7.5. Markov chain Monte Carlo approach

In a Bayesian framework, Markov Chain Monte Carlo (MCMC) methods are a powerful tool for calibration. They are especially valuable when the statistical model cannot be solved analytically, such as when the prior distribution has a complex structure or the model is nonlinear. Unlike many classical approaches, MCMC does not require the assumption of Gaussian errors: it remains applicable even when the likelihood is non-Gaussian.

Rather than providing a single “best-fit” solution (as in minimisation techniques), MCMC generates a collection of parameter samples that represent the full posterior distribution (similar to ABC methods). However, these methods also come at the cost of potentially high computational demand, since long sampling chains may be required to achieve convergence and reliable estimates. Users should therefore interpret results as distributions and ensure that convergence diagnostics are checked before drawing conclusions (more details are given in the following sections).

• 7.5.1. Markov chain principle
• 7.5.2. The MCMC algorithms
  • 7.5.2.1. Metropolis-Hastings algorithm
  • 7.5.2.2. Component-wise Metropolis-Hastings
• 7.5.3. Assessing convergence