# Approximate Bayesian Computation (ABC)

## Notation

• parameters
• generated samples from model with parameters
• denotes observed data
• is the domain of the observations
• is a metric on

## Overview

• Cases where computing the likelihood of the observed data is intractable
• ABC uses approximation of the likelihood obtained from simulation

## Rejection ABC

Let be a similarity threshold, and be the notion of distance, e.g. premetric on domain of observations.

The rejection ABC proceeds as follows:

1. Sample multiple model parameters .
2. For each , generate psuedo-dataset from
3. For each psuedo-datset , if , accept the generated , otherwise reject .

Result: Exact sample from approximated posterior , where

Choice of is crucial in the design of a n accurate ABC algorithm.

## Soft ABC

One can interpret the approximate likelihood in rejection ABC as the convolution of the true likelihood and the "similarity" kernel

In fact, one can use any similarity kernel parametrised by satisfying

which gives rise to the Soft ABC methods:

Soft ABC is an extension of rejection ABC which instead weights the parameter samples from the model instead of rejecting or accepting.

An example is using the Gaussian kernel:

Which results in the weighted sample

which can be directly utilized in estimating posterior expectations, i.e. for a test function