This equation determines the total energy in a Boltzmann-Machine for a specific state \( \htmlClass{sdt-0000000046}{\mathbf{x}} \). It is really similar to the Energy of a state in a Hopfield Network equation and is used for learning and inference in a Boltzmann-Machine. This equation is designed so that the probability of a state in a BM is higher for configurations that have a lower energy definition.
| \( j \) | 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. |
Consider the a fully visible Boltzmann machine with 3 units:
And the states of the units being:
For \(i < j\), we need to consider the pairs \((1,2)\), \((1,3)\), and \((2,3)\). The energy is therefore equal to \[\htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000091}{\mathbf{s}}) = -(\htmlClass{sdt-0000000092}{w}_{12}\htmlClass{sdt-0000000091}{\mathbf{s}}_{1}\htmlClass{sdt-0000000091}{\mathbf{s}}_{2} + \htmlClass{sdt-0000000092}{w}_{13}\htmlClass{sdt-0000000091}{\mathbf{s}}_{1}\htmlClass{sdt-0000000091}{\mathbf{s}}_{3} + \htmlClass{sdt-0000000092}{w}_{23}\htmlClass{sdt-0000000091}{\mathbf{s}}_{2}\htmlClass{sdt-0000000091}{\mathbf{s}}_{3})\]
Filling these values in the yields the following:
\[\htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000091}{\mathbf{s}}) = -(0.5 \cdot 1 \cdot (-1) + (-0.3) \cdot 1 \cdot 1 + 0.8 \cdot (-1) \cdot 1) = -(-1.6) = 1.6\]
So, the energy of the states \(\htmlClass{sdt-0000000091}{\mathbf{s}} = (1,-1,1)\) is \(\htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000091}{\mathbf{s}}) = 1.6\)