Bayesian updating — posterior ∝ prior × likelihood

Bayes' theorem

Turning "what we believe" plus "what we saw" into "what we now believe"

We want the probability of unknown parameters θ given the data — but our model tells us the reverse, the probability of the data given the parameters. Bayes' theorem inverts it:

p(θ | data) = p(data | θ) · p(θ)p(data) // posterior = (likelihood × prior) / evidence

The denominator p(data) does not depend on θ — it is just a constant that makes the posterior integrate to 1. So the shape of the answer comes entirely from the numerator, which is why the whole of Bayesian inference fits in one proportionality:

p(θ | data) ∝ p(data | θ) × p(θ) // posterior ∝ likelihood × prior

Why bother with a prior?

Encode knowledge. You often know things — an incubation period is a few days, a reporting probability is between 0 and 1, R(t) rarely jumps from 0.8 to 3 overnight. The prior is where that epidemiological knowledge enters the model.
Regularise. Maximum likelihood can overfit badly on sparse data (3 cases today, 6 tomorrow → "R(t) = 2.0 exactly"). A prior pulls wild estimates back toward the plausible, especially when data are thin.
Honest uncertainty. Bayes returns a whole distribution over θ, not a single number — so you can ask "what is the probability the outbreak is growing?" rather than just reading off a point estimate.

A prior is not cheating — it is the explicit, auditable statement of what you assumed. With lots of data it barely matters; with little data it is doing the honest work of stopping the model from believing nonsense.

In this course we sample the posterior

For the toy examples on this page the posterior has a tidy closed form. Real epidemic models have hundreds of parameters and no such formula, so the normalising integral p(data) is intractable. Instead of solving it we draw samples from the posterior using MCMC (in Stan). Each sample is one plausible parameter set — one possible version of the epidemic — and the cloud of samples is the posterior. Everything below is the conjugate, exactly-solvable shadow of that sampling machinery.

Updating a probability Beta–Binomial

Estimating a reporting / ascertainment probability θ from "k of n" data

Suppose θ is a probability we want to learn — the chance a true case is reported, say, or a test comes back positive. We hold a prior belief about θ, then observe data: k successes in n trials. The likelihood of that data is Binomial, ∝ θ^k(1−θ)^n−k. A Beta(a, b) prior, ∝ θ^a−1(1−θ)^b−1, is the natural partner — multiply the two and the exponents simply add:

prior Beta(a, b) × data (k of n) → posterior Beta(a + k, b + n − k)

So updating is just bookkeeping: add the successes to a and the failures to b. The figure evaluates each Beta density on a fine grid of θ ∈ [0, 1] and normalises it numerically (the trapezoid rule) — no special functions needed. Watch how the posterior sits between prior and likelihood, and how it slides toward the data as n grows.

prior Beta(a, b) likelihood (scaled) posterior Beta(a+k, b+n−k)

prior a: 2.0 prior b: 2.0 trials n: 10 successes k: 7

prior mean = — data k/n = — posterior mean = — 95% CrI = —

—

Watch the data take over, one draw at a time

Press play to stream Bernoulli draws (each a "success" with true probability 0.7) and feed them in sequentially. The prior is fixed at whatever a, b you set above. Early on, the posterior barely moves from the prior; as the trials pile up the likelihood dominates and the posterior tightens around the true 0.7 — the data swamp the prior.

streamed n = 0, k = 0

The "play" stream overrides the n and k sliders and uses the live prior a, b. Press reset stream to return to the manual sliders.

A continuous mean Normal–Normal

The posterior mean is a precision-weighted average of prior and data

Now θ is a continuous quantity — a mean we want to estimate (an average delay, a log-incidence level). We hold a prior N(μ₀, σ₀²) and observe n data points with sample mean x̄ and known observation noise σ. With a Normal prior and Normal likelihood the posterior is again Normal, and its mean is the two sources blended by their precisions (precision = 1/variance):

μ_post = μ₀/σ₀² + n·x̄/σ²1/σ₀² + n/σ² 1σ_post² = 1σ₀² + nσ²

Read it as a weighted tug-of-war: whichever side is more precise (less variance, or — for the data — more observations) pulls the posterior toward itself. A confident prior (small σ₀) holds firm; lots of data (large n) overwhelms it. And the posterior is always sharper than either input — precisions add, so it can only narrow.

prior N(μ₀, σ₀²) likelihood (from data) posterior

prior mean μ₀: 3.0 prior sd σ₀: 1.5 data mean x̄: 1.5 obs noise σ: 1.5 n data points: 5

prior mean = — data mean = — posterior mean = — posterior sd = —

—

Set n = 0 and the posterior is the prior exactly — no data, no update. Crank n up and the posterior slides onto the data mean and shrinks, almost ignoring the prior. That is the same convergence MCMC shows: with enough data the choice of (reasonable) prior stops mattering.

Takeaways

The posterior is a compromise. It lives between the prior and the likelihood, never outside the tug-of-war between them.
Data eventually swamp the prior. As n grows the likelihood sharpens and dominates; the posterior converges on the data regardless of a (reasonable) prior.
A stronger prior pulls harder. A confident prior (Beta with large a+b, or Normal with small σ₀) holds the estimate near its belief until enough data arrive to move it.
Priors regularise and encode knowledge. They stop sparse data from implying nonsense and let your epidemiological expertise enter the model explicitly.
You get a whole distribution, not a point. Credible intervals and "probability of growth" fall straight out of the posterior.
In practice we sample it. Real models have no closed form, so Stan's MCMC draws thousands of posterior samples — each one a plausible epidemic — and those samples become the forecasts.