Bayesian updating for a mean, the Normal-Normal

Question

You believe an asset's mean daily return proxy is $\mu \sim \mathcal{N}(\mu_0, \tau^2)$ with $\mu_0 = 100$ and $\tau^2 = 16$ . You then observe $n = 4$ independent readings $x_i \sim \mathcal{N}(\mu, \sigma^2)$ with known variance $\sigma^2 = 64$ and sample mean $\bar x = 106$ .

Derive the posterior for $\mu$ , give its mean and variance, and interpret the prior's weight.

Accepted Answer

With a Normal prior on the mean and Normal data of known variance, the posterior is again Normal (Normal is conjugate to itself here). The clean way to combine them is in terms of precision (one over variance), because precisions add. Precisions. Prior precision is \frac{1}{	au^2} = \frac{1}{16}. The data carry precision \frac{n}{\sigma^2} = \frac{4}{64} = \frac{1}{16} (averaging n readings multiplies precision by n). Posterior precision is the sum: \frac{1}{	au^2} + \frac{n}{\sigma^2} = \frac{1}{16} + \frac{1}{16} = \frac{1}{8}, \qquad 	ext{so posterior variance} = 8,\ 	ext{sd} \approx 2…

Bayesian updating for a mean, the Normal-Normal

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