Bayesian temperature estimate with a Normal prior

Question

You estimate the true temperature \mu of a furnace in degrees. Your prior is Normal with mean 300 and variance 9 (standard deviation 3). A calibrated probe reads 312, with known noise variance 3. Both the prior and the probe noise are Normal. Using Bayesian updating, what is the posterior mean estimate of the temperature \mu?

Accepted Answer

Intuition first. Normal prior plus Normal data gives a Normal answer (conjugate). The updated mean is a precision-weighted tug-of-war, where precision means "one over variance." Here the single probe reading is sharper (variance 3) than the prior (variance 9), so it should pull harder. The rule. With prior mean \mu0 and variance 	au0^2, and measurement x with variance \sigma^2: 	ext{posterior mean} = \frac{\dfrac{\mu0}{	au0^2} + \dfrac{x}{\sigma^2}}{\dfrac{1}{	au0^2} + \dfrac{1}{\sigma^2}}, a precision-weighted average. Plug in. Prior precision = 1/9 \approx 0.1111; probe precision = 1/3 …

Bayesian temperature estimate with a Normal prior

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