Extended Variational Message Passing for Automated Approximate Bayesian Inference
Abstract
:1. Introduction
2. Problem Statement
Variational Message Passing on ForneyStyle Factor Graphs
 Choose a variable ${z}_{k}$ from the set ${z}_{1:K}$.
 Compute the incoming messages.$$\begin{array}{cc}\hfill {\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right)& \propto exp\left({\u2329log{f}_{a}\left({z}_{1:k}\right)\u232a}_{q\left({z}_{1:k1}\right)}\right)\hfill \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{cc}\hfill {\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)& \propto exp\left({\u2329log{f}_{b}\left({z}_{k:K}\right)\u232a}_{q\left({z}_{k+1:K}\right)}\right)\hfill \end{array}$$
 Update the posterior.$$\begin{array}{c}\hfill q\left({z}_{k}\right)=\frac{{\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right){\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)}{\int {\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right){\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)\mathrm{d}{z}_{k}}.\end{array}$$
 Update the local free energy (for performance tracking), i.e., update all terms in $\mathcal{F}$ that are affected by the update (4):$$\begin{array}{c}\hfill {\mathcal{F}}_{k}={\u2329log\frac{q\left({z}_{k}\right)}{{f}_{a}\left({z}_{1:k}\right){f}_{b}\left({z}_{k:K}\right)}\u232a}_{q\left({z}_{1:K}\right)}.\end{array}$$
3. Specification of EVMP Algorithm
3.1. Distribution Types
 (1)
 The standard Exponential Family (EF) of distributions, i.e., the following:$$p\left(z\right)=h\left(z\right)exp\left(\varphi {\left(z\right)}^{\u22ba}\eta {A}_{\eta}\left(\eta \right)\right),$$
 (2)
 Distributions that are of the following exponential form:$$p\left(z\right)\propto exp\left(\varphi {\left(g\left(z\right)\right)}^{\u22ba}\eta \right),$$
 (3)
 A List of Weighted Samples (LWS), i.e., the following:$$p\left(z\right):=\left\{\left({w}^{\left(1\right)},{z}^{\left(1\right)}\right),\dots ,\left({w}^{\left(N\right)},{z}^{\left(N\right)}\right)\right\}\phantom{\rule{0.166667em}{0ex}}.$$
 (4)
 Deterministic relations are represented by delta distributions, i.e., the following:$$p\left(x\rightz)=\delta (xg\left(z\right))\phantom{\rule{0.166667em}{0ex}}.$$Technically, the equality factor $f(x,y,z)=\delta (zx)\delta (zy)$ also specifies a deterministic relation between variables.
3.2. Factor Types
3.3. Message Types
3.4. Posterior Types
3.5. Computation of Posteriors
 (1)
 In the case that the colliding forward and backward messages both carry EF distributions with the same sufficient statistics $\varphi \left(z\right)$, then computing the posterior simplifies to a summation of natural parameters:$$\begin{array}{cc}\hfill {\overrightarrow{m}}_{z}\left(z\right)& \propto exp\left(\varphi {\left(z\right)}^{\u22ba}{\eta}_{1}\right)\hfill \\ \hfill {\overleftarrow{m}}_{z}\left(z\right)& \propto exp\left(\varphi {\left(z\right)}^{\u22ba}{\eta}_{2}\right)\hfill \\ \hfill q\left(z\right)& \propto {\overrightarrow{m}}_{z}\left(z\right)\xb7{\overleftarrow{m}}_{z}\left(z\right)\propto exp\left(\varphi {\left(z\right)}^{\u22ba}({\eta}_{1}+{\eta}_{2})\right).\hfill \end{array}$$In this case, the posterior $q\left(z\right)$ will also be represented by the EF distribution type. This case corresponds to classical VMP with conjugate factor pairs.
 (2)
 The forward message again carries a standard EF distribution. The backward message carries either an NEF distribution or a nonconjugate EF distribution.
