Chapter 15 Bayesian Statistics

Learning Objectives

  1. Use Bayes’ theorem to calculate simple conditional probabilities.
  2. Explain what is meant by a prior distribution, a posterior distribution and a conjugate prior distribution.
  3. Derive the posterior distribution for a parameter in simple cases.
  4. Explain what is meant by a loss function.
  5. Use simple loss functions to derive Bayesian estimates of parameters.
  6. Explain what is meant by the credibility premium formula and describe the role played by the credibility factor.
  7. Explain the Bayesian approach to credibility theory and use it to derive credibility premiums in simple cases.
  8. Explain the empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
  9. 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:

  1. 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.
  2. 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).
  3. Derive the posterior distribution for a parameter in simple cases.
    • General Steps (Continuous Prior):
      1. Write down the prior PDF.
      2. Write down the likelihood function (probability/PDF of observed data).
      3. Use the formula: Posterior PDF ∝ Likelihood × Prior PDF.
      4. Identify the posterior distribution by its form (e.g., Gamma for λ, Beta for p, Normal for μ).
    • General Steps (Discrete Prior):
      1. The prior and posterior distributions have the same support.
      2. 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.
  4. 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.
  5. 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.
    • Absolute Error Loss:
      • Formula: Loss = |θ̂ - θ|.
      • Bayesian Estimate: The median of the posterior distribution.
    • 0/1 (All-or-Nothing) Loss:
      • Formula: Loss = 0 if θ̂ = θ, Loss = 1 if θ̂ ≠ θ.
      • Bayesian Estimate: The mode of the posterior distribution (the value that maximises the posterior PDF).
  6. 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.
      • : 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 () 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).
  7. 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) or Z = 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.
  8. 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 risk i in year j) are conditionally independent given a risk parameter θ_i.
      • Estimates variance within risk groups E[s²(θ)] and variance between risk groups var[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 risk i.
        • : Overall average claims across all risks and years.
    • 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 risk i and influenced by its risk volume.
      • Credibility Premium: Z_i * x̄_i + (1 - Z_i) * x̄ (where x̄_i and are weighted averages by risk volume).
  9. 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.
    • 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.
      • 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).

Keep reviewing these core concepts, and remember, practice makes perfect for these types of derivations and explanations!

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