Metropolis-Hastings

Concept

Theoretically, Metropolis-Hastings is a Markov Chain Monte Carlo (MCMC) algorithm for generating random samples from a target probability distribution when direct sampling is impractical or impossible. It extends the original Metropolis algorithm by allowing candidate values to be proposed from an arbitrary proposal distribution, with acceptance determined by an acceptance probability that ensures convergence to the target distribution. In health economics, the algorithm is widely used for Bayesian parameter estimation, evidence synthesis, and probabilistic modelling where posterior distributions have no closed-form solution.

Mathematically, the algorithm constructs a Markov chain whose stationary distribution is the target posterior distribution. At each iteration, a candidate parameter value is generated from a proposal distribution and accepted with a probability derived from the ratio of the target density and the proposal density. Repeated iterations produce samples that converge asymptotically to the target distribution, allowing posterior expectations and uncertainty measures to be estimated through Monte Carlo integration.

In practice, Metropolis-Hastings is implemented within Bayesian software such as JAGS, WinBUGS, OpenBUGS and custom statistical programs. The algorithm is applied to estimate posterior distributions of clinical, epidemiological and economic model parameters, particularly where Gibbs sampling cannot be applied directly. Posterior samples are subsequently used in probabilistic sensitivity analysis, Bayesian network meta-analysis and decision-analytic modelling.

---

Purpose

Used to generate samples from complex posterior probability distributions for Bayesian estimation, uncertainty quantification, probabilistic sensitivity analysis, and evidence synthesis in health economic evaluation.

---

Mathematical Formulae

Primary Formula

For a current state ? and proposed state ?*, the acceptance probability is

�(?, ?) = min(1, (�(?) q(? | ?)) / (�(?) q(? | ?)))

where:

• �(?) = target probability density
• q(? | ?)* = proposal distribution
• � = probability of accepting the proposed state

Supporting Formulae

Candidate generation:

? ~ q(? | ?)**

Monte Carlo estimate:

E[f(?)] � (1/N) �???? f(??)

Special case for symmetric proposal distributions:

� = min(1, �(?*) / �(?))

Related Mathematical Methods

• Markov Chain Monte Carlo
• Bayesian inference
• Gibbs sampling
• Hamiltonian Monte Carlo
• Monte Carlo integration
• Bayesian network meta-analysis
• Probabilistic sensitivity analysis

---

Example

A Bayesian network meta-analysis estimates the relative treatment effect of a new oncology intervention.

The current parameter estimate is

? = 0.42

A candidate value

? = 0.48*

is proposed from a normal proposal distribution.

The posterior density ratio and proposal ratio produce

� = 0.81

A uniform random number of 0.37 is generated.

Since

0.37 < 0.81

the proposed value is accepted and becomes the next state of the Markov chain. Repeating this process for many iterations produces posterior samples used to estimate treatment effects, costs and QALYs.

---

Excel Implementation

Function	Example Formula	Health Economics Application
RAND	=RAND()	Generate uniform random numbers for acceptance decisions
NORM.INV	=NORM.INV(RAND(),CurrentValue,ProposalSD)	Generate candidate parameter values
MIN	=MIN(1,AcceptanceRatio)	Calculate acceptance probability
IF	=IF(RAND()<AcceptanceProbability,Proposal,CurrentValue)	Accept or reject candidate value
AVERAGE	=AVERAGE(B2:B50001)	Estimate posterior mean after convergence

---

VBA (Optional)

Automate Metropolis-Hastings iterations, convergence diagnostics and storage of posterior samples for subsequent probabilistic sensitivity analysis.

---

Sources

• Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of State Calculations by Fast Computing Machines. Journal of Chemical Physics. 1953;21:1087-1092.
• Hastings WK. Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika. 1970;57(1):97-109.
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. CRC Press; 2013.
• Brooks S, Gelman A, Jones GL, Meng XL, eds. Handbook of Markov Chain Monte Carlo. CRC Press; 2011.
• Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.