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Derive the Beta-Binomial Update and Use It for an Email Campaign

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An email team has run five earlier campaigns with an average click rate of about 4 percent. They encode that as a \text{Beta}(8, 192) prior on the click probability p of the next campaign. The campaign is sent to 500 people and 30 click.

Derive the posterior from Bayes' rule, showing why the Beta prior is conjugate to the Binomial likelihood.
Give the posterior, its mean, and show the mean as a weighted average of the prior mean and the observed rate. What is the weight on the data and what does "worth 200 pseudo-visitors" mean?
A manager asks for the number of clicks to expect when the campaign goes to 1,000 more people. Why is plugging in the posterior mean the wrong calculation, and what is the right distribution?
How do the answers change with a flat \text{Beta}(1,1) prior, and what does that tell you about prior influence?

Solution

1. Derivation. Bayes' rule says posterior is proportional to likelihood times prior. With k clicks in n sends:

p(p \mid k) \propto \underbrace{p^{k}(1-p)^{n-k}}_{\text{Binomial likelihood}} \cdot \underbrace{p^{\alpha-1}(1-p)^{\beta-1}}_{\text{Beta}(\alpha,\beta)\ \text{prior}} = p^{\alpha + k - 1}(1-p)^{\beta + n - k - 1}

The right-hand side has exactly the shape of a Beta density with parameters (\alpha + k,\ \beta + n - k). The binomial coefficient and the Beta function's normalising constant do not depend on p, so they are absorbed into the proportionality constant. Same family in, same family out: that is conjugacy. The update is "add successes to \alpha, add failures to \beta".

2. Posterior. \text{Beta}(8 + 30,\ 192 + 470) = \text{Beta}(38, 662), mean 38/700 = 0.0543, sd 0.0086, 95 percent equal-tailed interval 0.0388 to 0.0722 (verified). As a weighted average:

\frac{38}{700} = \underbrace{\frac{500}{700}}_{0.714}\cdot\underbrace{0.060}_{\text{observed}} + \underbrace{\frac{200}{700}}_{0.286}\cdot\underbrace{0.040}_{\text{prior mean}} = 0.0543

The prior has \alpha + \beta = 200, so it counts as 200 pseudo-sends (8 pseudo-clicks); the data are 500 real sends, so the data carry 500/700 = 71 percent of the weight. The observed 6.0 percent is shrunk to 5.4 percent because five earlier campaigns suggested 4 percent.

3. Predicting 1,000 more sends. Plugging in 0.0543 gives \text{Binomial}(1000, 0.0543) with mean 54.3 and sd 7.2, as if p were known. It is not: the posterior sd of p alone contributes 1000 \times 0.0086 = 8.6 clicks of uncertainty. The right object is the posterior predictive, a Beta-Binomial(1000, 38, 662): mean 54.3, sd 11.2, 95 percent range 34 to 78 clicks (verified). Binomial noise and uncertainty about p add in quadrature (\sqrt{7.2^2 + 8.6^2} \approx 11.2).

4. Flat prior. \text{Beta}(1,1) gives \text{Beta}(31, 471) with mean 0.0618 and interval 0.0424 to 0.0844 (verified): almost the MLE, because the prior is worth 2 pseudo-sends against 500 real ones. The informative prior moves the mean by 0.8 points and narrows the interval. Conclusion: with a few hundred observations, a prior worth 200 sends is a real influence, so state it, justify it from history, and show the flat-prior answer next to it. As n grows the two posteriors converge.

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