Beginning Anomaly Detection Using

probability function that will output the 

probability of the network having a specific 

(v,h). To elaborate on v and h, v is a vector 

that represents the states of each node in the input layer, and 

h is a vector that represents 

the states of each node in the hidden layer.

Chapter 5   Boltzmann maChines

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Z is defined as shown in Figure 

5-10

.

The 

probability function is shown in Figure 

5-9

, given a specific (v,h).

Figure 5-9.  The probability function that is associated with the visible layer v and 

the hidden layer h

Figure 5-10.  Z performs the operation over every possible v and h in the data set, 

so you can see how it forms a probability function. (Say you want a probability of 

all hearts in a card deck. This is 13/52, with 13 being all of the hearts and 52 being 

the total number of cards.)

Z is the sum of the function e

−E(v, h)

 over every single pair of input and hidden layer 

state vectors (a vector representing the states of the layer). The parameters passed into 

p(v,h) are supposed to be vectors representing a specific configuration of the two layers 

in terms of what neurons are activated.

You can see how this forms a probability function, since we want to find e

−E(v, h)

 for 

some 

v, h over the sum of e

−E(v, h)

 for all possible pairs of 

v, h.

We can go a step further and define formulas for the probability of