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I'm trying to expand my experience with restricted Boltzmann machines to a more general class of graphical models and currently learning about belief propagation using message passing algorithms. One thing I can't understand is whether BP may be used to learn parameters of a probabilistic network or is just used to, well, propagate probability given single instance of data. Let me explain it by example.

For demonstration purposes I will use RBMs that I know well, although same principles should be applicable to most graphical models. RBM is a kind of bipartite graph with 2 kinds of nodes - evidence and hidden (both - Bernoulli random variables) - and parametrized by matrix of weights connecting these groups of nodes and 2 corresponding groups of bias terms:

enter image description here

The most common algorithm to learn RBM, i.e. estimate weights and biases, looks like this:

  1. Take batch of eviddata $\textbf{E}$.
  2. Make several loops of Gibbs sampling between evidence units $E$ and hidden units $H$ to obtain equilibrium distribution.
  3. Adjust weights and biases to minimize Kullback-Leibler divergence.
  4. Repeat.

After a number of iterations RBM is considered trained and we can use it to calculate $P(H|E)$, i.e. perform inference.

Now what about belief propagation?

I presume RBM may be transformed into a factor graph like the following:

enter image description here

And here's how I understand loopy belief propagation for this graph:

  1. Take a single vector of evidence data $E_i$ and pass it to factor nodes $\phi_i$.
  2. Pass beliefs from factor nodes $\phi_i$ to hidden $H_j$ and vice versa.
  3. Repeat (2) until convergence.

It sounds like inference of $P(H|E)$, but not like learning of network parameters.

I'm pretty much sure that my understanding of at least part of this process is incorrect, but being a newbie in LBF I need some extra help to figure out exactly what.

Here's a couple of concrete questions that I have most doubts about:

  • is it correct that in message passing we always take only a single vector of evidence data (i.e. not a batch)?
  • do we have network parameters analogous to weights and biases in standard RBM?
  • is the energy or any other cost function that we try to minimize/maximize?
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  • $\begingroup$ I have exactly the same question. Could you please share what do you think about it now? $\endgroup$
    – Moonwalker
    Oct 4, 2020 at 15:34
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    $\begingroup$ As far as I remember, I never found an answer to this question and later lost interest in general probabilistic graph models. However, there are many modern models with similar properties and lots of literature about them. For example, Graph Neural Networks (GNNs) use similar concepts of information exchange between nodes, Variational Autoencoders and flow-based models (e.g. RealNVP) extend on RBMs ideas, and so on. $\endgroup$
    – ffriend
    Oct 4, 2020 at 17:53

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