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# Bayesian estimate of the mean of a Normal distribution with unknown standard deviation

Assume that we have a set of n data samples from a Normal distribution with unknown mean m and unknown standard deviation s. We would like to estimate the mean together with the appropriate level of uncertainty. A Normal distribution can have a mean anywhere in [-∞, +∞], so we could use a Uniform improper prior p(m) = k. The uninformed prior for the standard deviation should be p(s) = 1/s to ensure invariance under a linear transformation. The likelihood function is given by the Normal distribution density function:

Multiplying the priors together with the likelihood function and integrating over all possible values of s, we arrive at the posterior distribution for m:

(1)

where x and are the mean and sample standard deviation of the data values. The Student-t distribution with n degrees of freedom has probability density:

(2)

Equation 1 and 2 are the same functions if we set n = n – 1 and   $y=\frac{\sqrt{n}(\bar{x}-\mu )}{\hat{\sigma}}$.

Thus:

where t(n-1) represents the Student-t (0,1,n-1) distribution with (n-1) degrees of freedom. Therefore, with Crystal Ball 7.0+, we can make the following model as also shown in the model Estimate_mean_and_stdev_for_Normal_distribution_when_neither_known

m = Student (x, s/SQRT(n), n-1)

This is the same result used in classical statistics.

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