 (a)
 If the forward message is Gaussian, i.e., ${\overrightarrow{m}}_{z}\left(z\right)=\mathcal{N}(z;{\mu}_{1},{V}_{1})$, we use a Laplace approximation to compute the posterior:$$\begin{array}{cc}\hfill \mu & =arg\underset{z}{max}\left(log{\overrightarrow{m}}_{z}\left(z\right)+log{\overleftarrow{m}}_{z}\left(z\right)\right),\hfill \\ \hfill V& ={(\nabla {\nabla}_{z}(log{\overrightarrow{m}}_{z}\left(z\right)+log{\overleftarrow{m}}_{z}\left(z\right)){}_{z=\mu})}^{1}\hfill \\ \hfill q\left(z\right)& \propto {\overrightarrow{m}}_{z}\left(z\right)\xb7{\overleftarrow{m}}_{z}\left(z\right)=\mathcal{N}(z;\mu ,V)\hfill \end{array}$$
 (b)
 Otherwise (${\overrightarrow{m}}_{z}\left(z\right)$ is not a Gaussian), we use Importance Sampling (IS) to compute the posterior:$$\begin{array}{cc}\hfill {z}^{\left(1\right)},\dots ,{z}^{\left(N\right)}& \sim {\overrightarrow{m}}_{z}\left(z\right),\hfill \\ \hfill {\tilde{w}}^{\left(i\right)}& ={\overleftarrow{m}}_{z}\left({z}^{\left(i\right)}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,N\hfill \\ \hfill {w}^{\left(i\right)}& ={\tilde{w}}^{\left(i\right)}/\sum _{j=1}^{N}{\tilde{w}}^{\left(j\right)}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,N\hfill \\ \hfill q\left(z\right)& \propto {\overrightarrow{m}}_{z}\left(z\right)\xb7{\overleftarrow{m}}_{z}\left(z\right)=\left\{\left({w}^{\left(1\right)},{z}^{\left(1\right)}\right),\dots ,\left({w}^{\left(N\right)},{z}^{\left(N\right)}\right)\right\}.\hfill \end{array}$$
 (3)
 The forward message carries an LWS distribution, i.e., the following:$${\overrightarrow{m}}_{z}\left(z\right):=\left\{\left({w}_{1}^{\left(1\right)},{z}_{1}^{\left(1\right)}\right),\dots ,\left({w}_{1}^{\left(N\right)},{z}_{1}^{\left(N\right)}\right)\right\}\phantom{\rule{0.166667em}{0ex}},$$$$\begin{array}{cc}\hfill {\tilde{w}}^{\left(i\right)}& ={w}_{1}^{\left(i\right)}{\overleftarrow{m}}_{z}\left({z}_{1}^{\left(i\right)}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,N\hfill \\ \hfill {w}^{\left(i\right)}& ={\tilde{w}}^{\left(i\right)}/\sum _{j=1}^{N}{\tilde{w}}^{\left(j\right)}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,N\hfill \\ \hfill {z}^{\left(1\right)},\dots ,{z}^{\left(N\right)}& ={z}_{1}^{\left(1\right)},\dots ,{z}_{1}^{\left(N\right)}\hfill \\ \hfill q\left(z\right)& \propto {\overrightarrow{m}}_{z}\left(z\right)\xb7{\overleftarrow{m}}_{z}\left(z\right)=\left\{\left({w}^{\left(1\right)},{z}^{\left(1\right)}\right),\dots ,\left({w}^{\left(N\right)},{z}^{\left(N\right)}\right)\right\}.\hfill \end{array}$$
3.6. Computation of Messages
 (1)
 If factor ${f}_{a}({z}_{1},{z}_{2},\dots ,{z}_{k})$ is a soft factor of the form (see Figure 3a)$$\begin{array}{c}\hfill {f}_{a}\left({z}_{1:k}\right)=p\left({z}_{k}\right{z}_{1:k1})={h}_{a}\left({z}_{k}\right)exp\left({\varphi}_{a}{\left({z}_{k}\right)}^{\u22ba}{\eta}_{a}\left({z}_{1:k1}\right){A}_{a}\left({z}_{1:k1}\right)\right).\end{array}$$$$\begin{array}{cc}\hfill {\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right)& \propto {h}_{a}\left({z}_{k}\right)exp\left({\varphi}_{a}{\left({z}_{k}\right)}^{\u22ba}{\u2329{\eta}_{a}\left({z}_{1:k1}\right)\u232a}_{q\left({z}_{1:k1}\right)}\right).\hfill \end{array}$$If rather ${z}_{1}$ (or ${z}_{2},\dots ,{z}_{k1}$) than ${z}_{k}$ is the output variable of ${f}_{a}$, i.e., if the following is true:$$\begin{array}{c}\hfill {f}_{a}\left({z}_{1:k}\right)=p\left({z}_{1}\right{z}_{2:k})={h}_{a}\left({z}_{1}\right)exp\left({\varphi}_{a}{\left({z}_{1}\right)}^{\u22ba}{\eta}_{a}\left({z}_{2:k}\right){A}_{a}\left({z}_{2:k}\right)\right).