM
---|---|---
P01BL12SXL | 31.21 | 100.92
V00LEXWHY1 | 7.94 | 7.56 $\%$
V00XRCHML | 23.01 | 31.84 $\%$
P01LXRKOA | 65.78 | 90.30 $\%$

#### Acknowledgements

This work was support by Award Number R01ES017436 from the National Institute
of Environmental Health Sciences and DARPA MSEE. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institute of Environmental Health Sciences or the
National Institutes of Health or DARPA MSEE.

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#  Gravitational lensing in a topologically charged Eddington-inspired Born-
Infeld spacetime

A. R. Soares ID<EMAIL_ADDRESS>Grupo de Estudos e Pesquisas em
Laboratório de Educação matemática, Instituto Federal de Educação Ciência e
Tecnologia do Maranhão, R. Dep. Gastão Vieira, 1000, CEP 65393-000 Buriticupu,
MA, Brazil.  R. L. L. Vitória  ID<EMAIL_ADDRESS><EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará, Av.
Augusto Corrêa, Guamá, 66075-110, Belém, PA, Brazil.  C. F. S. Pereira  ID
<EMAIL_ADDRESS>Departamento de Física e Química, Universidade
Federal do Espírito Santo, Av.Fernando Ferrari, 514, Goiabeiras, Vitória, ES
29060-900, Brazil

###### Abstract

In the present paper, we study several aspects of gravitational lensing caused
by a topologically charged Monopole/Wormhole, both in the weak field limit and
in the strong field limit. We calculate the light deflection and then use it
to determine the observables, with which one can investigate the existence of
these objects through observational tools. We emphasize that the presence of
the topological charge produces changes in the observables in relation to the
case of General Relativity Ellis-Bronnikov wormhole.

##  I Introduction

It is known that despite its enormous success, the Einstein’s General
Relativity (GR) has some weaknesses that may point to a broader gravitational
theory. In particular, we highlight the singularity problem, i.e., the
termination of a geodesic in a black hole singularity. The theory also does
not naturally deal with the accelerating expansion of the universe and the
cosmological problem of the Big Bang singularity I1 , I2 , I3 , I4 , I5 , I6 .
These, among other questions, motivated physicists to search for a theory
capable of avoiding such problems f1 , f2 , f3 , f4 , f5 . Among these
theories, we highlight the so-called Eddington-inspired Born-Infeld
modification of gravity (EiBI gravity)f5 . The structure of this theory is
inspired by the nonlinear electrodynamics of Born and Infeld E1  and, when
approached by the affine-metric formalism, avoids problems such as phantom
degrees of instability olmo2011 . Even at a classical level (without quantum
corrections), such a theory provides singularity-free black holes solutions
and sustained wormholes without the need for exotic matter o1 , o2 , o3 ,
Soares2020 , Soares2019 . In fact, these new theoretical possibilities,
combined with the technological tools arising from large international
collaborations, have rekindled the search for regular black holes and
wormholes.

One of the forms of investigation in cosmology and gravitation are
gravitational lensing Liebes , Mellier1999 , Schneider2001 , Kaiser1993 ,
Chowdhuri174 , Einstein himself used this method to make GR effectively
relevant in the scientific community Einstein1936 , Crispino2019 .
Gravitational lensing can occur both in the weak field limit, which is when
the light ray passes very far from the source responsible for the lens, or in
the so-called strong field limit, which is when the light passes very close,
in this case the deflection of light it is actually very big Atkinson1965 ,
Darwin . Although the mathematical treatment of lensing in the strong field
regime is more complicated, recent works have made it possible to treat the
subject even analytically. In Boz-Cap2001 , Bozza2002 , the authors obtained
an expression to calculate the deflection of light in the strong field limit
and applied it to the case of the Scharszchild spacetime. In Tsukamoto2017-2 ,
Tsukamoto presents an improved expression of light deflection in static and
asymptotically flat spacetimes as well in ultrastatic spacetimes. In
Virbhadra-Ellis-2000 , Virbhadra and Ellis showed that in the strong field
limit there is, in addition to the first and second images, an infinity of
images called relativistic images that do not discriminate between themselves.
In Perlick2004 , Perlick considered gravitational lensing exactly in a
spherically symmetric and static spacetime, exemplifying with o Barriola-
Vilenkin monopole and Ellis-Bronnikov wormhole. Since then, gravitational
lensing has been investigated in several contexts involving black holes
Eiroa2002 , Eiroa2004 , Tsukamoto2017 , rotacao , Aazami , azquez , Bozza-
Sereno , Bozza-Scarpetta , VBz , Virbhadra-064038 , Virbhadra-2204.01792 ,
wormholes Nakajima85 , Chetouani , Nandi , Dey-sen , Gibbons-Vyska ,
Tsukamoto-Harada , Nandi-Potapov , Tsukamoto2017-95 , Tsukamoto-Harada-2017 ,
Shaikh-Banerjee , Gao2211.17065 , Huang2302.13704 , Abe2010 , topological
defects Cheng , Sharif2015 , Cheng92 , Cheng28 , modified theories of
gravitation Sotani92 , Wei75 , Bhadra , Eiroa73 , Sarkar23 , Mukherjee39 ,
Gyulchev75 , Chen80 , Shaikh96  and regular black holes Eiroa28 , Eiroa88 ,
soaresBbounce , Ghosh006 .

