Gamma Severity Models with Gamma Rate Priors

Introduction

A gamma severity model can describe positive claim amounts with a shape that is treated as known and a rate that varies across risks or portfolios. Bayesian updating then combines a prior distribution for that rate with observed severities. The essential data summaries are the number of observed claims and their total severity.

The update is straightforward once the gamma parameterization is explicit. A frequent source of errors is to use a scale where a rate is intended, or to update the prior parameters using the wrong data summary. This tutorial derives the update and works through both issues.

Definition: Use the shape-rate parameterization throughout. A random variable $Z\sim\operatorname{Gamma}(a,b)$, with shape $a>0$ and rate $b>0$, has density $$ f_Z(z)=\frac{b^a}{\Gamma(a)}z^{a-1}e^{-bz},\qquad z>0. $$ Its mean is $a/b$, and its variance is $a/b^2$. In particular, the second parameter is a rate, not a scale.

Severity Model and Prior

Let $X_1,\ldots,X_n$ be independent, fully observed claim severities, measured in a fixed monetary unit. Suppose each severity, conditional on an unknown rate $\theta$, follows a gamma distribution with known shape $k>0$:

X_i\mid\theta\sim\operatorname{Gamma}(k,\theta),\qquad f_{X_i\mid\theta}(x_i\mid\theta) =\frac{\theta^k}{\Gamma(k)}x_i^{k-1}e^{-\theta x_i},\quad x_i>0.

Here $\theta$ is a rate per unit of severity. The shared rate describes the severity distribution for all the observations in this model; conditional on that rate, the severities are independent. Assign $\theta$ a gamma prior with shape $a>0$ and rate $b>0$:

\theta\sim\operatorname{Gamma}(a,b),\qquad \pi(\theta)=\frac{b^a}{\Gamma(a)}\theta^{a-1}e^{-b\theta},\quad \theta>0.

Write $T=\sum_{i=1}^n X_i$ for the total observed severity. Conditional on the number $n$ of observed severities, the likelihood for $\theta$ depends on the data through $n$ and $T$. The claim count is the number of severity observations in this likelihood; it is not, by itself, an additional frequency-model observation about $\theta$.

Theorem: Under the independent gamma severity model with known shape $k$, and a $\operatorname{Gamma}(a,b)$ prior on the rate $\theta$, the posterior distribution given $n$ fully observed severities with total $T$ is $$ \theta\mid X_1,\ldots,X_n\sim\operatorname{Gamma}(a+nk,\ b+T). $$ The first posterior parameter is the shape and the second is the rate.

Why the Update Has This Form

The result follows by multiplying the prior density by the likelihood and identifying the resulting gamma kernel. This also shows exactly which parts of the observations affect the posterior.

Write the likelihood.
Independence gives $L(\theta)=\prod_{i=1}^n f_{X_i\mid\theta}(x_i\mid\theta)$. Factors that do not involve $\theta$ can be collected separately, leaving a likelihood proportional to $\theta^{nk}e^{-\theta T}$.

Multiply by the prior.
The prior contributes $\theta^{a-1}e^{-b\theta}$. Therefore the unnormalized posterior is proportional to $\theta^{a+nk-1}e^{-(b+T)\theta}$.

Normalize the kernel.
This is the gamma density kernel with shape $a+nk$ and rate $b+T$. Its normalizing constant is $(b+T)^{a+nk}/\Gamma(a+nk)$, giving the stated posterior.

More explicitly, the likelihood and prior product are

L(\theta) =\frac{\theta^{nk}}{\Gamma(k)^n} \left(\prod_{i=1}^n x_i^{k-1}\right)e^{-\theta T}, \qquad L(\theta)\pi(\theta)\ \propto\ \theta^{a+nk-1}e^{-(b+T)\theta}.

The product $\prod x_i^{k-1}$ and the gamma-function constants do not depend on $\theta$, so they do not change the posterior shape or rate. The factor $\theta^{nk}$ explains why the claim count contributes $nk$ to the posterior shape; the exponential factor $e^{-\theta T}$ explains why total severity contributes $T$ to the posterior rate.

Worked Updates

Worked Example: Four Severities in Thousands of Dollars

A hypothetical portfolio has severity model $X_i\mid\theta\sim\operatorname{Gamma}(2,\theta)$, where losses are measured in thousands of dollars. The prior is $\theta\sim\operatorname{Gamma}(3,0.5)$. Four fully observed losses are $1,2,3,$ and $4$ thousand dollars. Find the posterior distribution and its mean. State the method and a common mistake it avoids.

Method. Apply the gamma-rate conjugate update using the known severity shape, number of severities, and total severity.

Step 1: Calculate the sample size and total. There are $n=4$ severities, and $T=1+2+3+4=10$ thousand dollars.

Step 2: Update the shape. The prior shape is $a=3$, and the severity shape is $k=2$. Thus $a+nk=3+4(2)=11$.

Step 3: Update the rate. The prior rate is $b=0.5$, so $b+T=0.5+10=10.5$, with the severity amounts measured in thousands.

Therefore $\theta\mid X_1,\ldots,X_4\sim\operatorname{Gamma}(11,10.5)$. Its posterior mean is $11/10.5=1.0476$ per thousand-dollar unit, to four decimal places. The posterior shape and rate are both positive, as required.

Common mistake avoided: Adding the four claims only to the prior shape would give $3+4=7$, which ignores the known severity shape $k=2$. The likelihood contributes $nk=8$, not merely $n=4$, to the shape.

