Summary of Expectation Propagation

Expectation propagation (EP) is an approximate inference method used to infer approximate distributions of latent variables. In many probabilistic models, the joint distribution of the data \(X\) and latent variables \(\theta\) takes the form of a product of factors \(f_{i}\) \[p(X,\theta)=\prod_{i=1}^{m}f_{i}(X|\theta).\] EP can be used to estimate the posterior distribution \(p(\theta|X)\). With \(X\) observed, the idea of EP is to approximate the posterior with \[q(\theta)\propto\prod_{i=1}^{m}\tilde{f}_{i}(X|\theta)\] where \(\tilde{f}_{i}\) is an approximate factor corresponding to \(f_{i}\) with the constraint that \(\tilde{f}_{i}\) belongs to a chosen parametric form (e.g., Gaussian) in the exponential family. Instead of considering each factor \(f_{i}\) individually and finding its corresponding \(\tilde{f}_{i}\), EP takes into account the fact that the final quantity of interest is the posterior \(q(\theta)\) which is given by the product of all estimated factors. Hence, in finding each approximate factor \(\tilde{f}_{i}\), EP uses other approximate factors \(q^{\backslash i}(\theta):=\prod_{j\neq i}\tilde{f}_{j}(X|\theta)\) as a context.

Given that \(X\) is observed to be \(X_{0}\), EP iteratively refines \(\tilde{f}_{i}\) for each \(i\) with the following update

\[\begin{aligned} q(\theta) & =\arg\min_{q}KL\left(f_{i}(X_{0}|\theta)q^{\backslash i}(\theta)\,\|\, q(\theta)\right)\\ & =\arg\min_{q}\int d\theta\, f_{i}(X_{0}|\theta)q^{\backslash i}(\theta)\log\left(\frac{f_{i}(X_{0}|\theta)q^{\backslash i}(\theta)}{q(\theta)}\right).\end{aligned}\]

For notational convenience, define \(\text{proj}\left[\cdot\right]\) operator as \[\text{proj}\left[p\right]=\arg\min_{q}KL\left(p\,\|\, q\right)\] where \(q\) takes a prespecified form in the exponential family. Since \(q(\theta)=q^{\backslash i}(\theta)\tilde{f}_{i}(X_{0}|\theta)\), the update for each \(\tilde{f}_{i}\) is given by \[\tilde{f}_{i}=\frac{\text{proj}\left[f_{i}(X_{0}|\theta)q^{\backslash i}(\theta)\right]}{q^{\backslash i}(\theta)}.\] In the EP literature, \(q^{\backslash i}\) is known as a cavity distribution and \(f_{i}(X_{0}|\theta)q^{\backslash i}(\theta)\) is known as tilted distribution. Each factor \(f_{i}\) is known as a site.

Why \(\tilde{f}_{i}\in\text{ExpFam}\) ?

The main reason to require \(q\) to be in the exponential family is to make the KL minimization easy. Assume \(q\) is in the exponential family, \[q(\theta)=h(\theta)g(\eta)\exp\left(\eta^{\top}u(\theta)\right)\] where \(u(\theta)\) is the sufficient statistic of \(q\) and \(\eta\) is the natural parameter. It can be shown (by expanding the definition of KL divergence, taking the derivative with respect to \(\eta\), and setting it to 0) that \(q^{*}=\arg\min_{q}KL(p\,\|\, q)\) is the one such that \[\mathbb{E}_{q^{*}(\theta)}\left[u(\theta)\right]=\mathbb{E}_{p(\theta)}\left[u(\theta)\right].\] That is, the projection of \(p\) onto the exponential family is given by the exponential family distribution \(q^{*}\) that has the same moment parameters as the moments under \(p\). This procedure is known as moment matching. For example, if \(q\) is chosen to be a Gaussian, then \(u(\theta)=\left(\theta,\theta^{2}\right)^{\top}\). So the projection of \(p\) amounts to finding a Gaussian \(q\) that has the same first two moments.

If each \(\tilde{f}_{i}\) is in the exponential family, then \(q\) is also in the exponential family as the family is close under multiplication and division. Division of two exponential family distributions amounts to subtraction of their natural parameters followed by a renormalization.

Message View of EP

Besides factor-based treatment, one can view EP as a message passing algorithm by considering the following changes.

  1. The cavity \(q^{\backslash i}\) can be thought of as a message carrying information about \(\theta\) to the factor \(i\). We denote it by \(m_{\theta\rightarrow f_{i}}(\theta)\).
  2. \(\tilde{f}_{i}\) can be treated as a message from \(f_{i}(X_{0}|\theta)\) to \(\theta\). We denote it by \(m_{f_{i}\rightarrow\theta}(\theta)\).
  3. In the KL objective, one can replace \(f_{i}(X_{0}|\theta)\) with \(\int dX\, m_{X\rightarrow f_{i}}(X)f_{i}(X|\theta)\) where \(m_{X\rightarrow f_{i}}(X):=\delta(X-X_{0})\) is a message from \(X\) carrying information about \(X\) to the factor \(f_{i}\). In this case, it indicates that the distribution of \(X\) is a point mass at \(X_{0}\). In general, the observed evidence does not have to be an exact value. We may, say, observe that \(X\) is non-negative which gives \(m_{X\rightarrow f_{i}}(X)=p(X|X>0)\).

With these changes, the factor update can be rewritten as

\[m_{f_{i}\rightarrow\theta}(\theta)=\frac{\text{proj}\left[\int dX\, f(X|\theta)m_{X\rightarrow f_{i}}(X)m_{\theta\rightarrow f_{i}}(\theta)\right]}{m_{\theta\rightarrow f_{i}}(\theta)}.\label{eq:ep_msg_update}\]

The estimated posterior then becomes \(q(\theta)=\prod_{i=1}^{m}m_{f_{i}\rightarrow\theta}(\theta)\). For comparison purpose, a belief propagation (BP) message from \(f_{i}\) to \(\theta\) is \[m_{f_{i}\rightarrow\theta}(\theta)=\int dX\, f(X|\theta)m_{X\rightarrow f_{i}}(X).\] Two notable differences between EP and BP are

  1. In BP, no projection is involved in computing a message.
  2. In BP, to send a message from \(f_{i}\) to \(\theta\), there is no need to gather a message from \(\theta\) to \(f_{i}\). EP needs it to compute an outgoing message to give a context for the global posterior and then divide it out after the projection.

General EP Messages

Let \(f\) be a factor and \(\mathcal{V}(f)=\left\{ v_{1}^{f},\ldots,v_{D}^{f}\right\} \) be the set of variables connected to \(f\). In general, an EP message from \(f\) to a variable \(v'\in\mathcal{V}(f)\) takes the following form

\[m_{f\rightarrow v'}(v')=\frac{\text{proj}\left[\int d\mathcal{V}\backslash\{v'\}\, f(\mathcal{V})\prod_{v\in\mathcal{V}(f)}m_{v\rightarrow f}(v)\right]}{m_{v'\rightarrow f}(v')}\label{eq:ep_msg_update_general}\]

where \(\int d\mathcal{V}\backslash\{v'\}\) denotes an integral over all variables except \(v'\). In the previous example, \(\mathcal{V}(f)=\{X,\theta\}\). Note that the division is easy to carry out since both numerator and denominator messages are in the exponential family.

EP Update Sequence

As \(X=X_{0}\), we will sometimes drop \(X_{0}\) (constant) from \(\tilde{f}_{i}\).

\(q(\theta)\) is the posterior of the latent \(\theta\) that we are after.