Probability of A Unit's State
The difference in the global energy that results from a single unit being 0 (off) versus 1 (on), written, is given by:
This can be expressed as the difference of energies of two states:
We then substitute the energy of each state with its relative probability according to the Boltzmann Factor (the property of a Boltzmann distribution that the energy of a state is proportional to the negative log probability of that state):
where is Boltzmann's constant and is absorbed into the artificial notion of temperature . We then rearrange terms and consider that the probabilities of the unit being on and off must sum to one:
We can now finally solve for, the probability that the -th unit is on.
where the scalar is referred to as the temperature of the system. This relation is the source of the logistic function found in probability expressions in variants of the Boltzmann machine.
Read more about this topic: Boltzmann Machine
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