# Approximate Bayesian Computation in Insurance

## 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

## 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

\pi(\theta | \boldsymbol{x}) ?