Bayesian fair price with a Normal prior

Question

You estimate the fair price \mu of an asset. Your prior belief is Normal with mean 20 and variance 4 (so a standard deviation of 2). A single expert quote arrives at 26, and you know the quote is noisy with variance 12. Both the prior and the quote's noise are Normal. Using Bayesian updating, what is the posterior mean estimate of the fair price \mu?

Accepted Answer

Intuition first. When a Normal prior meets Normal data, the answer is again Normal (they are conjugate). The updated mean is a tug-of-war between your prior guess and the new measurement, and the winner is whoever is more precise. Precision just means "one divided by variance": small variance equals high precision equals more pull. The rule. Let the prior have mean \mu0 and variance 	au0^2, and the measurement have value x and variance \sigma^2. Convert each to a precision, then take a precision-weighted average: 	ext{posterior mean} = \frac{\dfrac{\mu0}{	au0^2} + \dfrac{x}{\sigma^2}}{\dfr…

Bayesian fair price with a Normal prior

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