Comparison of the Backpropagation Network and a Generative Binary Stochastic Graphical Model

Consider a generative graphical model consisting of probabilistic bi-state (on or off) variables; each variable contains two nodes, one for each of the two possible states.

The variables are arranged into top-down layers; each 'on' node also has a set of implied probabilities for all of the nodes (and therefore all of the variables) in the next layer down. Thus, a variable sends a vector of implied probabilities to the next layer only when in its 'on' state.

It follows that nodes receive implied probabilities from the layer immediately above, apart from nodes in the top layer that have probabilities directly assigned, which are effectively inputs to the model. Alternatively, actual binary states can be applied to the top layer nodes; either way, the nodes in all other layers always receive an implied probability from above.

A top-down pass consists of sampling binary states for the variables in a layer from their implied probabilities; propagating implied probabilities to the next layer, and repeating until the bottom layer is reached. The states observed at the bottom layer, and their distribution, are the model outputs.

Note that a node may receive zero, one or multiple implied probabilities from above, therefore some means of integrating those probabilities is required in order to arrive at a final probability for the owning variable to sample from. It is also assumed that there is no guarantee that the implied probabilities of the two nodes within each variable will sum to one, therefore the probabilities are normalised.

Consider two connected layers `X` and `Y` in which the state of the variables in layer `X` and their associated implied probabilities for layer `Y` determine the state of a single variable `y_0` in layer `Y`.

Figure 1.

Two variables `x_0`, `x_1` in layer `X` (top) send implied probabilities

to the nodes of a single variable `y_0` in layer `Y` (bottom).

Notation:

$X, Y$ Layers `X` and `Y`. $x_i$ The variables in layer `X`. $x_{i,j}$ The `j` nodes/states that make up variable `x_i`. $P(x_i)$ The probability that variable `x_i` is in its on state (state 0). $y_0$ The single variable in layer `Y`. $y_{0,j}$ The `j` nodes/states that make up variable `y_0`. $Q(y_{0,j})$ The non-normalised probability that variable `y_0` is in state `j`. Calculated per downward pass (i.e. per model sample). $Q_0, Q_1$ Aliases for `Q(y_{0,0})` and `Q(y_{0,1})`. $P(y_0)$ The probability that variable `y_0` is in its on state. Calculated per downward pass (i.e. per model sample). $u_{i,j}$ The implied probability from node `x_{i,0}` to node `y_{0,j}`.

In a single downward pass of the model (a single model sample), the implied probabilities arriving at each `y_{0,j}` are combined by taking their product (noting that the empty product has value 1), giving the non-normalised probabilities `Q_0`, `Q_1`:

\begin{align} Q_0 = \prod_{i:x_i = on} u_{i,0} \tag{1} \\[1em] Q_1 = \prod_{i:x_i = on} u_{i,1} \tag{2} \\[1em] \end{align}

Or equivalently (assuming the $x_i$ have values 0, 1 when off and on respectively):

\begin{align} Q_0 = \prod_i u_{i,0}^{x_i} \tag{3} \\[1em] Q_1 = \prod_i u_{i,1}^{x_i} \tag{4} \\[1em] \end{align}

Normalising to obtain $P(y_0)$:

$$ P(y_0) = \frac{Q_0}{Q_0 + Q_1} \tag{5} $$

Re-arranging:

\begin{align} P(y_0) &= \left[\frac{Q_0 + Q_1}{Q_0}\right]^{-1} \\[1em] &= \left[1 + \frac{Q_1}{Q_0}\right]^{-1} \\[1em] &= \left[1 + e^{\log \frac{Q_1}{Q_0}} \right]^{-1} \\[1em] P(y_0) &= \left[1 + e^{-\log \frac{Q_0}{Q_1}} \right]^{-1} \tag {6} \end{align}

The logistic function (sigmoid) has the form:

$$ \sigma(x) = \frac{1}{1 + e^{-x}} $$

Therefore:

$$P(y_0) = \sigma \left(\log \frac{Q_0}{Q_1} \right) \tag{7} $$

Now consider a traditional backpropagation network with logistic sigmoid activation function at the nodes. Use of the logistic function results in nodes with an activation level range of [0,1] which we can interpret as the probability that a node is in an 'on' state.

Our backprop network consists of two layers `X` and `Y`, in which the state of the variables in layer `X` and their associated connection weights `w_i`, determine the state of a single variable `\bar y` in layer `Y`.

Figure 2.

A backprop network in which two nodes in layer `X` (top) send activation signals through

weighted connections `w_0`, `w_1` to the nodes of a single node `\bar y` in layer `Y` (bottom).

The activation level of `\bar y` is given by:

$$ \bar y = \sigma \left(\sum_i \bar x_i w_i \right) \tag{8}$$

If we interpret `\bar y` as being the probability that node `\bar y` is in an on state, then we can consider possible equivalency between the backprop model of figure 2 with the probabilistic model of figure 1.

\begin{align} \sigma \left(\sum_i \bar x_i w_i \right) &= \sigma \left(\log \frac{Q_0}{Q_1} \right) \tag{9} \\[1em] \sum_i \bar x_i w_i &= \log \frac{Q_0}{Q_1} \tag{10} \\[1em] \end{align}

