Posterior variance in conjugate models | Poisson-gamma

15 hours ago

Posterior variance can increase under the beta-binomial model when unexpected data occurs, though precision always improves.
In the normal-normal model, precision (reciprocal of variance) increases with each new data point, consistently reducing posterior variance.
For the Poisson-gamma model, posterior variance rises with each observed event (k) and decreases quadratically with time (t) between events.
An illustration using simulated Poisson data with a gamma(1,1) prior demonstrates how posterior variance behaves over time.

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