In implementing the MCMC (Markov Chain Monte Carlo) sampling algorithm, the goal is generating an independent identically distributed (i.i.d.) sample \( \htmlClass{sdt-0000000046}{\mathbf{x}}_1 \), \( \htmlClass{sdt-0000000046}{\mathbf{x}}_2 \), ... , \( \htmlClass{sdt-0000000046}{\mathbf{x}}_n \) from some given probabilistic landscape. If some \( \htmlClass{sdt-0000000131}{X}measureFunction \) is equivalent or proportional to the target distribution, it can be used in the acceptance function formulation. Note that the normalization constant can be ignored as it cancels out.
| \( \mathbf{x}^* \) | This symbol represents a random proposal for the next state in the sampling sequence. |
| \( X \) | This symbol describes the Z-Transform, a mathematical tool used in digital signal processing and control systems to analyze discrete-time signals. |
| \( \mathbf{x} \) | This symbol represents a state of the dynamical system at some time point. |
For simpler representation, let the ratio:
\[ r = \frac{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000081}{\mathbf{x}^*})}{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}_n)} \]
This allows us to write the acceptance probability in terms of \( r \) as follows:
\[ \htmlClass{sdt-0000000131}{X}acceptanceProbability(\htmlClass{sdt-0000000081}{\mathbf{x}^*} \,\vert\; \htmlClass{sdt-0000000046}{\mathbf{x}}_n) = \frac{ r }{ r + 1 } \]
Assume we would like to sample from the Boltzmann distribution:
\[p(\htmlClass{sdt-0000000091}{\mathbf{s}}) = \frac{1}{\htmlClass{sdt-0000000077}{Z}} \exp\left\{ - \frac{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000091}{\mathbf{s}}) }{ \htmlClass{sdt-0000000029}{T} } \right\}\]
We can assign the measure function to the (normalized or not) probability value:
\[ \htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}) = \exp\left\{ - \frac{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000046}{\mathbf{x}}) }{ \htmlClass{sdt-0000000029}{T} } \right\} \]
This gives a value for the ratio \( r \):
\[ \begin{align*} r &= \frac{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000081}{\mathbf{x}^*})}{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}_n)} = \frac{ \exp\left\{ - \frac{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) }{ \htmlClass{sdt-0000000029}{T} } \right\} }{ \exp\left\{ - \frac{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000046}{\mathbf{x}}_n) }{ \htmlClass{sdt-0000000029}{T} } \right\} } \\ &= \exp\left\{ - \frac{1}{\htmlClass{sdt-0000000029}{T}} \left[ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) - \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000046}{\mathbf{x}}_n) \right] \right\} \\ &= \exp\left\{ \frac{1}{\htmlClass{sdt-0000000029}{T}} \left[ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000046}{\mathbf{x}}_n) - \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) \right] \right\} \\ r &= \exp\left\{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000046}{\mathbf{x}}_n) - \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) \right\}^{ \frac{ 1 }{ \htmlClass{sdt-0000000029}{T} }} \end{align*} \]
The Boltzmann acceptance probability for \( \htmlClass{sdt-0000000081}{\mathbf{x}^*} \) can now be easily computed, knowing the energy of the two states and the temperature of the system.