7.4. Approximate Bayesian Computation techniques (ABC)
This section covers methods grouped under the acronym ABC, which stands for Approximate Bayesian Computation. The core idea is to perform Bayesian inference without explicitly evaluating the model likelihood function. For this reason, these methods are also refered to as likelihood-free algorithms [Wil13].
As a reminder, the principle of the Bayesian approach is summarized in the equation \( \pi_{post} (\theta|\mathbf{y}) = \frac{L(\mathbf{y}|\theta) \pi_{prior} (\theta)} {\pi(\mathbf{y})} \propto L(\mathbf{y}|\theta) \pi_{prior} (\theta),\) where \(L(\mathbf{y}|\theta)\) is the conditional probability of the observations given the parameter values \(\theta\), \(\pi_{prior}(\theta)\) is the a priori probability density of \(\theta\) (the prior), and \(\pi(\mathbf{y})\) is the marginal likelihood of the observations, which is constant here. It does not depend on the values of \(\theta\) but only on its prior, as \(\pi(\mathbf{y})=\int_{\Theta} L(\mathbf{y}|\theta) \pi_{prior} (\theta)d\theta\) making it a normalizing factor. For more details, see Introduction to Bayesian approach .
