{"id":19655,"date":"2024-09-21T05:00:00","date_gmt":"2024-09-21T05:00:00","guid":{"rendered":"https:\/\/www.jtrive.com\/posts\/introduction-to-metropolis-hastings\/introduction-to-metropolis-hastings.html"},"modified":"2024-09-21T05:00:00","modified_gmt":"2024-09-21T05:00:00","slug":"introduction-to-markov-chain-monte-carlo-the-metropolis-hastings-algorithm","status":"publish","type":"post","link":"https:\/\/python-bloggers.com\/2024\/09\/introduction-to-markov-chain-monte-carlo-the-metropolis-hastings-algorithm\/","title":{"rendered":"Introduction to Markov Chain Monte Carlo – The Metropolis-Hastings Algorithm"},"content":{"rendered":"
\r\nThis article was first published on \r\n The Pleasure of Finding Things Out: A blog by James Triveri <\/a><\/strong>, and kindly contributed to python-bloggers<\/a>. (You can report issue about the content on this page here<\/a>)\r\nWant to share your content on python-bloggers? click here<\/a>.<\/i>\r\n<\/div>\n\n

Markov Chain Monte Carlo (MCMC) is a class of algorithms used to sample from probability distributions when direct sampling is difficult or inefficient. It leverages Markov chains to explore the target distribution and Monte Carlo methods to perform repeated random sampling. MCMC algorithms are widely used in the insurance industry, particularly in areas involving risk assessment, pricing, reserving, and capital modeling. Markov Chain Monte Carlo is an alternative to rejection sampling, which can be inefficient when dealing with high-dimensional probability distributions. MCMC is considered a Bayesian approach to statistical inference since it incorporates both prior knowledge and observed data into the estimation of the posterior distribution.<\/p>\n

The Metropolis-Hastings<\/em> algorithm is a method used to generate a sequence of samples from a probability distribution for which direct sampling might be difficult. It is a particiular variant of MCMC, which approximates a desired distribution by creating a chain of values that resemble samples drawn from that distribution. The algorithm generates a sequence of samples by proposing new candidates and deciding whether to accept or reject them based on a ratio of probabilities from the target distribution.<\/p>\n

Before getting into the details of Metropolis-Hastings, a few key definitions:<\/p>\n

Likelihood<\/strong>:
The apriori assumption specifying the distribution from which the data are assumed to originate. For example, if we assume losses follow an exponential distribution within unknown parameter , this is equivalent to specifying an exponential likelihood. Symbolically, the likelihood is represented as .<\/p>\n

Prior<\/strong>:
Sticking with the exponential likelihood example, once we\u2019ve proposed the likelihood, we need to specify a distribution for each parameter of the likelihood. In the case of the exponential there is only a single parameter, . Typically when selecting prior distributions, it should have the same domain as the parameter itself. When parameterizing the exponential distribution, we know that , so the prior distribution should be valid on . Valid distributions for are gamma, lognormal, pareto, weibull, etc. Invalid distributions would be any discrete distribution or the normal distribution. Symbolically, the prior is represented as .<\/p>\n

Posterior<\/strong>:
This is the expression which encapsulates the power, simplicity and flexibility of the Bayesian approach and is given by:<\/p>\n

<\/p>\n

The posterior is represented as , so the above expression becomes:<\/p>\n

<\/p>\n

We can update the proportionality to direct equality by the inclusion of a normalizing constant, which ensures the expression on the RHS integrates to 1:<\/p>\n

<\/p>\n

\n

Metropolis-Hastings Outline<\/h3>\n

Suppose we have a collection of , to which we would like to add a new value . We generate a sample from our transition kernel which is nearby .<\/p>\n