Normal-Inverse-Gamma Updating with Unknown Variance

Introduction

When claim costs are modeled as normal but both their average level and their variability are uncertain, an update must account for both sources of uncertainty. A normal-inverse-gamma prior provides a conjugate model: after observing conditionally independent normal costs, the posterior for the unknown mean and variance remains normal-inverse-gamma. The calculation depends on more than the sample average; it also uses the observed spread and the distance between the sample average and the prior mean.

The model is useful when a normal approximation is appropriate for the claim-cost quantity being analyzed. A normal distribution can assign positive probability to negative values, so it should not be used uncritically for raw nonnegative severities when that feature is material. The examples below use hypothetical claim-cost measurements, in thousands of dollars, as a modeling approximation.

Model and Prior Parameterization

Let \(X_1,\ldots,X_n\) be conditionally independent observations given an unknown mean \(\mu\) and variance \(\sigma^2\). The sampling model is \(X_i\mid\mu,\sigma^2\sim N(\mu,\sigma^2)\), where the second parameter is the variance. The prior specifies a normal distribution for \(\mu\) conditional on \(\sigma^2\), and an inverse-gamma distribution for \(\sigma^2\).

The conditional prior variance of \(\mu\) is \(\sigma^2/\kappa_0\); a larger \(\kappa_0\) therefore concentrates the prior mean around \(m_0\), relative to the unknown observation variance. The parameters \(\alpha_0\) and \(\beta_0\) control the prior distribution of \(\sigma^2\). Keep the measurement units consistent: if observations are in thousands of dollars, means are in thousands of dollars and variance-related quantities are in squared thousands of dollars.

Sufficient Statistics and Posterior Update

For the normal sampling model, the data enter the update through the sample size \(n\), the sample mean \(\bar X\), and the within-sample sum of squares \(W=\sum_{i=1}^n(X_i-\bar X)^2\). Equivalently, one can use \(n\), \(\sum_i X_i\), and \(\sum_i X_i^2\), since \(W=\sum_i X_i^2-n\bar X^2\). The posterior parameters are:

The posterior mean parameter \(m_n\) is a weighted average of the prior mean and the sample mean. The weight on the observed sample mean is \(n/(\kappa_0+n)\), while the weight on \(m_0\) is \(\kappa_0/(\kappa_0+n)\). The updated inverse-gamma parameter \(\beta_n\) has two data-related additions: within-sample variation, \(W/2\), and a nonnegative adjustment for disagreement between the sample mean and prior mean.

Why the Update Has These Forms

The likelihood kernel, as a function of \(\mu,\sigma^2\), is proportional to \((\sigma^2)^{-n/2}\exp[-\sum_i(X_i-\mu)^2/(2\sigma^2)]\). The joint prior kernel contributes \((\sigma^2)^{-\alpha_0-3/2}\exp[-\{\beta_0+\kappa_0(\mu-m_0)^2/2\}/\sigma^2]\). Combining the exponents uses the identity

This identity follows by writing \(\sum_i(X_i-\mu)^2=W+n(\bar X-\mu)^2\), then completing the square in \(\mu\). The coefficient of \(\mu^2\) is \(n+\kappa_0\), so the center of the completed square is \(m_n=(n\bar X+\kappa_0m_0)/(n+\kappa_0)\). The remaining constant is \(W+\kappa_0n(\bar X-m_0)^2/(\kappa_0+n)\), which yields the two data terms in \(\beta_n\) after division by 2. The power of \(\sigma^2\) increases the inverse-gamma shape by \(n/2\), giving \(\alpha_n\).

Worked Updates

Interpretation and Exam Traps

The update separates two types of information in the data. The sample mean informs the location parameter, while the within-sample sum of squares informs the variance parameter. The disagreement adjustment also increases the posterior variance scale when the observed center differs from the prior center. This is why a sample with little internal spread can still increase \(\beta_n\) substantially if its average is far from \(m_0\).

Several checks help prevent errors:

• Use the stated inverse-gamma convention. Here the density kernel contains \(\exp(-\beta/\sigma^2)\), so \(\beta\) updates by adding half of a sum of squared deviations.
• Distinguish total squared values from within-sample variation. Use \(W=\sum_iX_i^2-n\bar X^2\), not \(\sum_iX_i^2\) alone.
• Include both components in \(\beta_n\). The within-sample term and prior-data disagreement term answer different questions and neither can generally be dropped.
• Keep units consistent. If costs are recorded in thousands of dollars, \(\beta_0\), \(W\), and \(\beta_n\) use squared-thousands units, while \(m_0\) and \(m_n\) use thousands.
• Respect the model assumptions. The result requires conditionally independent observations with a common normal mean and variance. Dependence, changing variances, or a materially unsuitable normal model requires a different likelihood or model.

For this conjugate model, \(n\), \(\bar X\), and \(W\) contain all the information in the observations needed for the posterior update. The update establishes the posterior distributions for \(\mu\) and \(\sigma^2\); it does not by itself specify a future-observation calculation. This material is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.