A Beginners Guide to Bayesian Modelling

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A Beginners Guide to Bayesian Modelling

• Peter England, PhD
• EMB
• GIRO 2002

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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

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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

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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

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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)

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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

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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

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One Parameter ProblemHarder 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?

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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

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Scollnik Example 1
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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

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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)

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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

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Conceptual Framework
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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
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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
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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
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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

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Parameter Uncertainty in DFA

• Often, in DFA, forecasts are obtained using
  simulation, assuming the underlying parameters
  are fixed (for example, a standard application of
  Wilkies 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!

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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)

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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.