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Metropolis Acceptance Function

Description

This function is another example of an implementation of acceptance probability when designing a MCMC (Markov Chain Monte Carlo) sampling algorithm. This formulation always accepts proposals which are "more likely" than the previous state under some measure function \( \htmlClass{sdt-0000000131}{X}measureFunction \).

\[\htmlClass{sdt-0000000131}{X}acceptanceProbability(\htmlClass{sdt-0000000081}{\mathbf{x}^*} \,\vert\; \htmlClass{sdt-0000000046}{\mathbf{x}}_n) = \begin{cases} 1, &\textup{ if } \htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) \geq \htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}_n) \\ \frac{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000081}{\mathbf{x}^*})}{\htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}_n)}, &\textup{ if } \htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000081}{\mathbf{x}^*}) < \htmlClass{sdt-0000000131}{X}measureFunction(\htmlClass{sdt-0000000046}{\mathbf{x}}_n)\end{cases}\]

Symbols Used:

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.

References

  1. Jaeger, H. (n.d.). Neural Networks (AI) (WBAI028-05) Lecture Notes BSc program in Artificial Intelligence. Retrieved June 9, 2024, from https://www.ai.rug.nl/minds/uploads/LN_NN_RUG.pdf