\end{array}$$$$\begin{array}{cc}\hfill {\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)& \propto exp\left({\u2329{\varphi}_{a}\left({z}_{1}\right)\u232a}_{q\left({z}_{1}\right)}^{\u22ba}{\u2329{\eta}_{a}\left({z}_{2:k}\right)\u232a}_{q\left({z}_{2:k1}\right)}{\u2329{A}_{a}\left({z}_{2:k}\right)\u232a}_{q\left({z}_{2:k1}\right)}\right).\hfill \end{array}$$In this last expression, we chose to assign a backward arrow to ${\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)$ since it is customary to align the message direction with the direction of the factor, which in this case points to ${z}_{1}$.Note that the message calculation rule for ${\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right)$ requires the computation of expectation ${\u2329{\eta}_{a}\left({z}_{1:k1}\right)\u232a}_{q\left({z}_{1:k1}\right)}$, and for ${\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)$ we need to compute expectations ${\u2329{\varphi}_{a}\left({z}_{1}\right)\u232a}_{q\left({z}_{1}\right)}$ and ${\u2329{\eta}_{a}\left({z}_{2:k}\right)\u232a}_{q\left({z}_{2:k1}\right)}$. In the update rules to be shown below, we will see these expectations of statistics of z appear over and again. In Section 3.8 we detail how we calculate these expectations and in Appendix A, we further discuss the origins of these expectations.
 (2)
 In the case that ${f}_{\delta}$ is a deterministic factor (see Figure 3b):$$\begin{array}{c}\hfill {f}_{\delta}(x,{z}_{1:k})=p\left(x\right{z}_{1:k})=\delta (xg\left({z}_{1:k}\right)).\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\overrightarrow{m}}_{x}\left(x\right)=\left\{\left(\frac{1}{N},g\left({z}_{1:k}^{\left(1\right)}\right)\right),\dots ,\left(\frac{1}{N},g\left({z}_{1:k}^{\left(N\right)}\right)\right)\right\},\hfill \\ & \mathrm{where}\phantom{\rule{4.pt}{0ex}}{z}_{j}^{\left(i\right)}\sim {\overrightarrow{m}}_{{z}_{j}}\left({z}_{j}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}j=1:k.\hfill \end{array}$$For the computation of the backward message toward ${z}_{k}$, we distinguish two cases:
 (a)
 If all forward incoming messages from the variables ${z}_{1:k}$ are Gaussian, we first use a Laplace approximation to obtain a Gaussian joint posterior $q\left({z}_{1:k}\right)=\mathcal{N}({z}_{1:k};{\mu}_{1:k},{V}_{1:k})$; see Appendix B.1.2 and Appendix B.2.2 for details. Then, we evaluate the posteriors for individual random variables, e.g., $q\left({z}_{k}\right)=\int q\left({z}_{1:k}\right)\mathrm{d}{z}_{1:k1}=\mathcal{N}({z}_{k};{\mu}_{k},{V}_{k})$. Finally, we send the following Gaussian backward message:$$\begin{array}{c}\hfill {\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)\propto q\left({z}_{k}\right)/{\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right).\end{array}$$
 (b)
 Otherwise (the incoming messages from the variables ${z}_{1:k}$ are not all Gaussian), we use Monte Carlo and send a message to ${z}_{k}$ as a NEF distribution:$$\begin{array}{c}\hfill {\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)\approx \frac{1}{N}\sum _{i=1}^{N}{\overleftarrow{m}}_{x}\left(g({z}_{1:k1}^{\left(i\right)},{z}_{k})\right),\phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}{z}_{j}^{\left(i\right)}\sim {\overrightarrow{m}}_{{z}_{j}}\left({z}_{j}\right).\end{array}$$Note that if ${f}_{\delta}$ is a single input deterministic node, i.e., ${f}_{\delta}(x,{z}_{k})=p\left(x\right{z}_{k})=\delta (xg\left({z}_{k}\right))$, then the backward message simplifies to ${\overleftarrow{m}}_{{z}_{k}}\left({z}_{k}\right)={\overleftarrow{m}}_{x}\left(g\left({z}_{k}\right)\right)$ (Appendix B.1.1).
