Track 16 · Advanced Foundations · Lesson 2 of 8

Beta–binomial updating

Update a Beta prior for a proportion with binomial data, interpret the posterior, check its sensitivity to the prior, and predict future outcomes with the posterior predictive distribution.

40 minAdvanced8 graded exercisesMembers

A clinic wants to know its no-show rate for a new appointment type. After the first 20 bookings, 7 people did not turn up — but its own records for similar services show no-show rates of around 20% (an illustrative scenario). The Beta–binomial model combines both in two additions.

A proportion $\theta$ (a conversion rate, a no-show rate, a defect rate) lies between 0 and 1. The Beta distribution is a flexible family on that interval, with two positive parameters $a$ and $b$ . It can be flat, peaked, skewed or U-shaped, so it can express most prior beliefs about a rate.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Express a belief about a proportion as a Beta prior (mean and effective sample size)
Update a Beta prior with binomial data by adding successes and failures
Explain the posterior mean as a weighted compromise between prior and data
Show that sequential and batch updating agree
Check prior sensitivity and use the posterior predictive for future outcomes

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Formula sheet

The Beta distribution
$\theta \sim \text{Beta}(a, b):\quad p(\theta) \propto \theta^{a-1}(1-\theta)^{b-1},\qquad \E[\theta] = \frac{a}{a+b},\qquad \SD[\theta] = \sqrt{\frac{ab}{(a+b)^2(a+b+1)}}$
Conjugate updating: add successes to a, failures to b
$\text{Beta}(a, b) \;+\; k \text{ successes in } n \text{ trials} \;\longrightarrow\; \text{Beta}(a + k,\; b + n - k),\qquad \E[\theta \given \text{data}] = \frac{a + k}{a + b + n} = w\,\frac{k}{n} + (1 - w)\,\frac{a}{a + b},\quad w = \frac{n}{a + b + n}$
Beta–binomial posterior predictive distribution
$P(K = j) = \binom{m}{j}\frac{B(a' + j,\; b' + m - j)}{B(a', b')},\qquad j = 0, 1, \ldots, m$

Reference

Beta–binomial updating
With a Beta(a, b) prior for a success probability and s successes and f failures, the posterior is Beta(a + s, b + f) — the prior acts like a + b earlier observations.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.