# Approximate Bayesian Computation

## Background

• 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 $$U_1, \dots, U_N \overset{\mathrm{i.i.d.}}{\sim} f_U( \,\cdot\, ; \boldsymbol{\theta}_{\mathrm{sev}} )$$.

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?

Easier question: Given $$(X_1, N_1), \dots, (X_n, N_n)$$ summaries & counts, what is $$\boldsymbol{\theta}$$?

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.

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

For a sample of $$n$$ i.i.d. observations the joint likelihood is

$$p_{\boldsymbol{X}}(\boldsymbol{x} \mid \boldsymbol{\theta}) = \prod_{i=1}^n p_{X_i}(x_i; \boldsymbol{\theta}) \,.$$

If $$n$$ increases, then $$p_{\boldsymbol{X}}(\boldsymbol{x} \mid \boldsymbol{\theta}) = \prod \text{Small things} \overset{\dagger}{=} 0$$, or just takes a long time to compute, then $$\texttt{Intractable Likelihood}$$!

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

## Bayesian statistics

Prior distribution $$\pi(\boldsymbol{\theta})$$

Likelihood $$\pi( \boldsymbol{x} \mid \boldsymbol{\theta} )$$

Posterior distibution

$$\pi(\boldsymbol{\theta} \mid \boldsymbol{x} ) = \frac{ \pi(\boldsymbol{\theta}) \pi( \boldsymbol{x} \mid \boldsymbol{\theta} ) }{ \pi( \boldsymbol{x} ) }$$

## 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}) ?

## Exact matching algorithm

Given some observations $$\boldsymbol{x}_{\text{obs}}$$, repeat:

• generate a potential parameter from the prior distribution $$\boldsymbol{\theta}^{\ast} \sim \pi(\boldsymbol{\theta})$$;
• simulate some 'fake data' $$\boldsymbol{x}^{\ast}$$ from the model $$\boldsymbol{\theta}^{\ast}$$;
• if $$\boldsymbol{x}_{\text{obs}} = \boldsymbol{x}^{\ast}$$, then store $$\boldsymbol{\theta}^{\ast}$$.

The resulting $$\boldsymbol{\theta}^{\ast}$$s are an i.i.d. sample from the posterior $$\pi(\theta \mid \boldsymbol{x}_{\text{obs}})$$.