So far, the simplest solution ever obtained in EiBI gravity is the Global
Monopole/Wormhole (GM/WH for short) Soares2020 , lambaga2018 . This solution
interpolates a modified global monopole and a wormhole like Ellis-Bronnikov
with topological charge. We must emphasize that said solution was obtained
with a source of matter that does not violate the energy conditions and,
impressively, it is as simple as the well-known Ellis-Bronnikov solution ellis
, bronn . Given its simplicity and potential applications to understand
systems linked to condensed matter, some studies related to the quantum
dynamics of particles in this space-time have already been published me2 , me3
, CARB , Ahmed141 , Aounallah1013 . With regard to gravitational lensing, in
lambaga2018 , the authors calculated the deflection of light in the EiBI-GM in
the weak field limit and in Soares2020  the authors obtained a general
expression for the light deflection in the topologically charged WH.

In the present paper, we will do a more general study of the topologically
charged EiBI spacetime. Let’s consider both possibilities, WH and GM, and
let’s treat them in the same theoretical framework. We will address the
deflection of light not only in the weak field limit, but also in the strong
field limit for the first time. Furthermore, we calculate the observables and
discuss the possibility of their detection. It should also be noted that the
results obtained here can also shed light on the optical properties of liquid
crystals and crystalline lattices with topological defects Moraes103 .

The paper is organized as follows: In section II we present the metric that
describes the GM/WH in EiBI gravity and calculate the light deflection in the
weak field limit. In section III we calculate the light deflection in the
strong field limit and investigate the lens equation only in the wormhole
spacetime. In section IV, we present some observational perspectives of the
wormhole and its detection plausibility and, finally, in section V, we make
the last discussions and conclusions of the paper, presenting our future
perspectives.

##  II The GM/WH metric and Lensing in the Weak Field Limit

The line element describing the GM/WH in EiBI gravity, in spherical
coordinates ($t,r,\theta$, $\phi$), is given by Soares2020 :

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-(1-\kappa^{2}\eta^{2})dt^{2}+\frac{r^{2}dr^{2}}{(1-\kappa^{2}\eta^{2})(r^{2}+\varepsilon\kappa^{2}\eta^{2})}$
(1) $\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\ ,$

where $\kappa^{2}=8\pi G$, being $G$ the gravitational constant, $\eta$ the
energy scale associated with spontaneous symmetry breaking and $\varepsilon$,
the parameter associated with the non-linearity of the EiBI gravity. For
simplicity, let us make just a few reparametrizations: $t\to
t\sqrt{1-\kappa^{2}\eta^{2}}$, $\alpha=\sqrt{1-\kappa^{2}\eta^{2}}$ and
$\varepsilon\to\varepsilon(1-\alpha^{2})$. With that, the Eq.(1) becomes

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}\big{(}1+\frac{\varepsilon}{r^{2}}\big{)}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\ .$ (2)

