Beginning Anomaly Detection Using

v or h given h or v 

(see Figure 

5-11

 and Figure 

5-12

).

Figure 5-11.  Formula for the probability of the hidden layer being in the state h 

given the visible layer being in the state v

Figure 5-12.  Formula for the probability of the hidden layer being in the state v 

given the visible layer being in the state h

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185

The 

Π works similarly to Σ, except with multiplication instead of addition. 

Essentially, p(h | v) is the multiplication of every p(h

i

, v) that exists. In these cases, 

m is 

the number of hidden nodes, and 

n is the number of visible nodes.

This could be a bit complex, so just know that the formulas in Figure 

5-11

 and 

5-12

 

are basically to find the probabilities of 

v or h being in their states given their respective 

h or v layer counterparts.

From there, we can define two more formulas regarding the probability that a 

particular node v

i

 or h

j

 activates given the vector h or v, respectively (see Figure 

5-13

 and 

Figure 

5-14

).

Figure 5-15.  The formula for a sigmoid function

Figure 5-13.  The probability of one particular node v

i

 activating given the 

multiplication of the weights between v

i

 and every single hidden node added with 

the bias

Figure 5-14.  The probability of one particular node h

j

 activating given the 

multiplication of the weights between h

j

 and every single visible node added with 

the bias

The 

σ represents the sigmoid function, defined by the formula in Figure 

5-15

.

Finally, given training inputs, we want to maximize the joint probability of the inputs, 

given by the formula in Figure 

5-16

.

Figure 5-16.  We are maximizing the joint probability of every possible visible 

node (the inputs) with respect to the weights

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186

Essentially, we will end up with a huge chain of multiplication of p(