Starting from a Beta(1,1) prior, what posterior follows from 8 clicks in 100 impressions?

## What Beta(1, 1) is The `Beta(a, b)` density is proportional to `theta^(a-1) * (1-theta)^(b-1)` on the interval from 0 to 1. Setting `a = b = 1` makes both exponents zero, so the density is constant: `Beta(1, 1)` is the uniform distribution on `(0, 1)`. It says every click-through rate between 0 and 1 is equally plausible before you see data. In pseudo-count language it is the weakest prior that still carries counts: one imagined success and one imagined failure, an effective prior sample size of 2. ## The update With a binomial likelihood, the conjugate update adds successes to the first parameter and failures to the second. Eight clicks in 100 impressions means `s = 8` successes and `n - s = 92` failures, so `Beta(1 + 8, 1 + 92) = Beta(9, 93)` Two numbers now summarise everything you know about the rate. ## Reading the posterior **Mean.** The mean of `Beta(a, b)` is `a / (a + b)`, here `9 / 102 = 0.0882`. The raw click-through rate is `8 / 100 = 0.08`. The posterior mean sits slightly higher because the prior's two pseudo-observations are split evenly, dragging the estimate a hair toward one half. With 100 real observations against 2 pseudo-observations, that pull is tiny — as it should be. **Mode.** The mode of `Beta(a, b)` for `a > 1, b > 1` is `(a - 1) / (a + b - 2)`, here `8 / 100 = 0.08`, exactly the maximum-likelihood estimate. This is a useful thing to know: a flat prior does not move the peak of the posterior at all, because multiplying the likelihood by a constant cannot shift where it is maximised. It moves the *mean*, because the posterior is a skewed distribution and the flat prior fattens the upper tail. **Spread.** The posterior is far from a point estimate. `Beta(9, 93)` puts most of its mass roughly between 0.04 and 0.15; 100 impressions is simply not much data for an 8% rate. The number of *successes*, not the number of trials, drives the precision here, which is why low-rate metrics need much more traffic than people expect. ## Why the count interpretation is the whole trick Because the update is addition, you never need the dataset again — only the running totals. If tomorrow brings 3 clicks in 60 impressions, you update `Beta(9, 93)` to `Beta(12, 150)` and you are done. The posterior becomes the next prior, and the arithmetic is identical whether the data arrives all at once or in dribs. ## The zero-success case The same machinery handles the case that breaks naive estimation. Suppose a new creative gets 0 clicks in 50 impressions. The raw rate is `0 / 50 = 0`, which asserts the rate is exactly zero — a claim 50 impressions cannot support. With a `Beta(1, 1)` prior the posterior is `Beta(1, 51)`, mean `1 / 52 = 0.019`. In general the posterior mean under a flat prior is `(s + 1) / (n + 2)`, a result old enough to have its own name: Laplace's rule of succession, and the same idea appears in classification as add-one smoothing. It never returns exactly 0 or exactly 1, which is precisely what you want when a downstream step takes a logarithm, divides by the estimate, or ranks items by it. ## Common mistakes - Writing `Beta(9, 101)`: adding the number of trials to the second parameter instead of the number of failures. - Writing `Beta(8, 92)`: dropping the prior's pseudo-counts. - Calling `9 / 102` the probability that the rate is 0.088. It is the posterior *mean*; the rate is a continuous quantity, so the probability of any single exact value is zero. - Treating `Beta(1, 1)` as "no prior". It is a genuine assumption — that every rate from 0 to 1 is equally likely, which for a click-through rate is a fairly strange belief, since rates above 0.5 are basically unheard of. It is weak, not absent. ## What to report Hand back the pair `(9, 93)` alongside the summary you care about. The two parameters are the complete state, the mean or mode is the headline number, and the width of the distribution is the honest statement about how little 100 impressions tells you.