Worked Example: Noninteger Severity Shape

For a hypothetical coverage, severities measured in hundreds of dollars have conditional distribution $\operatorname{Gamma}(1.5,\theta)$. The prior on the rate is $\operatorname{Gamma}(2,4)$. Three observed severities are $2,5,$ and $7$ hundred dollars. Find the posterior distribution and mean rate.

Method. Use the same conjugate update; the known shape need not be an integer.

Step 1: Summarize the data. The sample size is $n=3$, and the total is $T=2+5+7=14$ hundred dollars.

Step 2: Calculate the posterior shape. The prior shape is $a=2$, and $nk=3(1.5)=4.5$, giving $a+nk=6.5$.

Step 3: Calculate the posterior rate. The prior rate is $b=4$, so $b+T=4+14=18$, in the units implied by measuring severities in hundreds of dollars.

Thus $\theta\mid X_1,X_2,X_3\sim\operatorname{Gamma}(6.5,18)$, and its posterior mean is $6.5/18=0.3611$ per hundred-dollar unit, to four decimal places. The calculation uses $k=1.5$ for each claim, so the shape increment is $4.5$, even though it is not an integer.

Common mistake avoided: Rounding $k$ to 2 before updating would produce a different and unjustified posterior. The stated known shape is used as given.

Worked Example: Updating From Count and Total Only

A hypothetical severity model, with losses measured in hundreds of dollars, has known shape $k=1.5$ and prior $\theta\sim\operatorname{Gamma}(4,2)$. Five fully observed losses are $0.5,1,2,3,$ and $6$ hundred dollars. Find the posterior distribution and its mean. Explain whether the ordering of the losses matters.

Method. Reduce the data to $n$ and $T$, then apply the conjugate update.

Step 1: Check the data summary. There are $n=5$ losses, and $T=0.5+1+2+3+6=12.5$ hundred dollars.

Step 2: Update the shape. The shape increment is $nk=5(1.5)=7.5$. Adding the prior shape gives $4+7.5=11.5$.

Step 3: Update the rate. Add the observed total to the prior rate: $2+12.5=14.5$.

The posterior is $\theta\mid X_1,\ldots,X_5\sim\operatorname{Gamma}(11.5,14.5)$, with mean $11.5/14.5=0.7931$ per hundred-dollar unit, to four decimal places. The ordering does not matter: the likelihood contains the severities through their product, which is free of $\theta$, and their sum $T$, which determines the posterior rate. The posterior parameters are consistent with the positive prior parameters and positive total severity.

Common mistake avoided: Using the largest loss, or the average loss in place of the total, would not give the required likelihood update. Here the rate increment is the total $12.5$, not the mean $12.5/5=2.5$.

Interpretation, Conditions, and Exam Traps

The posterior shape and rate collect different contributions. The prior shape $a$ is increased by $nk$, while the prior rate $b$ is increased by the total $T$. The data therefore enter through two specific summaries, not through an arbitrary statistic such as the largest claim or the sample average alone.

The posterior mean rate, when wanted, is $(a+nk)/(b+T)$. This is a summary of uncertainty about the severity parameter $\theta$; it is not itself a claim severity mean. Conditional on $\theta$, the gamma severity mean is $k/\theta$, so substituting the posterior mean rate into that expression is not generally the same operation as taking a posterior expectation of $k/\theta$.

The conjugate result assumes that $k$ is known, the same $\theta$ applies to all observations, and the observed severities are conditionally independent, complete, and drawn from the stated gamma model. A truncated sample, censored severity, or a model in which the shape is also unknown has a different likelihood and does not justify this update without modification.

If no severities are observed, $n=0$ and $T=0$, so the severity data leave the prior unchanged. This does not mean that an observed absence of claims can never affect a Bayesian analysis: it means only that this severity likelihood, conditional on the observed severities, contains no severity observations to update $\theta$. A frequency model that links claim counts to a parameter would be a separate source of information.

Common exam trap: The posterior rate is $b+T$, not $b+n$, and the posterior shape is $a+nk$, not $a+n$. Also, for this shape-rate convention, the gamma density has $e^{-\theta x}$; a scale-parameter formula must not be substituted without converting parameterizations.

Before finalizing an update, check the parameterization, the units used for every severity, the number of observed severities, and their total. In particular, if the data are recorded in dollars but the model and prior use thousands of dollars, convert the observations before adding their total to the prior rate.

Check Your Understanding

For each problem, give the posterior distribution of the severity rate using the shape-rate convention.

A known-shape $k=2$ severity model has prior $\operatorname{Gamma}(2,3)$. Three observed severities total $6$. What is the posterior? (Answer: $\operatorname{Gamma}(8,9)$)
A severity model with known shape $k=1.5$ has prior $\operatorname{Gamma}(4,2)$. Four observed losses total $8$. What is the posterior? (Answer: $\operatorname{Gamma}(10,10)$)
A known-shape $k=3$ model has prior $\operatorname{Gamma}(1.5,0.25)$. Three losses are $1,2,$ and $4$ in the model's severity units. What is the posterior? (Answer: $\operatorname{Gamma}(10.5,7.25)$)
For known shape $k=2$, the prior is $\operatorname{Gamma}(3,1)$. Dataset A has two severities $1$ and $5$; dataset B has two severities $2$ and $4$. Do they give the same posterior? (Answer: Yes; both give $\operatorname{Gamma}(7,7)$, since each has $n=2$ and $T=6$.)
A severity model has known shape $k=2$ and prior $\operatorname{Gamma}(5,2)$. No severity observations are available. What posterior follows from the severity likelihood alone? (Answer: $\operatorname{Gamma}(5,2)$; with $n=0$ and $T=0$, the prior is unchanged.)

What You'll Learn