IAI Actuarial Core Principles · Actuarial Statistics

Bayesian Inference: Priors, Posteriors, Loss Functions and Credible Intervals

Bayesian inference treats a parameter as random. You choose a prior, combine it with the data's likelihood, and get a posterior: posterior ∝ likelihood × prior. You then pick an estimate that minimises expected loss, such as the posterior mean under quadratic loss, and quote a credible interval from the posterior.

What this chapter covers

This chapter covers Bayesian statistics, which sits in CS1 Actuarial Statistics. The 2026 syllabus gives Bayesian statistics 15% of CS1. The idea is simple. Classical inference treats the parameter θ as a fixed unknown. Bayesian inference treats θ as a random variable with a distribution that reflects what you know before and after seeing data.

The workflow has five steps. Start with Bayes' theorem in density form. Choose a prior f(θ). Multiply by the likelihood f(x | θ) to get the posterior f(θ | x) ∝ f(x | θ) × f(θ). Then choose a point estimate by minimising expected loss, and finally build a credible interval from the posterior. Conjugate priors, such as Beta for a binomial likelihood or Gamma for a Poisson likelihood, keep the algebra manageable.

The chapter connects to the rest of the paper through likelihood and distributions, which you need for the posterior. It also links to estimation and confidence intervals, which you will compare with credible intervals. Beyond CS1, the same ideas return in credibility theory and in risk modelling, where parameters are treated as random.

Bayesian questions are often short and mechanical once you know the pattern, so they are a reliable source of marks. A typical question asks you to find a posterior, state its mean, and then say which loss function gives that estimate. Written questions reward clear working: the kernel, the recognised distribution, the estimate and a sentence of interpretation. Because the algebra is repeated across Beta-binomial, Gamma-Poisson and Normal-Normal cases, a few hours of practice give you a method you can reuse. Candidates who skip this chapter lose marks that are easy to win elsewhere.

Bayesian inference: priors, posteriors, loss functions and credible intervals: topics in the order to study them

1Bayes' Theorem and Bayesian FrameworkEverything else builds on posterior ∝ likelihood × prior, so learn the framework and the vocabulary first.
2Prior Distributions and Conjugate PriorsYou need to know how priors are chosen, and which prior pairs with which likelihood, before you can compute a posterior quickly.
3Posterior Distributions and Bayesian EstimatorsHere you apply the framework with conjugate priors, find the posterior and read off its mean, mode or median.
4Loss Functions and Bayes EstimatorsThis explains why a particular estimate is chosen, and it only makes sense once you can find a posterior.
5Bayesian Credible IntervalsIntervals come last because they use the full posterior and are compared with the confidence intervals you already know.

How to prepare Bayesian inference: priors, posteriors, loss functions and credible intervals

Treat this chapter as one repeating method. Learn the method, then drill it on different distributions until the steps are automatic.

Write out Bayes' theorem for densities and say in words what each part means. Practise the line: posterior ∝ likelihood × prior.
Learn the kernel trick. Drop every factor that does not contain θ, then recognise the remaining shape as a known distribution. This saves you from computing the normalising constant.
Memorise the main conjugate pairs: Beta prior with binomial data, Gamma prior with Poisson data, and Normal prior with Normal data. For each, derive the posterior parameters once by hand, then practise using them.
Link each loss function to its Bayes estimate: quadratic loss gives the posterior mean, absolute loss gives the posterior median, and all-or-nothing (0-1) loss gives the posterior mode.
Practise credible intervals from the posterior using tables or R. Write one sentence interpreting the interval in terms of the parameter.
Do past-paper style questions under time. Write the likelihood, the kernel, the named posterior, the estimate and a conclusion every time, so you pick up method marks even if arithmetic slips.
In the computer-based paper, practise simulating or evaluating posteriors in R, such as quantiles of a Beta or Gamma distribution for credible intervals.

Common mistakes in Bayesian inference: priors, posteriors, loss functions and credible intervals

Including factors that do not depend on θ when finding the posterior, or forgetting that the proportionality constant is lost.
Fix: Drop everything free of θ, identify the distribution from the remaining form, and let the known distribution supply the constant.
Matching the wrong loss function to the estimate, for example giving the mean under absolute error loss.
Fix: Remember the pairs: quadratic gives mean, absolute gives median, 0-1 gives mode. Write the loss function and the estimate together in your answer.
Updating the parameters of a conjugate posterior incorrectly, such as using n instead of Σxᵢ for a Gamma-Poisson update.
Fix: Derive each update once from the likelihood and prior. Check by asking what a larger sample should do to the posterior.
Interpreting a credible interval as a confidence interval, or the reverse.
Fix: For a credible interval, state that the posterior probability that θ lies in the interval is the stated level. A confidence interval is about repeated sampling.
Using the wrong parameterisation of the Gamma or other distribution when reading mean and variance.
Fix: Check the parameterisation in the question or the Tables. For a Gamma with rate λ, the mean is α ÷ λ.
Giving a numerical answer with no statement of the posterior distribution or the assumptions.
Fix: Always write the likelihood, the posterior kernel, the named posterior with its parameters, then the estimate and a one-line conclusion.

Last-day revision: Bayesian inference: priors, posteriors, loss functions and credible intervals

Bayes for parameters: f(θ | x) = f(x | θ) f(θ) ÷ ∫ f(x | θ) f(θ) dθ, so posterior ∝ likelihood × prior.
In Bayesian inference, θ is random and the data x are fixed once observed.
Only terms containing θ matter when you find the posterior kernel.
A conjugate prior gives a posterior in the same family as the prior.
Beta(α, β) prior with Binomial(n, θ) data and x successes gives a Beta(α + x, β + n − x) posterior.
Gamma(α, λ) prior with Poisson data x₁,…,xₙ gives a Gamma(α + Σxᵢ, λ + n) posterior.
Quadratic loss gives the posterior mean as the Bayes estimate.
Absolute error loss gives the posterior median; 0-1 loss gives the posterior mode.
The Bayes estimate minimises the expected posterior loss.
A 95% credible interval is a range with posterior probability 0.95 that contains θ.
A credible interval is not the same as a frequentist confidence interval, and the interpretations differ.
A posterior mean can be written as a weighted average of the prior mean and the data estimate, with weights that shift towards the data as n grows.

Bayesian inference: priors, posteriors, loss functions and credible intervals practice questions

Bayesian inference: priors, posteriors, loss functions and credible intervals in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Bayesian inference: priors, posteriors, loss functions and credible intervals: frequently asked questions

What is the difference between a prior and a posterior?

The prior is your distribution for the parameter before you see the data. The posterior is the updated distribution after you combine the prior with the likelihood of the observed data, using posterior ∝ likelihood × prior.

Why do we use conjugate priors?

A conjugate prior gives a posterior in the same family, so you can write the posterior by updating parameters instead of doing a difficult integral. In the exam this saves time and reduces errors.

Which loss function gives the posterior mean?

Quadratic (squared error) loss. The Bayes estimate under this loss is the posterior mean. Absolute error loss gives the posterior median, and all-or-nothing loss gives the posterior mode.

How is a credible interval different from a confidence interval?

A credible interval is taken from the posterior, so you can say the parameter lies in it with the stated posterior probability. A confidence interval describes the long-run behaviour of the method over repeated samples, not the probability for one fixed interval.

Do I need R for this chapter?

It helps for the computer-based paper. You can use R to find quantiles of the posterior for credible intervals and to check posterior means and variances. For the written paper, you still need to show the algebra by hand.