How does a Gamma prior on a support-ticket rate update after observing daily counts?

## The model Counts of events in a fixed window — support tickets per day, crashes per hour, defects per batch — are commonly modelled as Poisson with an unknown rate lambda. Observing `y_1, ..., y_n` counts over `n` equal-length periods gives a likelihood proportional to `lambda^(sum of y_i) * exp(-n * lambda)` Only two numbers from the data appear: the total count and the number of periods. Those are the sufficient statistics, and their appearance is the reason a conjugate update exists. ## The conjugate prior The Gamma family on positive lambda has density proportional to `lambda^(alpha - 1) * exp(-beta * lambda)` in the **shape-and-rate** parametrisation, where `alpha` is the shape and `beta` is the rate. Its mean is `alpha / beta` and its variance is `alpha / beta^2`. Multiplying prior by likelihood: `lambda^(alpha - 1) * exp(-beta * lambda) * lambda^y * exp(-n * lambda) = lambda^(alpha + y - 1) * exp(-(beta + n) * lambda)` where `y` is the total count. That is the kernel of `Gamma(alpha + y, beta + n)`. The shape absorbs events; the rate parameter absorbs exposure. ## Pseudo-events over pseudo-days The prior mean `alpha / beta` has the form of a count divided by an exposure, and the posterior mean `(alpha + y) / (beta + n)` is literally total events over total exposure once you count the imagined ones. So a Gamma prior is fully described by the sentence "I have already seen alpha tickets in beta days". Work an example. You expect about 5 tickets a day, and you would trust that about as much as four days of observation: `alpha = 20`, `beta = 4`, giving `Gamma(20, 4)` with prior mean 5. Ten real days bring 30 tickets. The posterior is `Gamma(20 + 30, 4 + 10) = Gamma(50, 14)`, mean `50 / 14 = 3.57`. The observed rate is `30 / 10 = 3.0`. The posterior sits between 5 and 3, nearer the data because ten days of real exposure outweigh four days of imagined exposure. As exposure accumulates, `beta` becomes negligible against `n` and the posterior mean converges on the observed rate. The prior's influence is measured in days, which makes it unusually easy to defend or attack in a design review: "this prior is worth four days of traffic" is a claim a stakeholder can actually argue with. ## Why the parametrisation matters Gamma is written two ways. Shape-and-rate uses `exp(-beta * lambda)` with mean `alpha / beta`; shape-and-scale uses `exp(-lambda / scale)` with mean `alpha * scale`. The conjugate update above is stated in the rate form. In the scale form the same update reads `scale_new = 1 / (1 / scale_old + n)`, which is far easier to get wrong. Interviewers do notice when a candidate writes the rule without saying which convention they are in — announcing it first is a cheap signal of care. ## Uneven exposure Days are rarely comparable. If period `i` has exposure `e_i` — hours open, active users, machine-hours — model the count as Poisson with mean `lambda * e_i`. The likelihood becomes proportional to `lambda^(sum y_i) * exp(-lambda * sum e_i)`, so the update is `Gamma(alpha + sum of y_i, beta + sum of e_i)`. Exposure, not the number of rows, goes into the second parameter. Forgetting this is the most common real-world error with this pair: someone adds the number of days when the days were half-length. ## Assumptions worth naming The Poisson likelihood asserts that events arrive independently at a constant rate within the window, which implies the variance equals the mean. Support tickets often violate this: an incident produces a burst, so the observed variance exceeds the mean — overdispersion. The Gamma-Poisson pair still updates cleanly, but the posterior will be too narrow, because the model has been told the data is less variable than it is. The standard response is to move to a model that allows extra variance, at which point the simple conjugate update no longer applies and you have traded closed form for fidelity. The other assumption is a fixed rate over the whole observation window. If ticket volume genuinely doubled after a release, a posterior accumulated since launch will lag badly, because every old day still counts as much as every new one. ## What to say "Gamma is conjugate to Poisson; in shape-and-rate form add the total events to the shape and the total exposure to the rate; the prior mean alpha over beta reads as pseudo-events over pseudo-exposure." Then flag overdispersion, and you have covered the mechanics and the main failure mode.