After this page you can
Update a gamma prior with Poisson claim data, read off the posterior mean, and show it is a credibility weighted average of prior and data
The change of viewpoint
Everything in CS1 up to this point treats the parameter as a fixed unknown constant and the data as random. That is what makes a confidence interval a statement about a procedure rather than about a parameter.
Bayesian statistics turns this around. The parameter is treated as a random quantity, with a distribution describing what is believed about it; the data, once observed, are simply fixed facts. The question stops being "how would this estimate behave over repeated samples?" and becomes "given what I knew before, and what I have now seen, what should I believe?"
This is not a philosophical digression in an actuarial syllabus. It is the formal justification for credibility theory, which is how an insurer prices a risk that has too little of its own experience to speak for itself — and that is most risks.
The machinery
Three objects and one theorem.
The prior, π(θ). What is believed about the parameter before this data arrived. It might come from industry experience, from a larger related portfolio, or from previous years.
The likelihood, f(x | θ). The same function as in maximum likelihood: how probable the observed data are for each value of the parameter.
The posterior, π(θ | x). What should be believed afterwards. Bayes' theorem gives it:
In words: multiply what the data say by what you believed, and rescale so it is a distribution again.
The denominator does not involve θ at all — it is only the constant that makes the result integrate to one. This is why nearly all Bayesian work in the syllabus is done with the proportionality sign: get the shape right in θ, recognise the distribution, and the constant comes free.
Turning a posterior into a single number
The posterior is a whole distribution, which is more than an estimate. Reducing it to one number requires saying what it costs to be wrong, and the loss function decides the answer:
| Loss function | Estimate |
|---|---|
| Quadratic, (θ̂ − θ)² | posterior mean |
| Absolute, |θ̂ − θ| | posterior median |
| All-or-nothing | posterior mode |
Quadratic loss is the default and the one actuarial work almost always uses, so the Bayesian estimate is normally the posterior mean. It is worth knowing why the other two exist: squared error punishes large misses disproportionately, absolute error does not, and which is appropriate depends on whether a reserve being out by twice as much is twice as bad or four times as bad.
Conjugate priors
Multiplying a prior by a likelihood usually produces something with no name, which then has to be integrated numerically. A conjugate prior is one chosen so that the posterior belongs to the same family as the prior, and the entire update becomes a matter of adjusting parameters.
| Likelihood | Conjugate prior | Posterior |
|---|---|---|
| Poisson (rate λ) | Gamma(α, β) | Gamma(α + ∑x, β + n) |
| Binomial (probability p) | Beta(a, b) | Beta(a + successes, b + failures) |
| Normal, known variance | Normal | Normal |
| Exponential (rate λ) | Gamma(α, β) | Gamma(α + n, β + ∑x) |
Look at the first row and notice how little work the update is. Add the observed claims to the shape parameter, add the exposure to the rate parameter, and you are finished. There is no integral to do.
That also makes the prior interpretable in units of data. A Gamma(α, β) prior for a Poisson rate behaves exactly like having already seen α claims in β units of exposure — so stating a prior is the same as stating how much pretend experience your beliefs are worth. A prior you cannot express that way is probably a prior you cannot defend.
The credibility formula, which is already here
This is the result that makes the topic actuarial rather than merely statistical. Work out the posterior mean for the gamma-Poisson pair:
and it rearranges, exactly and with no approximation, into
In words: the answer is a weighted average of what the data say and what you believed before, and the weight on the data is decided by how much data there is.
That Z is the credibility factor, and every property credibility theory asks of it is visible in the formula. It lies between 0 and 1. It rises towards 1 as n grows, so a large portfolio is eventually trusted on its own evidence. It is near 0 when n is small, so a thin portfolio is held close to the prior. And β sets how fast the transition happens — a confident prior resists the data longer.
No new theory is needed to get credibility: it falls out of Bayes' theorem with a conjugate prior. That is also why the posterior mean is sometimes called the Bayesian credibility premium.
Questions students ask
Is it not unscientific to put beliefs into the calculation?
The objection assumes the alternative is belief-free, and it is not: choosing a model, a distribution and a significance level are all judgements, simply made less visibly. The Bayesian approach puts one of them in the open where it can be stated, criticised and tested by trying a different prior. In insurance the prior is rarely a personal opinion anyway — it is industry data, or last year's portfolio, which is information it would be careless to discard.
What happens if the prior is badly wrong?
Data win in the end. As n grows, Z goes to 1 and the posterior mean converges on the data mean, which is the maximum likelihood estimate. A bad prior costs accuracy when evidence is thin and becomes irrelevant when it is plentiful. The danger case is a prior placing zero probability on the true value, because multiplying by zero leaves zero whatever the data say.
