# Approximate Bayesian Computation in Insurance

### Patrick J. Laub and Pierre-Olivier Goffard

## Background

- Undergrad (UQ): Software engineering & math

- PhD (Aarhus & UQ): Computational applied probability

- Post-doc #1 (ISFA): Insurance applications

- Post-doc #2 (UoM): Empirical dynamic modelling

## A Software Engineer's Toolkit for Quant. Research

## Motivation

Have a random number of claims \(N \sim p_N( \,\cdot\, ; \boldsymbol{\theta}_{\mathrm{freq}} )\) and the claim sizes \(U_1, \dots, U_N \sim f_U( \,\cdot\, ; \boldsymbol{\theta}_{\mathrm{sev}} )\).

We aggregate them somehow, like:

- aggregate claims: \(X = \sum_{i=1}^N U_i \)
- maximum claims: \(X = \max_{i=1}^N U_i \)
- stop-loss: \(X = ( \sum_{i=1}^N U_i - c )_+ \).

**Question**: Given a sample \(X_1, \dots, X_n\) of the summaries, what is the \(\boldsymbol{\theta} = (\boldsymbol{\theta}_{\mathrm{freq}}, \boldsymbol{\theta}_{\mathrm{sev}})\) which explains them?

E.g. a reinsurance contract

## Likelihoods

For simple rv's we know their likelihood (normal, exponential, gamma, etc.).

When simple rv's are combined, the resulting thing rarely has a likelihood.

$$ X_1, X_2 \overset{\mathrm{i.i.d.}}{\sim} f_X(\,\cdot\,) \Rightarrow X_1 + X_2 \sim ~ \texttt{Unknown Likelihood}! $$

Have a sample of \(n\) i.i.d. observations. As \(n\) increases,

$$ p_{\boldsymbol{X}}(\boldsymbol{x} \mid \boldsymbol{\theta}) = \prod \text{Small things} \overset{\dagger}{=} 0, $$

or just takes a long time to compute, then \(\texttt{Intractable}\) \(\texttt{Likelihood}\)!

*Usually it's still possible to simulate these things...*

## Approximate Bayesian Computation

**Example**: Flip a coin a few times and get \((x_1, x_2, x_3) = (\text{H, T, H})\); what is