The above metric can describe a global monopole, when $\varepsilon>0$, or a
topologically charged wormhole, when $\varepsilon<0$. It should be noted that
solution (2) was originally obtained, in Soares2020 , lambaga2018 , from a
energy source corresponding to the external region of the Barriola-Vilenkin
global monopole core Barriola-Vilenkin . For that reason, in the case
$\varepsilon>0$, (2) also describes a spacetime external to the global
monopole core, but this time within the EiBI gravity. And, when
$\varepsilon=0$, the metric reduces to on the well-known outside core global
monopole. For $\varepsilon<0$, we can compare the Eq.(2) with the more general
case of a wormhole, proposed by Morris and Thorne MorrisThorne , to conclude
that the redshift function is null, $\Phi(r)=0$, and the form function is
$b(r)=r-\alpha^{2}\big{(}r-\frac{|\varepsilon|}{r}\big{)}$. Because of
topological charge, the EiBI wormhole is not asymptotically flat, unlike the
Ellis-Bronnikov wormhole:
$\lim\limits_{r\to\infty}\frac{b(r)}{r}=1-\alpha^{2}$.

Let us now get the geodesics associated with spacetime (2), using the
variational method. For a smooth curve on a space with metric (2), the length,
$S$, of the curve is

$S=\int\sqrt{\bigg{(}g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}\bigg{)}}d\lambda\
,$ (3)

where $\lambda$ is the affine parameter of the curve. Taking $S$ as an affine
parameter in (3), we can show that the curves that minimize (3), $\delta S=0$,
also minimize:

$\int\bigg{(}g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}\bigg{)}d\lambda\
=\int\mathcal{L}d\lambda\ .$ (4)

Therefore, for $\theta=\frac{\pi}{2}$, the lagrangian $\mathcal{L}$ becomes:

$\mathcal{L}=-\bigg{(}\frac{dt}{d\lambda}\bigg{)}^{2}+\frac{1}{\alpha^{2}\big{(}1+\frac{\varepsilon}{r^{2}}\big{)}}\bigg{(}\frac{dr}{d\lambda}\bigg{)}^{2}+r^{2}\bigg{(}\frac{d\phi}{d\lambda}\bigg{)}^{2}\
.$ (5)

The corresponding Euler-Lagrange equation for the coordinates $t$ and $\phi$
leads to the following conserved quantities

$E=\frac{dt}{d\lambda}\hskip 28.45274pt\text{and}\hskip
28.45274ptL=r^{2}\frac{d\phi}{d\lambda}\ .$ (6)

In terms of the quantities $E$ and $L$, the lagrangian, Eq.(5), becomes:

$\mathcal{L}=-E^{2}+\frac{1}{\alpha^{2}\big{(}1+\frac{\varepsilon}{r^{2}}\big{)}}\bigg{(}\frac{dr}{d\lambda}\bigg{)}^{2}+\frac{L^{2}}{r^{2}}\
.$ (7)

It is known that for null geodesics, $\mathcal{L}=0$. In this case, the (7)
leads to

$\bigg{(}\frac{dr}{d\lambda}\bigg{)}^{2}=\alpha^{2}\bigg{(}1+\frac{\varepsilon}{r^{2}}\bigg{)}\bigg{(}E^{2}-\frac{L^{2}}{r^{2}}\bigg{)}\
.$ (8)

This equation can be interpreted as describing the one-dimensional motion of a
particle of energy $E$ subject to an effective potential $V_{ef}=L^{2}/r^{2}$.
In Fig. 1, we plot the effective potential for some values of $L$. Of course,
there is only radial motion when $\frac{dr}{d\lambda}>0$, and the smaller the
angular momentum $L$ greater the approximation. Therefore, the turning point
$r_{0}$ occurs for $\frac{dr}{d\lambda}=0$, i.e., $r_{0}=L^{2}/E^{2}$. It is
worth mentioning that in the case of the wormhole, $\varepsilon<0$, the
solution has a minimum radius given by $r=\sqrt{|\varepsilon|}$, thus, photons
with sufficient energy pass to the other side of the wormhole.

Figure 1: Effective potential for some values of angular momentum $L$.

From (6), it can be shown that Eq.(8) becomes

$\frac{d\phi}{dr}=\frac{\beta}{\alpha\sqrt{(r^{2}+\varepsilon)(r^{2}-\beta^{2})}}\
,$ (9)

where $\beta=\frac{L}{E}$. We want to find the change in coordinate $\phi$,
i.e., $\Delta\phi=\phi_{-}-\phi_{+}$. By symmetry, the contributions to
$\Delta\phi$ before and after the turning point are equal, so Eq.(9) leads to

$\Delta\phi=2\int_{\beta}^{\infty}\frac{\beta\
dr}{\alpha\sqrt{(r^{2}+\varepsilon)(r^{2}-\beta^{2})}}\ .$ (10)