 (3)
 The third factor type that leads to a special message computation rule is the equality node; see Figure 3c. The outgoing message from an equality node$${f}_{=}(z,{z}^{\prime},{z}^{\u2033})=\delta (z{z}^{\prime})\delta (z{z}^{\u2033})$$$$\begin{array}{cc}\hfill {\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right)& =\int \underset{\mathrm{node}\phantom{\rule{4.pt}{0ex}}\mathrm{function}}{\underset{\u23df}{\delta ({z}_{k}{z}_{k}^{\prime})\delta ({z}_{k}{z}_{k}^{\u2033})}}\underset{\mathrm{incoming}\phantom{\rule{4.pt}{0ex}}\mathrm{messages}}{\underset{\u23df}{{\overrightarrow{m}}_{{z}_{k}^{\prime}}\left({z}_{k}^{\prime}\right){\overrightarrow{m}}_{{z}_{k}^{\u2033}}\left({z}_{k}^{\u2033}\right)}}\mathrm{d}{z}_{k}^{\prime}\mathrm{d}{z}_{k}^{\u2033}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\overrightarrow{m}}_{{z}_{k}^{\prime}}\left({z}_{k}\right){\overrightarrow{m}}_{{z}_{k}^{\u2033}}\left({z}_{k}\right).\hfill \end{array}$$
3.7. Computation of Free Energy
 If $q\left(z\right)$ is a represented by a standard EF distribution, i.e.,$$q\left(z\right)=h\left(z\right)exp\left(\varphi {\left(z\right)}^{\u22ba}\eta {A}_{\eta}\left(\eta \right)\right),$$$$\begin{array}{c}\hfill {\mathcal{H}}_{z}={\u2329log\left(h\right(z\left)\right)\u232a}_{q\left(z\right)}{\u2329\varphi \left(z\right)\u232a}_{q\left(z\right)}^{\u22ba}\eta +{A}_{\eta}\left(\eta \right).\end{array}$$
 Otherwise, if $q\left(z\right)$ is represented by a LWS, i.e.,$$q\left(z\right):=\left\{\left({w}^{\left(1\right)},{z}^{\left(1\right)}\right),\dots ,\left({w}^{\left(N\right)},{z}^{\left(N\right)}\right)\right\},$$$$\begin{array}{c}\hfill {\mathcal{H}}_{z}={\widehat{\mathcal{H}}}_{z}^{1}+{\widehat{\mathcal{H}}}_{z}^{2},\end{array}$$
3.8. Expectations of Statistics
 (1)
 We have two cases when $q\left(z\right)$ is coded as an EF distribution, i.e.,$$q\left(z\right)=h\left(z\right)exp\left(\varphi {\left(z\right)}^{\u22ba}\eta {A}_{\eta}\left(\eta \right)\right):$$
 (a)
 If $\Phi \left(z\right)\in \varphi \left(z\right)$, i.e., the statistic $\Phi \left(z\right)$ matches with elements of the sufficient statistics vector $\varphi \left(z\right)$, then ${\u2329\Phi \left(z\right)\u232a}_{q\left(z\right)}$ is available in closed form as the gradient of the logpartition function (this is worked out in Appendix A.1.1, see (A14) and (A15)):$${\u2329\Phi \left(z\right)\u232a}_{q\left(z\right)}\in {\nabla}_{\eta}{A}_{\eta}\left(\eta \right).$$
 (b)
 Otherwise ($\Phi \left(z\right)\notin \varphi \left(z\right)$), then we evaluate$${\u2329\Phi \left(z\right)\u232a}_{q\left(z\right)}\approx \frac{1}{N}\sum _{i=1}^{N}\Phi \left({z}^{\left(i\right)}\right)\phantom{\rule{0.166667em}{0ex}},$$
 (2)
 In case $q\left(z\right)$ is represented by a LWS, i.e., the following:$$q\left(z\right)=\left\{\left({w}^{\left(1\right)},{z}^{\left(1\right)}\right),\dots ,\left({w}^{\left(N\right)},{z}^{\left(N\right)}\right)\right\}\phantom{\rule{0.166667em}{0ex}},$$$${\u2329\Phi \left(z\right)\u232a}_{q\left(z\right)}\approx \sum _{i=1}^{N}{w}^{\left(i\right)}\Phi \left({z}^{\left(i\right)}\right)\phantom{\rule{0.166667em}{0ex}}.$$
3.9. PseudoCode for the EVMP Algorithm
Algorithm 1 Extended VMP (Meanfield assumption) 

