Chapter 15 Bayesian Statistics
Learning Objectives
- Use Bayes’ theorem to calculate simple conditional probabilities.
- Explain what is meant by a prior distribution, a posterior distribution and a conjugate prior distribution.
- Derive the posterior distribution for a parameter in simple cases.
- Explain what is meant by a loss function.
- Use simple loss functions to derive Bayesian estimates of parameters.
- Explain what is meant by the credibility premium formula and describe the role played by the credibility factor.
- Explain the Bayesian approach to credibility theory and use it to derive credibility premiums in simple cases.
- Explain the empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
- Explain the differences between the two approaches and state the assumptions underlying each of them.
Theory
Chapter: Bayesian Statistics
15.0.0.1 Learning Objectives Summary:
- Use Bayes’ theorem to calculate simple conditional probabilities.
- Definition: Bayes’ Theorem updates the probability of a hypothesis (H) given new evidence (E).
- For discrete events: P(H|E) = P(E|H)P(H) / P(E).
- For continuous distributions: posterior ∝ likelihood × prior.
- Application: It’s used to update beliefs about a parameter after observing data.
- Example (Discrete): To find the posterior distribution of a Poisson mean (λ) after observing claims, given a discrete prior distribution for λ.
- Definition: Bayes’ Theorem updates the probability of a hypothesis (H) given new evidence (E).
- Explain what is meant by a prior distribution, a posterior distribution and a conjugate prior distribution.
- Prior Distribution: Represents initial beliefs about a parameter before observing any data. It can be discrete or continuous.
- Posterior Distribution: Represents the updated beliefs about a parameter after observing data. It is the conditional distribution of the parameter given the data. The posterior PDF is proportional to the likelihood function multiplied by the prior PDF.
- Conjugate Prior Distribution: A prior distribution is “conjugate” if, when combined with the likelihood function, it results in a posterior distribution that belongs to the same family as the prior distribution.
- Examples:
- Gamma prior for a Poisson mean (λ).
- Beta prior for a binomial probability (p).
- Normal prior for a Normal mean (with known variance).
- Examples:
- Derive the posterior distribution for a parameter in simple cases.
- General Steps (Continuous Prior):
- Write down the prior PDF.
- Write down the likelihood function (probability/PDF of observed data).
- Use the formula: Posterior PDF ∝ Likelihood × Prior PDF.
- Identify the posterior distribution by its form (e.g., Gamma for λ, Beta for p, Normal for μ).
- General Steps (Discrete Prior):
- The prior and posterior distributions have the same support.
- Calculate the conditional probabilities for each possible parameter value using Bayes’ Theorem.
- Simple Cases Covered:
- Poisson mean (λ) with Gamma prior.
- Binomial probability (p) with Beta prior.
- Normal mean (μ) with Normal prior.
- Exponential parameter (θ) with a specific prior.
- General Steps (Continuous Prior):
- Explain what is meant by a loss function.
- Definition: A loss function is a measure of the penalty incurred when the chosen Bayesian estimate differs from the true value of the parameter.
- It quantifies the cost of making an estimation error.
- Use simple loss functions to derive Bayesian estimates of parameters.
- Squared Error Loss:
- Formula:
Loss = (θ̂ - θ)^2
. - Bayesian Estimate: The mean of the posterior distribution.
- Formula:
- Absolute Error Loss:
- Formula:
Loss = |θ̂ - θ|
. - Bayesian Estimate: The median of the posterior distribution.
- Formula:
- 0/1 (All-or-Nothing) Loss:
- Formula:
Loss = 0
ifθ̂ = θ
,Loss = 1
ifθ̂ ≠ θ
. - Bayesian Estimate: The mode of the posterior distribution (the value that maximises the posterior PDF).
- Formula:
- Squared Error Loss:
- Explain what is meant by the credibility premium formula and describe the role played by the credibility factor.
- Credibility Premium: A premium that combines direct data from a specific risk (or group) with collateral data from similar risks (or a broader population).
- Formula:
Credibility Premium = Z * x̄ + (1 - Z) * μ₀
.x̄
: Estimate based on the direct data (sample mean).μ₀
: Prior or collective mean (based on collateral data/prior expectation).Z
: The Credibility Factor.
- Role of Credibility Factor (Z):
- Determines the weight given to the individual risk’s experience (
x̄
) versus the broader population’s experience (μ₀
). Z
ranges from 0 to 1.- A higher
Z
means more weight is given to the direct data, reflecting greater confidence in the individual experience. Z
increases as the amount of direct data (n
) increases, as more direct data provides a more reliable estimate from the individual experience.Z
decreases as the reliability (or precision) of the prior/collective information increases (e.g., as the variance of the prior distribution decreases).