Let us define the variables

$t=\frac{\beta}{r}\hskip 28.45274pt\text{and}\hskip
28.45274ptm=-\frac{\varepsilon}{\beta^{2}}\ ,$ (11)

in terms of which, the (10) becomes

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{2}{\alpha}\int_{0}^{1}\frac{dt}{\sqrt{(1-t^{2})(1-mt^{2})}}$
$\displaystyle\Delta\phi$ $\displaystyle=$ $\displaystyle\frac{2K(m)}{\alpha}\
.$ (12)

The above expression is valid both for the WH case ($\epsilon<0$) and the GM
case ($\epsilon>0$). Remembering that $\beta$ is the turning point, we will
have $0<m<1$ for the WH; in this case $K(m)$ is an incomplete elliptic
integral of the first type, given in terms of the parameter itself rather than
the modulus. In the GM case, we can write
$m=-\frac{\varepsilon}{\beta^{2}}=-a$, with
$a=\frac{\varepsilon}{\beta^{2}}>0$. We can show, Appendix A, that

$K(m)=K(-a)=\frac{1}{\sqrt{a+1}}K\bigg{(}\frac{a}{a+1}\bigg{)}\ .$ (13)

The light deflection, angle between the new direction of propagation and the
previous direction, as shown in the Fig. 2, it is given by
$\delta\phi=\Delta\phi-\pi$. Therefore,

$\delta\phi=\frac{2K(m)}{\alpha}-\pi\ .$ (14)

Figure 2: Scheme of the deflection of light, $\delta\phi$, due to
gravitational attraction.

The expression (14) is valid for both $m>0$ and $m<0$. But for didactic
reasons we will discriminate, below, the deflection for a WH and a GM.

$\delta\phi=\begin{cases}\frac{2K(-\varepsilon/\beta^{2})}{\alpha}-\pi,\quad\mbox{($\varepsilon<0$,
WH)},\\\
\frac{2}{\alpha\sqrt{(\varepsilon/\beta^{2})+1}}K\bigg{(}\frac{(\varepsilon/\beta^{2})}{(\varepsilon/\beta^{2})+1}\bigg{)}-\pi,\quad\mbox{($\varepsilon>0$,
GM)}\ .\end{cases}$

Of course, if $\varepsilon=0$ and $\alpha=1$, we will have $\delta\phi=0$,
which corresponds to the case of flat spacetime, where there is no angular
deflection since the gravitational attraction is null. In fig.3, we plot the
deflection for $\alpha=0.8$ in order to clarify some aspects related to
lensing in the GM/WH spacetime. Note that in the WH case ($m>0$), when the
turning point tends to the throat radius the deflection diverges, we call this
limit strong field. On the other hand, in the case of the modified GM ($m<0$),
the deflection is always finite.

Figure 3: Light deflection, $\delta\phi$, as a function of $m$, for
$\alpha=0.8.$

First, let us investigate the deflection in the weak field limit, i.e., for
$\beta\gg\varepsilon$, where $m\to 0$. In this limit, Eq.(14) can be written
as

$\delta\phi=\frac{\pi(1-\alpha)}{\alpha}-\frac{\pi}{4\alpha}\Big{(}\frac{\varepsilon}{\beta^{2}}\Big{)}+\mathcal{O}\big{(}\frac{\varepsilon}{\beta^{2}}\big{)}^{2}\
.$ (15)

The first term in the expression above corresponds to the deflection of a pure
GM Barriola-Vilenkin , and the second term is a contribution from EiBI
gravity. We emphasize that in the WH case ($\varepsilon<0$), the deflection
clearly becomes more pronounced than in the Barriola and Vilenkin GM case, the
opposite occurs for EiBI GM. For the EiBI GM case, the deflection was first
obtained by lambaga2018 .