4. Experiments
4.1. Filtering with the Hierarchical Gaussian Filter
4.2. Parameter Estimation for a Linear Dynamical System
4.3. EVMP for a Switching State Space Model
5. Related Work
6. Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
VMP  Variational Message Passing 
EVMP  Extended Variational Message Passing 
BP  Belief propagation 
EP  Expectation propagation 
FFG  Forneystyle Factor Graph 
EF  Exponential family 
NEF  Nonstandard exponential family 
LWS  List of Weighted Samples 
IS  Importance sampling 
MCMC  Markov Chain Monte Carlo 
HMC  Hamiltonian Monte Carlo 
ADVI  Automatic Differentiation Variational Inference 
PG  Particle Gibbs 
Appendix A. On the Applicability of VMP
 ${f}_{a}\left({z}_{1:k}\right)$ is an element of the exponential family (EF) of distributions, i.e.,$$\begin{array}{cc}\hfill {f}_{a}\left({z}_{1:k}\right)& =p\left({z}_{k}\right{z}_{1:k1})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={h}_{a}\left({z}_{k}\right)exp\left({\varphi}_{a}{\left({z}_{k}\right)}^{\u22ba}{\eta}_{a}\left({z}_{1:k1}\right){A}_{a}\left({z}_{1:k1}\right)\right).\hfill \end{array}$$In this equation, ${h}_{a}\left({z}_{k}\right)$ is a base measure, ${\eta}_{a}\left({z}_{1:k1}\right)$ is a vector of natural (or canonical) parameters, ${\varphi}_{a}\left({z}_{k}\right)$ are the sufficient statistics, and ${A}_{a}\left({z}_{1:k1}\right)$ is the logpartition function, i.e., ${A}_{a}\left({z}_{1:k1}\right)=log\left(\int {h}_{a}\left({z}_{k}\right)exp\left({\varphi}_{a}{\left({z}_{k}\right)}^{\u22ba}{\eta}_{a}\left({z}_{1:k1}\right)\right)d{z}_{k}\right)$. It is always possible to write the logpartition function as a function of natural parameters ${A}_{{\eta}_{a}}\left({\eta}_{a}\right)$, such that ${A}_{{\eta}_{a}}\left({\eta}_{a}\right)={A}_{a}\left({z}_{1:k1}\right)$. Throughout the paper, we sometimes prefer the natural parameter parameterization of the log partition.
 We differentiate a few cases for ${f}_{b}$:
 ${f}_{b}$ is also an element of the EF, given by the following:$$\begin{array}{cc}\hfill {f}_{b}\left({z}_{k:K}\right)& =p\left({z}_{K}\right{z}_{k:K1})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={h}_{b}\left({z}_{K}\right)exp({\varphi}_{b}{\left({z}_{K}\right)}^{\u22ba}{\eta}_{b}\left({z}_{k:K1}\right){A}_{b}\left({z}_{k:K1}\right))\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{c}\hfill {f}_{b}({z}_{k},{z}_{k+1:K})={h}_{b}\left({z}_{K}\right)exp({\varphi}_{a}{\left({z}_{k}\right)}^{\u22ba}{\eta}_{ba}\left({z}_{k+1:K}\right)+{c}_{ba}\left({z}_{k+1:K}\right)).\end{array}$$The crucial element of this rewrite is that both ${f}_{a}\left({z}_{1:k}\right)$ and ${f}_{b}\left({z}_{k:K}\right)$ are written as exponential functions of the same sufficient statistics function ${\varphi}_{a}\left({z}_{k}\right)$. This case leads to the regular VMP update equations, see Appendix A.1.Our Extended VMP does not need this assumption and derives approximate VMP update rules for the following extensions.
 ${f}_{b}$ is an element of the EF, but not amenable to the modification given in (A3), i.e., it cannot be written as an exponential function of sufficient statistics ${\varphi}_{a}\left({z}_{k}\right)$. Therefore, ${f}_{b}$ is not a conjugate pair with ${f}_{a}$ for ${z}_{k}$.
 ${f}_{b}\left({z}_{k:K}\right)$ is a composition of a deterministic node with an EF node, see Figure A1. In particular, in this case ${f}_{b}\left({z}_{k:K}\right)$ can be decomposed as follows:$$\begin{array}{cc}\hfill {f}_{b}\left({z}_{k:K}\right)& =\int \delta (xg\left({z}_{k}\right)){f}_{c}(x,{z}_{k+1:K})\mathrm{d}x\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& ={f}_{c}(g\left({z}_{k}\right),{z}_{k+1:K}),\hfill \end{array}$$$$\begin{array}{cc}\hfill {f}_{c}(x,{z}_{k+1:K})& =p\left({z}_{K}\rightx,{z}_{k+1:K1})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={h}_{c}\left({z}_{K}\right)exp({\varphi}_{c}{\left({z}_{K}\right)}^{\u22ba}{\eta}_{c}(x,{z}_{k+1:K1}){A}_{c}(x,{z}_{k+1:K1})).\hfill \end{array}$$We assume that the conjugate prior to ${f}_{c}$ for random variable x has sufficient statistics vector ${\widehat{\varphi}}_{c}\left(x\right)$, and hence (A5) can be modified as follows:$$\begin{array}{c}\hfill {f}_{c}(x,{z}_{k+1:K})={h}_{c}\left({z}_{K}\right)exp({\widehat{\varphi}}_{c}{\left(x\right)}^{\u22ba}{\widehat{\eta}}_{c}\left({z}_{k+1:K}\right)+{\widehat{c}}_{c}\left({z}_{k+1:K}\right)),\end{array}$$
Appendix A.1. VMP with Conjugate Soft Factor Pairs
Appendix A.1.1. Messages and Posteriors
Appendix A.1.2. Free Energy
Appendix A.2. VMP with NonConjugate Soft Factor Pairs
Appendix A.3. VMP with Composite Nodes
Appendix B. Derivation of Extended VMP
Appendix B.1. Deterministic Mappings with Single Inputs
Appendix B.1.1. NonGaussian Case
Appendix B.1.2. Gaussian Case
Appendix B.2. Deterministic Mappings with Multiple Inputs
Appendix B.2.1. Monte Carlo Approximation to the Backward Message
Appendix B.2.2. Gaussian Approximation to the Backward Message
Appendix B.3. NonConjugate Soft Factor Pairs
 If ${\overrightarrow{m}}_{{z}_{k}}\left({z}_{k}\right)$ is a Gaussian message, apply Laplace to approximate $q\left({z}_{k}\right)$ with a Gaussian distribution as in (A31a,b).
 Otherwise, use IS as in (A30a,b).
Appendix C. Free Energy Approximation
Appendix D. Implementation Details in ForneyLab
Appendix E. Bonus: Bootstrap Particle Filtering
Appendix F. Illustrative Example
 Initiate $q\left(x\right)$, $q\left(z\right)$ by Normal distributions and $q\left(w\right)$ by an LWS.
 Repeat until convergence the following three steps:
 