- Determines the weight given to the individual risk’s experience (
- Explain the Bayesian approach to credibility theory and use it to derive credibility premiums in simple cases.
- Bayesian Approach to Credibility: This approach formalises credibility by treating the underlying risk parameter as a random variable with a prior distribution. The “credibility premium” is derived as the mean of the posterior distribution (i.e., the Bayesian estimate under squared error loss).
- Derivation: The posterior mean is shown to be a weighted average of the sample mean (MLE) and the prior mean, which directly yields the credibility premium formula.
- Credibility Factor (Z) Examples:
- Poisson/Gamma Model:
Z = n / (n + β)
.n
: Number of observations (direct data).β
: Parameter of the Gamma prior, related to the reliability of the prior information.
- Normal/Normal Model:
Z = (n * σ₀²) / (n * σ₀² + σ²)
orZ = n / (n + σ² / σ₀²)
.n
: Number of observations.σ₀²
: Variance of the prior distribution (reflects uncertainty in prior beliefs).σ²
: Variance of the data within a risk group.
- Binomial/Beta Model:
Z = n / (n + α + β)
.n
: Sample size.α, β
: Parameters of the Beta prior, related to prior reliability.
- Poisson/Gamma Model:
- Explain the empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
- Empirical Bayes Credibility Theory (EBCT): This approach combines observed data with prior expectations to set premiums but, crucially, does not assume a specific parametric form for the prior distribution of the risk parameter (unlike full Bayesian credibility). Instead, the variance components needed for the credibility factor are estimated from the data itself.
- EBCT Model 1:
- Assumes observed claim amounts
X_ij
(for riski
in yearj
) are conditionally independent given a risk parameterθ_i
. - Estimates variance within risk groups
E[s²(θ)]
and variance between risk groupsvar[m(θ)]
from the data. - Credibility Factor (Z):
Z = n / (n + E[s²(θ)] / var[m(θ)])
. - Credibility Premium:
Z * x̄_i + (1 - Z) * x̄
.x̄_i
: Average claims for riski
.x̄
: Overall average claims across all risks and years.
- Assumes observed claim amounts
- EBCT Model 2:
- Extends Model 1 by incorporating a “risk volume” parameter (
P_j
) to account for different sizes of policies or businesses. This acknowledges that the number of observations can vary per risk. - Credibility Factor (
Z_i
): Specific to each riski
and influenced by its risk volume. - Credibility Premium:
Z_i * x̄_i + (1 - Z_i) * x̄
(wherex̄_i
andx̄
are weighted averages by risk volume).
- Extends Model 1 by incorporating a “risk volume” parameter (
- Explain the differences between the two approaches and state the assumptions underlying each of them.
- Key Differences:
- Prior Distribution:
- Bayesian Credibility: Assumes a specific, often parametric, prior distribution for the risk parameter (e.g., Gamma, Normal, Beta).
- Empirical Bayes Credibility: Does not assume a specific prior distribution for the risk parameter. Instead, it estimates the necessary parameters for the credibility factor directly from the observed data.
- Parameter Estimation:
- Bayesian Credibility: Integrates the likelihood with the specified prior to derive the posterior distribution, from which the premium (posterior mean) is calculated.
- Empirical Bayes Credibility: Uses the observed data to estimate the moments (means and variances) of the underlying distributions of the claims, and then uses these empirical estimates in the credibility formula.
- Information Usage:
- Bayesian Credibility: Uses a pre-specified prior and the current sample data.
- Empirical Bayes Credibility: Uses the collective experience from all observed risks/years to estimate variance components, even if a specific prior is unknown, and then combines this with individual risk experience.
- Prior Distribution:
- Underlying Assumptions:
- Bayesian Credibility:
- The risk parameter (e.g.,
λ
,p
,μ
) is a random variable. - A specific parametric form for the prior distribution of the risk parameter is assumed.
- The observations within each risk group are conditionally independent and identically distributed given the risk parameter.
- The risk parameter (e.g.,
- Empirical Bayes Credibility:
- The risk parameter is a random variable, but its specific prior distribution is unknown or not explicitly modelled.
- Observations are conditionally independent given the risk parameter (
X_ij | θ_i
are iid). - The variance of claims within a risk group may depend on the risk parameter (
var(X_j|θ) = s²(θ)
), for Model 1. - For Model 2, different risk volumes (
P_j
) are incorporated, allowing for varying exposure. - No assumption of normality for any random variables or parameters in the model (e.g., EBCT Model 1/2).
- Bayesian Credibility:
- Key Differences:
Keep reviewing these core concepts, and remember, practice makes perfect for these types of derivations and explanations!