Suppose I have a posterior $p(z, \theta | y, \eta)$ with $y$ observed data, $z$ are hidden variables and $\theta$ are parameters, and $\eta$ is a vector of hyperparameters. I construct a mean field approximation to the posterior using coordinate ascent, i.e. $$ q(z) \gets \exp\left\{E_q(\log p(z, \theta | y, \eta) | z) + \mbox{const}\right\} \\ q(\theta) \gets \exp\left\{E_q(\log p(z, \theta | y, \eta) | \theta) + \mbox{const}\right\} $$ where $\mbox{const}$ makes each integrate to $1$. Iterate until convergence. My question is, if I want to do empirical Bayes, and (approximately) profile out $\eta$, is it valid to just augment this with an EM step $$ \eta \gets \arg \max_\eta E_q \log(p(z, \theta | y, \eta))? $$ Based on the evidence lower bound $$ \log p(y | \eta) \ge E_q \log(p(z, \theta | y, \eta)) - E_q \log(q(z, \theta)) $$ it seems like I should be able to get away with this; optimizing the lower bound over $\eta$ is exactly the proposed EM step, so I'm still doing coordinate ascent on the lower bound.

I suspect that, if this works, it is the obvious thing to do for people who do variational inference regularly. Since variational methods are fast, my thought is that maybe I could do this to set hyperparameters before launching into a Gibbs sampler for exact (up to MCMC error) inference.

Update: I've toyed around with this a bit and found that adding the EM step can work okay, but in some situations it seems to increase sensitivity to bad local optima in the variational algorithm. Initializations of variational parameters that normally seem to work okay don't. If I instead only throw in the EM step every few iterations it works better, but then of course I slow down convergence. One approach that works okay is to "burn in" the variational algorithm using a method that I trust, and then add the EM step every iteration thereafter. I'm adding a bounty because I'm still looking for some good general advice.

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    $\begingroup$ I am currently struggling to understand a paper by Airoldi et al called "Mixed Membership Stochastic Blockmodels" in which they do exactly this (I think.) However, I am not very good with variational methods, so you might want to look at the paper yourself. See Figure 5 in their paper. A question I have been asking for a while is: Why is this EM approach not the same as trying to find a global maximum of the lower bound? And if it is, how can it succeed? It seems like I end up trying to optimise a function with way too many parameters and the answers I get depend too much on starting values. $\endgroup$ – Flounderer Sep 3 '12 at 23:53
  • $\begingroup$ @Flounderer thanks for the reference. I suspect the issues have to do with the fact that you are changing the marginal likelihood $p(y | \eta)$, so you are actually working with a bunch of different lower bounds. The intuition behind me staggering the updates/burning in was that to be truly analogous to EM I should completely run the algorithm before I maximize over $\eta$, then rerun the algorithm again, etc, but this is too costly. $\endgroup$ – guy Sep 4 '12 at 0:13
  • $\begingroup$ not my area of the literature, but are you aware of jaakola's work on variational methods for bayesian parameter estimation? Sorry if this is old news.. $\endgroup$ – user4733 Sep 4 '12 at 3:25
  • $\begingroup$ Note that the EM algorithm is a secial case of variational bayes, where the approximating distribution is restricted to be a delta function. So I would go down this path of constraining your approximate posterior to be $Q(\theta\eta z)=q_1(\theta)q_2(z)\delta(\eta - \hat{\eta})$, and solving the variational problem from there. I think your proposal is the answer to this variational problem. $\endgroup$ – probabilityislogic Sep 4 '12 at 7:43
  • $\begingroup$ @probabilityislogic that formulation, I think, implies that you are putting a point-mass prior on $\eta$. But then my update is effectively changing the prior on $\eta$, which is what concerns me. $\endgroup$ – guy Sep 4 '12 at 15:07

One way of deciding how to run variational MLE is to look at how the experts do it.

In Blei's LDA code (http://www.cs.princeton.edu/~blei/lda-c/lda-c-dist.tgz), within the "run_em" function, the "lda_inference" function (inside "doc_e_step") repeatedly maximizes with respect to each $q$ distribution until convergence. After the $q$'s converge, the algorithm maximizes with respect to the parameters in "lda_mle".

The justification for this order is that by maximizing with respect to the $q$'s until convergence you get a better estimate of the expectations of hidden variables (or marginalized parameters) needed to maximize with respect to the parameters.

