### What is ABC?

Approximate Bayesian computation (ABC) is a parameter inference method. It uses an accept-reject process which bypasses smartly the likelihood evaluation. ABC turns out to be a powerful tool for parameter constraints. To apply ABC, the model needs to be stochastic. It also requires the choice of a distance $|x-y|$ and a tolerance level $\epsilon$.

### How does ABC work?

Given an observation $x^\mathrm{obs}$, the ABC algorithm processes the following steps:

- Draw a parameter $\pi$ from the prior
- Draw a $x$ from the model $P(\cdot|\pi)$
- Accept $\pi$ if $|x - x^\mathrm{obs}|\leq\epsilon$
- Reject otherwise
- Repeat and reconstruct the distribution of accepted $\pi$

Why does this work? Because
\[
\definecolor{LincBlue}{RGB}{34,34,136}
\definecolor{LincDarkRed}{RGB}{102,0,34}
\definecolor{LincGreen}{RGB}{17,119,51}
\textcolor{LincDarkRed}{\mathrm{Distribution\ of\ accepted }\ \pi}\ \ \ \ \ \ \ \ \ \ \ \\
=\ \textcolor{LincBlue}{\mathrm{prior}}\ \times\ \textcolor{LincGreen}{\mathrm{green\ areas}}\\
\ \ \ \ \ \ \ \approx\ \textcolor{LincBlue}{\mathrm{prior}}\ \times\ 2\epsilon\ \times\ \mathrm{likelihood}\\
\propto\ \mathrm{posterior}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\]
Therefore, ABC is actually a sampling from the posterior. We only need to reconstruct the distribution of accepted $\pi$ to recover it.

### Combination with PMC

If $\epsilon$ is too large, the approximation is bad; if $\epsilon$ is too small, then the sampling will be inefficient. To solve this problem, ABC is often combined with the population Monte Carlo technique (PMC), an iterative solution for the choice of $\epsilon$. PMC ABC decreases $\epsilon$ little by little based on the previous iteration. When $\epsilon$ converges, the constraints stablize as well.

Applying the PMC ABC algorithm on weak-lensing peak counts, we have found a very good agreement between constraints from ABC and the Gaussian likelihood. In addition, ABC has reduced the computation time by two orders of magnitude.