 Choose w for updating.
 
 Calculate VMP message ${\overleftarrow{m}}_{w}\left(w\right)$ by (14). In this case,$$\begin{array}{cc}\hfill {\overleftarrow{m}}_{w}\left(w\right)& \propto exp\left({\left[\begin{array}{c}logw\\ w\end{array}\right]}^{\u22ba}\left[\begin{array}{c}0.5\\ \u2329x\u232ay0.5\left(\u2329{x}^{2}\u232a+{y}^{2}\right)\end{array}\right]\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \propto \mathcal{G}a\left(w;1.5,\u2329x\u232ay+0.5\left(\u2329{x}^{2}\u232a+{y}^{2}\right)\right),\hfill \end{array}$$
 
 Calculate ${\overrightarrow{m}}_{w}\left(w\right)$ by the following (16):$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\overrightarrow{m}}_{w}\left(w\right)=\left\{\left(\frac{1}{N},exp\left({z}^{\left(1\right)}\right)\right),\dots ,\left(\frac{1}{N},exp\left({z}^{\left(N\right)}\right)\right)\right\},\hfill \\ & \mathrm{where}\phantom{\rule{4.pt}{0ex}}{z}^{\left(i\right)}\sim {\overrightarrow{m}}_{z}\left(z\right)=\mathcal{N}(z;{\mu}_{z},{v}_{z}).\hfill \end{array}$$
 