Substituting in (3) and (4):

\begin{align} \sum_i \bar x_i w_i &= \log \frac{\prod_i u_{i,0}^{x_i}}{\prod_i u_{i,1}^{x_i}} \tag{11} \\[1em] &= \log \prod_i u_{i,0}^{x_i} - \log \prod_i u_{i,1}^{x_i} \tag{12} \\[1em] &= \sum_i \log u_{i,0}^{x_i} - \sum_i \log u_{i,1}^{x_i} \tag{13} \\[1em] &= \sum_i (x_i \log u_{i,0}) - \sum_i (x_i \log u_{i,1}) \tag{14} \\[1em] &= \sum_i x_i (\log u_{i,0} - \log u_{i,1}) \tag{15} \\[1em] \sum_i \bar x_i w_i &= \sum_i x_i \log \frac{u_{i,0}}{u_{i,1}} \tag{16} \\[1em] \end{align}

Hence the two models are equivalent in the specific context of a single top-down pass of binary input states. Specifically, each $w_i$ connection weight in the backprop model is equivalent to the log odds (log of the odds ratio) of the probabilities $u_{i,0}, u_{i,1}$

However, the equivalence does not appear to hold for the wider model, i.e. e.g. equation (16) does not equate to the expected value of $P(y_0)$ for a given input vector in the binary stochastic model.

The estimated value of $Q_i$ can be obtained by enumerating all possible combinations of the $x_i$, hence for two $x_i$ we get:

\begin{align} \Bbb E[Q_i] &= P(x_0)P(x_1)u_{0,1}u_{1,1} \\[1em] &= P(x_0)[1-P(x_1)]u_{0,i} \\[1em] &= P(x_1)[1-P(x_0)]u_{1,i} \\[1em] &= [1-P(x_0)][1-P(x_1)] \tag{17} \end{align}

Which simplifies and generalises to (i.e. for any number of $x_i)$:

\begin{align} \Bbb E[Q_i] &= \prod_j P(x_j)u_{j,i} - [1-P(x_j)] \\[1em] &= \prod_j P(x_j)u_{j,i} + P(x_j) - 1 \\[1em] &= \prod_j P(x_j)[u_{j,i} + 1] - 1 \tag{18} \end{align}

Alternatively, the geometric mean of $Q_i$ (the geometric expectation?) is given by the rather more elegant (and possibly more relevant):

$$ GM[Q_i] = \prod_j {u_{j,i}}^{P(x_j)} \tag{19}$$

Moving on to the expected value of $y_0$:

$$ \Bbb E[y_0] = \Bbb E\left[\frac{Q_0}{Q_0 + Q_1}\right] \tag{20}$$

The quotient in (20) is problematic, not least because the dividend and divisor are not independent (because $Q_0$ is present in both); otherwise we may at least have been able to factorise into $\Bbb E[Q_0] \Bbb E[1/(Q_0+Q_1)]$, i.e. the expectation of a product is equal to the product of the expectations; although even that would leave the potentially problematic expectation of a quotient.

Nonetheless, for two $x_i$ we can enumerate the four possible combinations of outcomes (as per equation (17)). Giving:

\begin{align} \Bbb E[y_0] = & P(x_0)P(x_1)\frac{u_{0,0}u_{1,0}}{u_{0,0}u_{1,0} + u_{0,1}u_{1,1}} \\[1em] + & P(x_0)[1-P(x_1)]\frac{u_{0,0}}{u_{0,0} + u_{0,1}} \\[1em] + & P(x_1)[1-P(x_0)]\frac{u_{1,0}}{u_{1,0} + u_{1,1}} \\[1em] + & [1-P(x_0)][1-P(x_1)]\frac{1}{2} \tag{21} \end{align}

For which there is no obvious (to me) factorisation, simplification or generalisation to $n$ $x_i$.

The geometric expectation of `y_0` for two `x_i`:

\begin{align} GM[y_0] = \left(\frac{u_{0,0}u_{0,1}}{u_{0,0}u_{0,1} + u_{1,0}u_{1,1}}\right)^{P(x_0)P(x_1)} \left(\frac{u_{0,0}}{u_{0,0} + u_{0,1}}\right)^{P(x_0)[1-P(x_1)]} \left(\frac{u_{1,0}}{u_{1,0} + u_{1,1}}\right)^{P(x_1)[1-P(x_0)]} \tag{22} \end{align}

We can at least express these enumerations over combinations of $x_i$ more compactly by defining an enumeration notation:

$[n]$ The set of all indexes into `x_i`, i.e. for $n$ $x_i$ we get $[n] = \{0,1,...,n\}$. $\mathcal{S}_k([n])$ The set of all unique subsets of size `k` within $[n]$, i.e. the set of all possible combinations rather than permutations (for which order matters). $\mathcal{S}_k^i([n])$ The $i$th subset. $\mathcal{S}_k^i$ Shorthand/alias for $\mathcal{S}_k^i([n])$ $s : \mathcal{S}_k$ Enumeration of the subsets defined by $\mathcal{S}_k$ $s, \hat s : \mathcal{S}_k$ Parallel enumeration of the subsets defined by $\mathcal{S}_k$, and the complement of each subset. I.e. for a given subset $s$ of the `x_i` that are in the on state, $\hat s$ gives the remaining set of `x_i` that are in the off state.

Equation (22) can now be generalised to $n$ $x_i$:

$$ GM[y_0] = \prod_{s:\mathcal S_k} \left[\frac{\prod_{i:s}u_{0,i}}{\prod_{i:s}u_{0,i} + \prod_{i:s}u_{1,i}} \right] ^ {\prod_{i:s} P(x_i) \prod_{j:\hat s} P(x_j)} \tag{23}$$

In a nutshell, the generative model is a richer model than the backprop model because for a given input vector we obtain a distribution of outputs rather than the single output vector response of the backprop model. The distribution of outputs is essentially a set of combinations of outputs (on and off combinations of the variable in Y) and their relative frequency of occurrence, hence any direct equivalence was only ever going to hold for a single $y_i$ at best.

Colin,

February 20th, 2017