Editing Boltzmann machine (section)

==Unit state probability==
The difference in the global energy that results from a single unit <math>i</math> equaling 0 (off) versus 1 (on), written <math>\Delta E_i</math>, assuming a symmetric matrix of weights, is given by:

:<math>\Delta E_i = \sum_{j>i} w_{ij} \, s_j + \sum_{j<i} w_{ji} \, s_j + \theta_i</math>

This can be expressed as the difference of energies of two states:

:<math>\Delta E_i = E_\text{i=off} - E_\text{i=on}</math>

Substituting 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)
yields:

:<math>
\Delta E_{i}
    = -k_{B} T \ln(p_\text{i=off})
       - (-k_{B} T \ln(p_\text{i=on})),
</math>

where <math>k_{B}</math> is the [[Boltzmann constant]] and is absorbed into the artificial notion of temperature <math>T</math>.
Noting that the probabilities of the unit being ''on'' or ''off'' sum to <math>1</math> allows for the simplification:

:<math>
-\frac{\Delta E_{i}}{k_{B}T}
    = -\ln(p_{i=\text{on}}) + \ln(p_{i=\text{off}})
    = \ln\Big(\frac{1 - p_{i=\text{on}}}{p_{i=\text{on}}}\Big)
    = \ln(p_{i=\text{on}}^{-1} - 1),
</math>

whence the probability that the <math>i</math>-th unit is given by

:<math>p_{i=\text{on}} = \frac{1}{1+\exp\Big(-\frac{\Delta E_{i}}{k_{B}T}\Big)},</math>

where the [[scalar (physics)|scalar]] <math>T</math> 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.