A Beginner’s Guide to Bayesian Modelling Peter England, PhD EMB GIRO 2002

Outline An easy one parameter problem A harder one parameter problem Problems with multiple parameters Modelling in WinBUGS Stochastic Claims Reserving Parameter uncertainty in DFA

Bayesian Modelling: General Strategy Specify distribution for the data Specify prior distributions for the parameters Write down the joint distribution Collect terms in the parameters of interest Recognise the (conditional) posterior distribution? –Yes: Estimate the parameters, or sample directly –No: Sample using an appropriate scheme Forecasting: Recognise the predictive distribution? –Yes: Estimate the parameters –No: Simulate an observation from the data distribution, conditional on the simulated parameters

A One Parameter Problem Data Sample [3,8,5,9,5,8,4,8,7,3] Distributed as a Poisson random variable? Use a Gamma prior for the mean of the Poisson Predicting a new observation? Negative Binomial predictive distribution

Poisson Example 1 – Estimation

Poisson Example 1 – Prediction

One Parameter Problem: Simple Case We can recognise the posterior distribution of the parameter We can recognise the predictive distribution No simulation required (We can use simulation if we want to)

Variability of a forecast Includes estimation variance and process variance Analytic solution: estimate the two components Bayesian solution: simulate the parameters, then simulate the forecast conditional on the parameters

Main Features of Bayesian Analysis Focus is on distributions (of parameters or forecasts), not just point estimates The mode of posterior or predictive distributions is analogous to “maximum likelihood” in classical statistics

One Parameter Problem: Harder Case Use a log link between the mean and the parameter, that is: Use a normal distribution for the prior What is the posterior distribution? How do we simulate from it?

Poisson Example 2 – Estimation

Poisson Example 2 Step 1: Use adaptive rejection sampling (ARS) from log density to sample the parameter Step 2: For prediction, sample from a Poisson distribution with mean, with theta simulated at step 1

A Multi-Parameter Problem From Scollnik (NAAJ, 2001) 3 Group workers compensation policies Exposure measured using payroll as a proxy Number of claims available for each of last 4 years Problem is to describe claim frequencies in the forecast year

Scollnik Example 1

Scollnik Example 1 Posterior Distributions

Scollnik Example 1 Use Gibbs Sampling –Iterate through each parameter in turn –Sample from the conditional posterior distribution, treating the other parameters as fixed Sampling is easy for Use ARS for

WinBUGS WinBUGS is an expert system for Bayesian analysis You specify –The distribution of the data –The prior distributions of the parameters WinBUGS works out the conditional posterior distributions WinBUGS decides how to sample the parameters WinBUGS uses Gibbs sampling for multiple parameter problems

Stochastic Claims Reserving Changes the focus from a “best estimate” of reserves to a predictive distribution of outstanding liabilities Most stochastic methods to date have only considered 2 nd moment properties (variance) in addition to a “best estimate” Bayesian methods can be used to investigate a full predictive distribution, and incorporate judgement (through the choice of priors). For more information, see England and Verrall (BAJ, 2002)

The Bornhuetter-Ferguson Method Useful when the data are unstable First get an initial estimate of ultimate Estimate chain-ladder development factors Apply these to the initial estimate of ultimate to get an estimate of outstanding claims

Conceptual Framework

Estimates of outstanding claims To estimate ultimate claims using the chain ladder technique, you would multiply the latest cumulative claims in each row by f, a product of development factors. Hence, an estimate of what the latest cumulative claims should be is obtained by dividing the estimate of ultimate by f. Subtracting this from the estimate of ultimate gives an estimate of outstanding claims:

The Bornhuetter-Ferguson Method Let the initial estimate of ultimate claims for accident year i be The estimate of outstanding claims for accident year i is

Comparison with Chain-ladder replaces the latest cumulative claims for accident year i, to which the usual chain-ladder parameters are applied to obtain the estimate of outstanding claims. For the chain-ladder technique, the estimate of outstanding claims is

Multiplicative Model for Chain-Ladder

BF as a Bayesian Model Put a prior distribution on the row parameters. The Bornhuetter-Ferguson method assumes there is prior knowledge about these parameters, and therefore uses a Bayesian approach. The prior information could be summarised as the following prior distributions for the row parameters:

BF as a Bayesian Model Using a perfect prior (very small variance) gives results analogous to the BF method Using a vague prior (very large variance) gives results analogous to the standard chain ladder model In a Bayesian context, uncertainty associated with a BF prior can be incorporated

Parameter Uncertainty in DFA Often, in DFA, forecasts are obtained using simulation, assuming the underlying parameters are fixed (for example, a standard application of Wilkie’s model) Including parameter uncertainty may not be straightforward in the absence of a Bayesian framework, which includes it naturally Ignoring parameter uncertainty will underestimate the true uncertainty!

Summary Bayesian modelling using simulation methods can be used to fit complex models Focus is on distributions of parameters or forecasts Mode is analogous to “maximum likelihood” It is a natural way to include parameter uncertainty when forecasting (e.g. in DFA)

References Scollnik, DPM (2001) Actuarial Modeling with MCMC and BUGS, North American Actuarial Journal, 5 (2), pages England, PD and Verrall, RJ (2002) Stochastic Claims Reserving in General Insurance, British Actuarial Journal Volume 8 Part II (to appear). Spiegelhalter, DJ, Thomas, A and Best, NG (1999), WinBUGS Version 1.2 User Manual, MRC Biostatistics Unit.