This equation describes the change in energy of a Boltzmann machine when one unit \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{\htmlClass{sdt-0000000018}{i}}\) changes it state from 0 to 1 while all other units \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{\htmlClass{sdt-0000000011}{j}}\) keep the same activation energy.
| \( j \) | This is a secondary symbol for an iterator, a variable that changes value to refer to a series of elements |
| \( i \) | This is the symbol for an iterator, a variable that changes value to refer to a sequence of elements. |
| \( \sum \) | This is the summation symbol in mathematics, it represents the sum of a sequence of numbers. |
| \( \mathbf{s} \) | This symbol represents a full description of the system taken at molecular level. |
| \( w \) | This symbol describes the connection strength between two units in an Boltzmann machine. |
| \( E \) | This symbol represents the energy. |
This is an example for the same Boltmann machine as in the example on the page:Energy of a Specific State in a Boltzmann Machine
Consider the a fully visible Boltzmann machine with 3 units:
And the states of the units being:
Now we compute the change in energy \(\Delta \htmlClass{sdt-0000000100}{E}\) when \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{1} = 1\) changes to \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{1} = -1\)
Using the formula:
\[- \Delta \htmlClass{sdt-0000000100}{E}_{\htmlClass{sdt-0000000018}{i}} = -\htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000011}{j}} \htmlClass{sdt-0000000092}{w}_{\htmlClass{sdt-0000000018}{i} \htmlClass{sdt-0000000011}{j}} \htmlClass{sdt-0000000091}{\mathbf{s}}_{\htmlClass{sdt-0000000011}{j}}\]
We get:
\[- \Delta \htmlClass{sdt-0000000100}{E}_{\htmlClass{sdt-0000000018}{i}} = -(\htmlClass{sdt-0000000092}{w}_{12}\htmlClass{sdt-0000000091}{\mathbf{s}}_{1} + \htmlClass{sdt-0000000092}{w}_{13}\htmlClass{sdt-0000000091}{\mathbf{s}}_{3}) \]
Filling these values in the yields the following:
\[- \Delta \htmlClass{sdt-0000000100}{E}_{\htmlClass{sdt-0000000018}{i}} = -(0.5 \cdot (-1) + -(0.3) \cdot 1) = 0.8\]
So, the energy the global system changes like this \(- \Delta \htmlClass{sdt-0000000100}{E}_{\htmlClass{sdt-0000000018}{i}} = 0.8\) if \(\Delta \htmlClass{sdt-0000000100}{E}\) when \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{1} = 1\) changes to \(\htmlClass{sdt-0000000091}{\mathbf{s}}_{1} = -1\)