##  III Wormhole Light deflection in the strong field limit

As already discussed, Eq.(15) is valid for both the WH and the GM. However, as
we can see in Fig. 3, the light deflection never diverges in the GM case
($\varepsilon>0$). On the other hand, the deflection diverges for the WH case
($\varepsilon<0$) when $m\to 1$. This limit, which corresponds to the
approximation of the turning point towards the wormhole throat, is called the
strong field. It is on this limit that we will focus from now on. For
simplicity, let’s take $\varepsilon=-a^{2}$; therefore, the parameter $a$
corresponds to the wormhole throat. Thus, from (14), we are left with

$\delta\phi=\frac{2}{\alpha}K\bigg{(}\frac{a^{2}}{\beta^{2}}\bigg{)}-\pi\ .$
(16)

In the strong field limit, $\beta\to a$, consequently,
$\frac{a^{2}}{\beta^{2}}\to 1$. According to Soares2020 , in this limit, the
deflection of light diverges logarithmically, that is 111We could also get
$\delta\phi$ in the strong field limit, taking into account that
$\lim\limits_{m\to
1}K(m)=-\frac{1}{2}\log(1-m)+\frac{3}{2}\log(2)+\mathcal{O}{(1-m)\log(1-m)}$
booktra .,

$\displaystyle\delta\phi$ $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\log\bigg{(}\frac{\beta}{a}-1\bigg{)}+\frac{3}{\alpha}\log(2)-\pi$
(17) $\displaystyle+\mathcal{O}{(1-a/\beta)\log(1-a/\beta)}\ .$

With this result, we can study gravitational lensing in the strong field
limit. To that end, let us review the lens equations at this limit in order to
clarify our findings. In Fig.4 we present the visual profile of lensing. The
light emitted by a source (S) is deflected by the lens (L), the wormhole,
towards the observer (O). With respect to the optical axis (LO), $\psi$ and
$\theta$ give the angular position of the source and the image (I),
respectively. Evaluating the figure, it can be shown that Virbhadra-Ellis-2000
:

$\tan\psi=\tan\theta-\frac{D_{LS}}{D_{OS}}[\tan\theta-\tan(\Lambda-\theta)]\
,$ (18)

where $D_{OS}=D_{OL}+D_{LS}$.

Figure 4: Lens diagram.

Let’s assume that the source and lens are almost perfectly aligned. In this
sense, although the angular positions of the source and the image are small,
the light ray circles the source several times before heading towards the
observer, thus, $\Lambda$ must be very close to a multiple of $2\pi$ Boz-
Cap2001 . Therefore, we can write $\Lambda=2\pi n+\Delta\Lambda_{n}$, where
$\Delta\Lambda_{n}$ is the deflection angle after the $n$ loops around the
lens, so $\tan(\Lambda-\theta)\sim\Delta\Lambda_{n}-\theta$. With these
observations, the lens equation becomes

$\psi=\theta-\frac{D_{LS}}{D_{OS}}\Delta\Lambda_{n}\ .$ (19)

Still, in agreement with what we have stated, we can then write the critical
impact parameter as

$\beta=D_{OL}\theta\ .$ (20)

With this, we can rewrite the angular deflection (17) as

$\Lambda(\theta)=-\frac{1}{\alpha}\log\bigg{(}\frac{D_{OL}\theta}{a}-1\bigg{)}+\frac{3}{\alpha}\log(2)-\pi\
.$ (21)

We must remember, however, that whoever enters the lens equation (19) is
$\Delta\Lambda_{n}$. To get $\Delta\Lambda_{n}$, we expand $\Lambda(\theta)$
em $\theta=\theta^{0}_{n}$, where $\Lambda(\theta^{0}_{n})=2\pi n$, i.e.,

$\displaystyle\Lambda(\theta)$ $\displaystyle=$
$\displaystyle\Lambda(\theta^{0}_{n})+\frac{\partial\Lambda}{\partial\theta}\Bigg{|}_{\theta=\theta^{0}_{n}}(\theta-\theta^{0}_{n})$
$\displaystyle\Delta\Lambda_{n}$ $\displaystyle=$
$\displaystyle\frac{\partial\Lambda}{\partial\theta}\Bigg{|}_{\theta=\theta^{0}_{n}}(\theta-\theta^{0}_{n})\
.$ (22)

Taking $\Lambda(\theta^{0}_{n})=2\pi n$ in (21), we find

$\theta_{n}^{0}=\frac{a}{D_{OL}}\left[1+\frac{8}{e^{(1+2n)\alpha\pi}}\right]\
.$ (23)

Substituting (21) and (1) in (III), we are left with

$\Delta\Lambda_{n}=-\frac{e^{(1+2n)\alpha\pi}D_{OL}}{8\alpha
a}\Delta\theta_{n}\ .$ (24)

Substituting (24) in the lens equation (19), we are left with

$\theta_{n}\simeq\theta^{0}_{n}+\frac{8\alpha
aD_{OS}(\psi-\theta^{0}_{n})}{D_{LS}D_{OL}e^{(1+2n)\alpha\pi}}\ .$ (25)

The Eq.(25) gives the angular position of the $n$th relativistic image. See
that they are influenced by the topological charge and when $\alpha\to 1$,
they fall back on the case of Ellis-Bronnikov Tsukamoto2016 . In general,
$\alpha<1$, which implies that in the case of the topologically charged, the
respective angular positions are greater than in the case of the Ellis-
Bronnikov WH, which has no topological charge Tsukamoto2016 .

