Basic Examples (3) 

Approximate a CauchyDistribution:

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Approximate a CauchyDistribution using the specified starting point x₀, burn and thin values:

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Approximate a MultivariateTDistribution in 2D:

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Options (9) 

Method (1) 

The default Method is Automatic, in which case the jump function is a NormalDistribution (or MultinormalDistribution):

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Scale (2) 

If f has multiple peaks, the algorithm can get stuck in the one nearest to the starting point:

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Use a larger "Scale" to explore the whole landscape:

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JumpDistribution (2) 

Use a discrete jump distribution to approximate discrete distributions:

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Multivariate example:

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WorkingPrecision (1) 

Use WorkingPrecision to get higher precision in parameter estimates:

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ShowRejectionRate (2) 

Use "ShowRejectionRate" to see the rejection rate of the process:

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"ShowRejectionRate" can be used to select an optimal "Scale" for the method ("RejectionRate" is usually advised to be between 50% and 80% [see References]):

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LogProbability (1) 

Use "LogProbability" to generate samples from a distribution defined by the Log of its PDF:

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Applications (4) 

Generate sample data using BernoulliDistribution:

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Define a prior distribution for the parameter (in this case, the Jeffreys prior):

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Use Bayes's rule and multiply the Likelihood with the prior to get an f function, which is proportional to the posterior distribution of the parameter:

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Use MetropolisHastingsSequence to generate one thousand samples of the θ parameter and create a Histogram:

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Infer the parameter of a UniformDistribution (also know as the German tank problem). First, generate a small sample of data using UniformDistribution:

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Define a prior distribution for the parameter (in this case, the non-normalizable Jeffreys prior):

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Use Bayes's rule and multiply the Likelihood with the prior to get a function that is proportional to the posterior distribution of the parameter:

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Use MetropolisHastingsSequence to generate one thousand samples of the θ parameter and generate a Histogram:

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Generate sample data using NormalDistribution:

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Define a prior distribution for the parameter (in this case, the scale-free non-normalizable prior):

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Use Bayes's rule and multiply the Likelihood with the prior to get a function that is proportional to the posterior distribution of the parameter:

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Use MetropolisHastingsSequence to generate one thousand samples of the {μ,σ} pair of parameters and generate a Histogram3D:

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Generating a sample from distribution with PDF defined on a circle:

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Properties and Relations (1) 

If f is normalized, it approximates RandomVariate[ProbabilityDistribution[f,{x,–∞,∞}],n]:

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Possible Issues (2) 

The parameter thin should be an integer greater than 0:

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The dimension of the parameter-dependent "JumpDistribution" must match the number of variables:

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