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Metropolis algorithm does not necessary require symmetric trial proposal probability. While typical canonical ensemble moves may happen to be symmetric for simplicity, biased methods and grand canonical insertion and deletion of particles are not. Thus, Metropolis algorithm does not require this. See 'Understanding molecular simulation' by Frenkel and Smit. The title of this section has (symmetric proposal...) which correctly indicates this discussion only applies to the symmetric case. |
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Let <math>f(x)</math> be a function that is proportional to the desired probability distribution <math>P(x)</math> (a.k.a. a target distribution).
# Initialization: Choose an arbitrary point <math>x_t</math> to be the first sample and choose an arbitrary probability density <math>g(x|y)</math> (sometimes written <math>Q(x|y)</math>) that suggests a candidate for the next sample value <math>x</math>, given the previous sample value <math>y</math>.
# For each iteration ''t'':
#* ''Generate'' a candidate <math>x'</math> for the next sample by picking from the distribution <math>g(x'|x_t)</math>.
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