In standard EM, of course, the expectations you are computing are exact - which is the main difference between standard and variational EM - so this is not a concern.

From the perspective of EM as a maximization algorithm over the function $F(q,\theta)$ (www.cs.toronto.edu/~radford/ftp/emk.pdf) or from the perspective of maximizing the evidence lower bound, it is not clear that maximizing over the q's until convergence is best in terms of computationally efficiency because the algorithm will reach a local maximum no matter the order of maximization steps.

  • $\begingroup$ Thanks, I'll take a peak at Blei's code and the Neal paper. $\endgroup$ – guy Sep 15 '12 at 16:49

Generally, in empirical Bayes, you maximise the marginal likelihood (also called model evidence, or the normalising constant of the posterior) with respect to the hyperparameters and plug this estimate of the hyperparameters into the posterior.

In Casella (2001) there is a derivation of an EM for empirical Bayes. Casella first writes the marginal likelihood $m(\eta ; y)$ as:

\begin{align} m(\eta ; y) = \frac{p(y, \theta, z | \eta) }{p(\theta, z| y, \eta)}. \end{align}

So the marginal likelihood is the joint distribution of model parameters $\theta$, latent parameters $z$ and data $y$ ($p(y,\theta,z| \eta)$) divided by the joint posterior distribution of the model and latent parameters given the data ($p(\theta,z| y, \eta$).

Take the logarithm and expectation with respect to the posterior $p(\theta, z| y, \eta^{(0)})$, for some starting value $\eta^{(0)}$:

$$ E [\log m(\eta ; y)| \eta^{(0)}] = E [\log p(y, \theta, z | \eta)|\eta^{(0)}] - E [\log p(\theta, z| y, \eta) | \eta^{(0)}]. $$

By Gibbs' inequality, $E [\log p(\theta, z| y, \eta) | \eta^{(0)}] \leq E [\log p(\theta, z| y, \eta^{(0)}) | \eta^{(0)}]$. So, maximising $E [\log p(\theta, z| y, \eta) | \eta^{(0)}]$ with respect to $\eta$ beyond $\eta^{(0)}$ increases the marginal likelihood such that the sequence

$$ \eta^{(k+1)} = \underset{\eta}{\arg \max} \, E [\log p(\theta, z| y, \eta) | \eta^{(k)}] $$

converges to a local maximum of the marginal likelihood. In general, this expectation is not available in closed form. In Casella (2001), the expectation is approximated using $M$ Monte Carlo samples from the posterior $p(\theta, z| y, \eta^{(k)})$:

\begin{align} E [\log p(\theta, z| y, \eta) | \eta^{(k)}] & \approx M^{-1} \sum_{m=1}^M \log p(y, \theta^{(m)}, z^{(m)} | \eta), \\ \theta^{(m)}, z^{(m)} & \sim p(\theta, z| y, \eta^{(k)}). \end{align}

However, you could of course use other approximating methods, such as variational Bayes. The expectation then becomes:

$$ E [\log p(\theta, z| y, \eta) | \eta^{(k)}] \approx E_Q [\log p(\theta, z| y, \eta) | \eta^{(k)}], $$

where the right-hand side expectation is now with respect to the variational posterior.

So, in the EM algorithm to find the hyperparameters that maximise the marginal likelihood, the E step is now a variational Bayes EM itself, so we have an EM within an EM. Which is basically what you describe.

That also explains why it doesn't work so well when you update the hyperparameters at every iteration: the expectation with respect to the variational Bayes posterior is not very accurate after one iteration. Updating the hyperparameters after the variational Bayes algorithm has converged gives better performance, since the expectation is more accurate.

  • $\begingroup$ Thanks for the answer. Actually, I was aware of the Casella paper back when I asked this question (6 years ago! time flies). And the algorithm I gave in my OP is the variational EM algorithm given by Blei et al in the original LDA paper. I still think it is not really justified from the variational lower bound since the thing we are lower-bounding changes every iteration. I think the intuition you have about letting the EM converge before updating the hyperparameters is what led me to delay updating them. $\endgroup$ – guy Jun 27 '18 at 16:46
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    $\begingroup$ Yeah, I realise it was 6 years ago, just thought I'll add it for other people who struggle with it as well and end up on this post. This interpretation helped me a lot when I was trying to figure it out. In my opinion, the changing lower bound doesn't really matter if you interpret the lower bound as an approximate expectation in an empirical Bayes EM. $\endgroup$ – munch Jun 29 '18 at 7:50

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