Updating a Gaussian belief with a Gaussian measurement

Question

Your prior belief about a stock's true expected daily return is $\mu \sim N(m_0, \tau_0^2)$ . You then observe a noisy estimate $x \sim N(\mu, \sigma^2)$ with known measurement noise $\sigma^2$ .

Derive the posterior distribution of $\mu$ and interpret the updated mean and variance.

Accepted Answer

Conjugacy. A Normal prior with a Normal likelihood yields a Normal posterior. Multiplying the two bell-shaped densities and completing the square gives \mu \mid x \sim N(m1, 	au1^2) with the clean rule that precisions (inverse variances) add: \frac{1}{	au1^2} = \frac{1}{	au0^2} + \frac{1}{\sigma^2}, \qquad m1 = 	au1^2\left(\frac{m0}{	au0^2} + \frac{x}{\sigma^2}ight). Interpreting the mean. The posterior mean is a precision-weighted average of the prior mean and the observation: m1 = \frac{	frac{1}{	au0^2}\, m0 + 	frac{1}{\sigma^2}\, x}{	frac{1}{	au0^2} + 	frac{1}{\sigma^2}}. Each…

Updating a Gaussian belief with a Gaussian measurement

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