Sequential Normal-Normal updating over two batches

Question

You estimate a signal's true level \mu. You start from a Normal prior with mean 0 and variance 1. Readings arrive in two batches, each reading having known noise variance 1: - Batch 1: 4 readings averaging 2 - Batch 2: 6 readings averaging 3 You update after each batch, feeding the first batch's posterior in as the second batch's prior. After both batches, what is the posterior mean estimate of \mu?

Accepted Answer

Intuition first. Normal updates also chain cleanly: the first batch's posterior becomes the second batch's prior, and the result equals updating on all readings at once. The mechanics run through precision, "one over variance," which simply adds up across the prior and every reading. Precision bookkeeping. Prior precision = 1/1 = 1. Each reading has precision 1/1 = 1, so a batch of n readings adds precision n and contributes n 	imes (	ext{batch average}) to the precision-weighted total. - Start: prior precision 1, weighted sum = 0	imes 1 = 0. - After Batch 1 (4 readings, average 2): precis…

Sequential Normal-Normal updating over two batches

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