Conjugate families and posterior updating

For $n$ conditionally independent Bernoulli trials sharing success probability $\theta$, a count $y$ has binomial likelihood proportional to $\theta^y(1-\theta)^{n-y}$. A beta prior with positive shape parameters has kernel $\theta^{\alpha-1}(1-\theta)^{\beta-1}$ on the unit interval. Multiplication adds exponents, giving another beta density. The binomial coefficient is constant in the parameter and cancels in posterior normalization, but it is required when calculating the marginal probability of observing a count.

The posterior mean combines the prior mean with the observed success fraction, with weights determined by $\alpha+\beta$ and the number of trials. This motivates the phrase prior effective count, but literal invented observations are not required. In particular a beta density's exponents involve shape minus one, whereas its mean uses the shapes themselves. Confusing these conventions changes modes and boundary behavior. Both shapes must be strictly positive for the ordinary beta prior to be proper, even when one wants a very weak prior.

For counts collected over known exposures $t_i$, suppose $Y_i\mid\lambda$ are conditionally independent Poisson variables with means $t_i\lambda$. A gamma prior in the shape-rate convention has kernel $\lambda^{a-1}e^{-b\lambda}$, with positive $a$ and $b$. Observed counts add to the shape and exposure adds to the rate. Exposure need not be an integer. Its units are time if the unknown rate is events per unit time, so the posterior rate parameter has those time units as well.

The gamma scale convention instead uses the reciprocal of the rate. A numerical implementation that receives a scale where a rate was intended can reverse the prior's meaning. State the convention beside the formula and verify the mean and variance as a unit check. Sequential updating agrees with updating all conditionally independent batches together because multiplication is associative. This equivalence requires that the batches supply new information; updating twice with the same observations counts the same evidence twice and produces artificially concentrated uncertainty.

A future Bernoulli success probability equals the posterior mean, but the probability of two future successes involves the posterior second moment. Integrating a shared unknown parameter creates positive dependence between those future trials. Plugging the posterior mean into a binomial distribution discards that dependence and generally understates predictive dispersion. The resulting beta-binomial predictive law is still analytically tractable, so this is a useful setting for seeing exactly what uncertainty a point estimate omits.

Conjugate calculations also provide reference answers for software. Check posterior shape changes after all successes, all failures, and zero new trials; test whether predictive probabilities sum to one. A normal likelihood with known variance and a normal prior gives a related precision-addition rule, but unknown variance changes the conjugate family. Algebraically convenient assumptions should be checked against outliers, dependence, changing rates, and prior predictive behavior. When those assumptions fail, a nonconjugate model can be more useful than an exact answer to the wrong question.