# Approximate Bayesian Computation (ABC)

## Table of Contents

## 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:

- Sample multiple model parameters .
- For each , generate psuedo-dataset from
- 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