The total flux from the $n$th lensed image is proportional to the
magnification $\mu_{n}$, which is given by
$\mu_{n}=\Big{[}\frac{\psi}{\theta}\frac{\partial\psi}{\partial\theta}\bigg{|}_{\theta_{n}^{0}}\Big{]}^{-1}$
Boz-Cap2001 . Therefore, from (25) and (1),we can show that

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\frac{8\alpha
D_{OS}}{\psi
D_{LS}}\Big{(}\frac{a}{D_{OL}}\Big{)}^{2}\frac{1}{e^{(1+2n)\alpha\pi}}$ (26)
$\displaystyle\times\left[1+\frac{8}{e^{(1+2n)\alpha\pi}}\right].$

As we can see, the magnification decreases rapidly with $n$, indicating that
the brightness of the first image, $\theta_{1}$, is higher than the others. On
the other hand, since in general $D_{OL}\gg a$, the magnification is always
small. We can also observe that the greater the alignment between the source
and the lens ($\psi\ll 1$), the stronger the magnification.

The equations (25) and (26) express the positions of the relativistic images
and the magnifications in terms of the parameters that characterize the
wormhole ($a$ and $\alpha$). We can think in reverse, that is, we can define
observables that can be measured via observational methods. Since these
observables can be resolved to obtain the parameters that characterize the
wormhole, they can present themselves as a good way to search for wormholes.
In addition, the such methodology can also be useful as a tool for research
beyond GR, since from the observables we can discriminate between a GR WH and
an EiBI WH. In Bozza2002 , Bozza defined the following observables:

$s=\theta_{1}-\theta_{\infty}\quad\text{and}\quad
R=\frac{\mu_{1}}{\sum_{n=2}^{\infty}\mu_{n}}\ .$ (27)

Where $s$ is the angular separation between the first and the rest of the
relativistic images and $R$ is the relationship between the flux of the first
image and the flux of all the others. From (25), we can show that $s$ which is
given by

$s\simeq\frac{8\theta_{\infty}}{e^{3\alpha\pi}}\ ,$ (28)

and from (26), we can also show that

$R=\frac{\left(e^{-3\pi\alpha}+8e^{-6\pi\alpha}\right)}{e^{-5\pi\alpha}+8e^{-10\pi\alpha}}\
.$ (29)

For $\alpha\simeq 1$, as we believe it should be, the expression (29) becomes

$R\simeq e^{2\pi\alpha}\ .$ (30)

Compared to the Ellis-Bronnikov WH, the presence of the topological charge
$\alpha$ decreases the value of the observable $R$ and increases the
observable $s$. Please, compare equations (30) and (28) with equations (5.15)
and (5.14) of Tsukamoto2016 , respectively.

##  IV Wormhole Lensing observational Aspects

In order to have an observational perspective of the EiBI WH, we will model
three astrophysical scenarios, one for the strong field limit and two for the
weak field limit; the choices were made in order to make the data clearer and
avoid too many approximations. First, let us assign well-motivated values to
the parameters $a$ and $\alpha$. Recent works jana , involving the velocity of
propagation of gravitational waves, estimated an upper limit for the Eddington
parameter ($\varepsilon$) that is, $|\varepsilon|\leq 10^{37}m^{2}$.
Remembering that the radius of the wormhole throat treated here is
$a=\sqrt{|\varepsilon|}$, it is estimated that $a\leq 10^{15}km$. Furthermore,
based on grand unification theories Barriola-Vilenkin , Soares2019 , we can
estimate $\alpha\sim\sqrt{1-10^{-5}}$.222According to section II,
$\alpha=\sqrt{1-\kappa^{2}\eta^{2}}$. On the other hand, as argued in
Barriola-Vilenkin , for a typical grand unification scale it is estimated that
$\eta\sim 10^{16}$Gev, which demonstrates the reasonableness of our estimate.

