For a two-order restricted Boltzmann machine with visible layer $\mathbf x$ and hidden layer $\mathbf h$, the conditional distribution can be written as:

$$p(\mathbf h|\mathbf x)=sigmoid(\mathbf{Wx}+\mathbf b)$$ $$p(\mathbf x|\mathbf h)=sigmoid(\mathbf{W}^T\mathbf{h}+\mathbf c)$$

where $\mathbf W$ is the connection matrix, $\mathbf b$ and $\mathbf c$ are the bias vectors for $\mathbf h$ and $\mathbf x$.

How does the above conditional probability generalize to a thrid-order restricted Boltzmann machine with visible layers $\mathbf x, \mathbf z$ and hidden layer $\mathbf h$?

I have read some articles using a third-order restricted Boltzmann machine, and they usually use a diagram which has a triangle in the middle with each corner connecting to one layer to represent the third-order interactions, but none of the articles that I've read explicitly provide formula of the conditional probabilities (e.g. $p(\mathbf h|\mathbf x,\mathbf z)$). This makes me hard to understand how does the Boltzmann machine work in practice.



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