How does this connect to empirical Bayes credibility?
Through the one thing this page assumed and did not justify: where α and β came from. Empirical Bayes credibility theory estimates the prior's parameters from the data of the whole collective rather than asserting them, and then applies the same weighted-average formula to each risk within it. The structure on this page is the structure there; what changes is that the prior is fitted instead of stated.
A worked example
An insurer is setting a claim frequency for a small scheme. Industry experience suggests a rate of about 0.08 claims per policy-year, with enough uncertainty that a rate anywhere between roughly 0.02 and 0.16 would not be surprising. This is expressed as a Gamma(α = 4, β = 50) prior. The scheme itself then reports 22 claims from 200 policy-years. All of these figures were invented for this example.
Reading the prior before using it
prior mean = α/β = 4/50 = 0.08
prior variance = α/β² = 4/2500 = 0.0016
prior standard deviation = 0.04A mean of 0.08 with a standard deviation of 0.04 is a genuinely vague prior — the uncertainty is half the mean. In the units-of-experience reading, Gamma(4, 50) is the same as having already seen 4 claims in 50 policy-years. Against the scheme's own 200 policy-years, that is modest, and the arithmetic below will show exactly how modest.
The update
Poisson data with a gamma prior is conjugate, so the posterior is gamma with the shape increased by the claims observed and the rate increased by the exposure observed:
posterior = Gamma(α + ∑x, β + n) = Gamma(4 + 22, 50 + 200) = Gamma(26, 250)Under quadratic loss the estimate is the posterior mean:
λ̂ = 26/250 = 0.104 claims per policy-yearCheck it by the credibility route
A completely different-looking calculation must give the same number. The scheme's own data say
x̄ = 22/200 = 0.11and the credibility factor is
Z = n/(n + β) = 200/(200 + 50) = 200/250 = 0.8So the data carry 80% of the weight and the prior 20%:
0.8 × 0.11 + 0.2 × 0.08 = 0.088 + 0.016 = 0.104Identical to the posterior mean, exactly rather than approximately. The two routes are the same formula written differently, and the agreement confirms both the update and the credibility factor.
Note what the answer has done. The scheme's own experience says 0.11; prior belief said 0.08; the estimate is 0.104, pulled about a fifth of the way back towards the prior. The scheme is mostly, but not entirely, trusted on its own evidence.
The rest of the posterior
posterior variance = 26/250² = 26/62,500 = 0.000416
posterior standard deviation = 0.0203961
posterior mode = (26 − 1)/250 = 0.100The standard deviation has fallen from 0.04 to 0.0204 — the data have roughly halved the uncertainty. The mode is 0.100 against a mean of 0.104, the gap being the right skew of the gamma; quoting the mode when quadratic loss was specified would be the wrong answer to the question asked.
On 200 policies, the expected number of claims next year is
0.104 × 200 = 20.8.How the weight moves with the data
Repeat the calculation with the same claim rate of 0.11 but different amounts of exposure, prior unchanged:
Policy-years Z Estimate 50 0.5000 0.0950 200 0.8000 0.1040 1,000 0.9524 0.1086 At 50 policy-years — exactly the β of the prior — the weight is precisely one half, which is what
Z = n/(n + β)must give whenn = β, and a useful check on the arithmetic. At 1,000 the estimate has moved to within 0.0014 of the scheme's own 0.11, and with enough data it would converge on it entirely. That convergence is the Bayesian answer to the objection that the prior has undue influence: it has influence exactly in proportion to how little the data have to say.
Where this fits in CS1
This page explains one idea from CS1 Actuarial Statistics. The full CS1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CS1 topics explained
- Moment generating functions, and how to get moments out of them
- The central limit theorem, and when the normal approximation is safe
- Maximum likelihood estimation, step by step
- Confidence intervals: what the 95% actually means
- Hypothesis testing and p-values, without the folklore
- Simple linear regression: least squares, fitted by hand
- Generalised linear models: the three parts, and the link function
- Conditional expectation, and the variance decomposition that comes with it
- Sampling distributions: what the sample mean and sample variance actually do
- The method of moments, and when maximum likelihood is worth the extra work
- The likelihood ratio test, and why the Wald and score tests exist too
- The chi-square goodness of fit test, and how to get its degrees of freedom right
- Contingency tables: testing whether two classifications are related
- Analysis of variance: comparing three or more means at once
- Multiple linear regression, and what controlling for another variable really does
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.