Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

7.4.2 METROPOLIS–HASTINGS ALGORITHM

The MH algorithm is almost a universal algorithm used to generate a Markov chain with a stationary distribution π(θ |x). It has been developed by Metropolis et al. (1953) in mechanical physics and generalized by Hastings (1970) in a statistical setting. It can be applied to a variety of problems since it requires the knowledge of the distribution of interest up to a normalizing constant only, i.e. as is typically the case in practice, the normalizing constant does not need to be known for the posterior for one to apply the following methods. Given a density π(θ |x), known up to a normalization constant, and a conditional density q(θ *|θ) the method generates the chain {θ⁽¹⁾, θ⁽²⁾, …} using the following algorithm.

Algorithm 7.2 (Metropolis–Hastings algorithm)

1. Initialize θ ^(l=0) with any value within a support of π(θ |x);
2. For l = 1,…, L
   1. Set θ ^(l) = θ ^(l–1);
   2. Generate a proposal θ * from q(θ *| θ ^(l));
   3. Accept a proposal with the acceptance probability
      (7.65)
      that is, simulate U from the uniform distribution function Uniform(0, 1) and set θ^(l) = θ * if U < α(θ ^(l), θ *). Note that the normalization constant of the posterior does not contribute here.
3. Next l (i.e., do an increment, l = l +1, and return to step 2).

Remark 7.11

• The density is called the target or objective density;
• q(θ * |θ) is called the proposal density and will ...