 Update $q\left(w\right)$ by Section 3.5 rule (3).
 
 Choose z for updating.
 
 Calculate ${\overleftarrow{m}}_{z}\left(z\right)$ by (18), which is a NEF distribution:$$\begin{array}{c}\hfill {\overleftarrow{m}}_{z}\left(z\right)={\overleftarrow{m}}_{w}(exp\left(z\right))\propto exp\left({\left[\begin{array}{c}z\\ exp\left(z\right)\end{array}\right]}^{\u22ba}\left[\begin{array}{c}0.5\\ \u2329x\u232ay0.5\left(\u2329{x}^{2}\u232a+{y}^{2}\right)\end{array}\right]\right).\end{array}$$
 
 The forward message is simply the prior: ${\overrightarrow{m}}_{z}\left(z\right)=\mathcal{N}(z;{\mu}_{z},{v}_{z})$
 
 Update $q\left(z\right)$ by Section 3.5 rule (2)(a).
 
 Choose x for updating.
 
 Calculate VMP message ${\overleftarrow{m}}_{x}\left(x\right)$ by (14). In this case,$$\begin{array}{cc}\hfill {\overleftarrow{m}}_{x}\left(x\right)& \propto exp\left({\left[\begin{array}{c}x\\ {x}^{2}\end{array}\right]}^{\u22ba}\left[\begin{array}{c}\u2329w\u232ay\\ 0.5\u2329w\u232a\end{array}\right]\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \propto \mathcal{N}\left(x;y,1/\u2329w\u232a\right).\hfill \end{array}$$
 
 The forward message is the prior:$$\begin{array}{c}\hfill {\overrightarrow{m}}_{x}\left(x\right)=\mathcal{N}(x;{\mu}_{x},{v}_{x})\propto exp\left({\left[\begin{array}{c}x\\ {x}^{2}\end{array}\right]}^{\u22ba}\left[\begin{array}{c}{\mu}_{x}/{v}_{x}\\ 0.5/{v}_{x}\end{array}\right]\right).\end{array}$$
 
 Update $q\left(x\right)$ by Section 3.5 rule (1), i.e., the following:$$\begin{array}{cc}\hfill q\left(x\right)& \propto exp\left({\left[\begin{array}{c}x\\ {x}^{2}\end{array}\right]}^{\u22ba}\left[\begin{array}{c}{\mu}_{x}/{v}_{x}+\u2329w\u232ay\\ 0.5(1/{v}_{x}+\u2329w\u232a)\end{array}\right]\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\mathcal{N}\left(x;\frac{{\mu}_{x}+\u2329w\u232a{v}_{x}y}{1+\u2329w\u232a{v}_{x}},\frac{{v}_{x}}{1+\u2329w\u232a{v}_{x}}\right).\hfill \end{array}$$
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Algorithm  Run Time (s) 

EVMP (ForneyLab)  $6.366\pm 0.081$ 
ADVI (Turing)  $91.468\pm 3.694$ 
Algorithm  Free Energy  Total Time (s) 

EVMP (ForneyLab)  135.837  $58.674\pm 0.467$ 
ADVI (Turing)  90.285  $47.405\pm 1.772$ 
NUTS (Turing)    $78.407\pm 4.206$ 
Algorithm  Free Energy  Total Time (s) 

EVMP (Meanfield)  283.991  $42.722\pm 0.197$ 
EVMP (Structured)  273.596  $51.684\pm 0.311$ 
HMCPG (Turing)    $116.291\pm 0.886$ 
NUTSPG (Turing)    $51.715\pm 0.441$ 
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Akbayrak, S.; Bocharov, I.; de Vries, B. Extended Variational Message Passing for Automated Approximate Bayesian Inference. Entropy 2021, 23, 815. https://doi.org/10.3390/e23070815
Akbayrak S, Bocharov I, de Vries B. Extended Variational Message Passing for Automated Approximate Bayesian Inference. Entropy. 2021; 23(7):815. https://doi.org/10.3390/e23070815
Chicago/Turabian StyleAkbayrak, Semih, Ivan Bocharov, and Bert de Vries. 2021. "Extended Variational Message Passing for Automated Approximate Bayesian Inference" Entropy 23, no. 7: 815. https://doi.org/10.3390/e23070815