###  IV.1 Strong field limit

At the strong field limit, we modeled the wormhole with data from Sagittarius
A*, at the center of our galaxy. The mass of SgrA* is estimated at $4.4\times
10^{6}M_{\odot}$ and is at an approximate distance of $D_{OL}=8.5$Kpc
Gezel2010 . As we have discussed before, it is estimated that the radius of
the EiBI WH throat has an upper bound, $a\leq 10^{15}$ km. In table 1 we
present the values of the observables, $\theta_{\infty}$, $s$ and $\tilde{R}$,
for some values of the throat radius $a$. The observable $\theta_{\infty}$ is
defined as the critical angle, of (20),
$\theta_{\infty}=\frac{\beta}{D_{OL}}$. The observable $s$ is given by (28)
and $\tilde{R}=2.5\log_{10}R$, where $R$ is given by (30), the latter
redefinition is useful to compare our results with those found in the
literature.

Table 1:  Observable $a(km)$ |  $\theta_{\infty}$($\mu$ arcsecs) | s($\mu$ arcsecs)
---|---|---
$10^{9}$ | $0.78634$ | $0.000507$
$10^{10}$ | $7.8634$ | $0.00507$
$10^{11}$ | $78.634$ | $0.0507$
$10^{12}$ | $786.34$ | $0.507$

In the case of the well-known Schwarzschild black hole, the values of these
observables for the scenario considered here are already known and are given
by Virbhadra-Ellis-2000 : $\theta_{\infty}$=26.547$\mu$arcsecs,
$s=0.03322$$\mu$arcsecs e $\tilde{R}=6.821$ magnitudes. According to table 1,
for $10^{10}km\leq a\leq 10^{11}km$, the order of magnitude of the critical
angle and angular separation have the same order of magnitude as in the
Schwarzschild case, but differ significantly for other values. In fact, the
data indicate that for $a\geq 10^{11}$ the values of the observables assume
values more compatible with the current observational range than in the
Schwarzschild case. As for the magnification (30), $\tilde{R}$, we observe
that it does not depend on the throat radius, and is given by
$\tilde{R}=6.821$ magnitudes; we note that it is similar to the Schwarzschild
case. However, we must emphasize that using the critical angle and angular
separation we can distinguish between the space of a Schwarzschild black hole
and an EiBI WH using the strong field limit lensing.

###  IV.2 Weak field limit

In the weak field limit, $a\ll 1$ and $a/\beta\ll 1$, there are no loops and
the deflection is given by (15), which for feasible topological charge values,
becomes

$\Lambda\sim\frac{\pi(1-\alpha)}{\alpha}+\frac{\pi}{4\alpha}\Big{(}\frac{a^{2}}{\beta^{2}}\Big{)}\
.$ (31)

It is worth remembering that, as we are dealing with the wormhole case, we are
taking $\epsilon=-a^{2}$ in (15). In perfect alignment, $\psi=0$. The angular
position, $\theta_{E}$, can be obtained from (19), (20), and (31) which leads
to the equation

$\theta_{E}^{3}+\frac{D_{LS}\pi(1-\alpha)}{D_{OS}\alpha}\theta^{2}-\frac{D_{LS}\pi
a^{2}}{D_{OS}4\alpha D_{OL}^{2}}=0\ ,$ (32)

whose real solution for values of $\alpha$ close to unity is given by

$\displaystyle\theta_{E}$ $\displaystyle=$ $\displaystyle\Bigg{[}\frac{\pi
D_{LS}a^{2}}{4\alpha D_{OS}D_{OL}^{2}}\Bigg{]}^{1/3}$ (33) $\displaystyle+$
$\displaystyle\frac{(1-\alpha)}{6D_{OS}}\Bigg{[}\bigg{(}\frac{a^{2}D_{LS}D_{OS}^{2}2\pi}{D_{OL}^{2}}\bigg{)}^{1/3}-2\pi
D_{LS}\Bigg{]}.$

From (20), we can calculate the Einstein radius, $R_{E}$,

$R_{E}=D_{OL}\theta_{E}\ .$ (34)

