How sharp is the posterior? Adding precisions

Question

You estimate an unknown quantity \mu with a Normal prior of variance 16. You then take 3 independent readings, each with known noise variance 48. Both the prior and the readings are Normal. What is the variance of the posterior distribution for \mu?

Accepted Answer

Intuition first. Uncertainty is easiest to combine when you flip it into precision, defined as "one over variance." Precision measures how much information you have: the prior carries some, each reading adds more, and Bayesian updating simply adds these precisions together. The posterior variance is then one over the total precision. The rule. With prior variance 	au0^2 and n readings each of variance \sigma^2: 	ext{posterior precision} = \frac{1}{	au0^2} + \frac{n}{\sigma^2}, \qquad 	ext{posterior variance} = \frac{1}{	ext{posterior precision}}. In words: add the prior's precision to th…

How sharp is the posterior? Adding precisions

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