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

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Poisson Example 1 – Prediction

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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

21

variance)estimation variance(processerror prediction

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?

Mean e

Poisson Example 2 – Estimation

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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

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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

Year Group 1 Group 2 Group 3 Group 1 Group 2 Group 3 Group 1 Group 2 Group 31 280 260 9 6 0.032 0.0232 320 275 145 7 4 8 0.022 0.015 0.0553 265 240 120 6 2 3 0.023 0.008 0.0254 340 265 105 13 8 4 0.038 0.030 0.0385 285 115

Average 0.029 0.019 0.039

Payroll P(i,j) Claims X(i,j) Probabilities

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Scollnik Example 1Posterior Distributions

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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

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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 2nd 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

P re d ic tive D istrib u tion

V a ria b ility(P re d ic tio n E rro r)

R e se rve e stim a te(M e a su re o f lo ca tio n)

10000 14000 18000 22000 26000 30000 34000

Total Reserves

Figure 1. Predictive Aggregate Distribution of Total Reserves

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:

1Estimated Ultimate 1

f

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 

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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

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Multiplicative Model for Chain-Ladder

1

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with 1

is the expected ultimate for origin year

is the proportion paid in development year

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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:

iiix ,t independen~

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 96-124.

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.