Let us now estimate the observables $R_{E}$ and $\theta_{E}$ taking into
account reasonable values for the model parameters. Following the example of
Abe2010 , let’s consider the lensing of a bulge star and a star in the Large
Magellanic Cloud (LMC). For a bulge star, the following parameters are
adopted: $D_{OS}=8kpc$ and $D_{OL}=4kpc$; while for the Large Magellanic
Cloud: $D_{OS}=50kpc$ and $D_{OL}=25kpc$. Based on these estimates, in the
tables 2 and 3 we present some values of the observables for the two presented
scenarios. We verified that, within the possibilities of EiBI gravity, the
present wormhole presents feasible theoretical possibilities of being detected
for values of the throat radius of the order of $10^{9}km$ or more, which is
well accommodated within the restrictions for the parameters of the mentioned
gravity theory. The angular deflection and the angle of the Einstein ring is
of the order of 10 arcsec, for realistic parameter values, and are certainly
within the observable range.

Table 2: Einstein Radii/angle for Bulge Lensings $a(km)$ | $R_{E}(km)$ | $\theta_{E}(mas)$
---|---|---
$10^{9}$ | $4.14\times 10^{10}$ | 69.31
$10^{10}$ | $1.36\times 10^{12}$ | $2.28\times 10^{3}$
$10^{11}$ | $7.53\times 10^{12}$ | $1.25\times 10^{4}$
$10^{12}$ | $3.61\times 10^{13}$ | $6.04\times 10^{4}$
Table 3: Einstein Radii/angle for LMC Lensings $a(km)$ | $R_{E}(km)$ | $\theta_{E}(mas)$
---|---|---
$6\times 10^{9}$ | $1.98\times 10^{11}$ | 53.95
$10^{10}$ | $1.09\times 10^{12}$ | $2.93\times 10^{2}$
$10^{11}$ | $1.24\times 10^{13}$ | $3.32\times 10^{3}$
$10^{12}$ | $6.51\times 10^{13}$ | $17.41\times 10^{3}$

##  V Conclusions and Perspectives

In this work, we explore the gravitational lensing of a GM/WH spacetime within
EiBI gravity. Initially, we calculate the light deflection in a general and
exact way, both for the case of a GM and for the case of a WH, and we
emphasize that the presence of a topological charge allows observationally to
discriminate whether such a solution is described by the RG or by the EiBI
gravity. We emphasize that in the latter case, the solution is physically more
attractive as it does not presuppose any source of exotic matter Soares2020 .
For the time being, we conjecture that a topologically charged wormhole may
arise within EiBI gravity as a result of the evaporation of a topologically
charged black hole, we hope to publish our theoretical research in this
direction soon. In the weak field limit, we show that the light deflection
increases for a wormhole and decreases for a GM compared to the Barriola and
Vilenkin GM. We argue that in the case of GM the light deflection never
diverges, on the other hand, in the case of WH, the light deflection
logarithmically diverges in the strong field limit, that is, when the impact
parameter tends to the radius of the WH throat. We studied the lensing of the
EiBI WH in the strong field limit and showed that this methodology allowed us
to distinguish between an EiBI WH and an Ellis-Bronnikov WH. We also performed
an analysis of the observables taking into account plausible and well-
motivated values for the parameters. The data suggest that at the weak field
limit the observables are within the observational range. Finally, we also
emphasize that the solution studied in this paper can collaborate in the
understanding of optical properties of liquid crystals, since the spatial
section of (2) adequately describes these condensed matter systems Moraes103 ,
Fumeron .

##  Appendix A  $K(m)$ Function for $m$ negative values.

From (II), we know that

$K(m)=\int_{0}^{1}\frac{dt}{\sqrt{(1-t^{2})(1-mt^{2})}}\ .$ (35)

Making the following substitution: $t=\sqrt{1-u^{2}}$, we are left with

$K(m)=\int_{0}^{1}\frac{du}{\sqrt{(1-u^{2})(1-m+mu^{2})}}\ .$ (36)

Therefore,

$\displaystyle K(-m)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{du}{\sqrt{(1-u^{2})(1+m-mu^{2})}}$
$\displaystyle K(-m)$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{m+1}}$
(37)
$\displaystyle\times\int_{0}^{1}\frac{du}{\sqrt{(1-u^{2})(1-\frac{m}{m+1}u^{2})}}\
.$

Then,

$K(-m)=\frac{1}{\sqrt{m+1}}K\bigg{(}\frac{m}{m+1}\bigg{)}\ .\ \square$ (38)

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