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 .

Thus:

where t(n-1) represents the Student-t distribution with (n-1) degrees of freedom. This is the same result used in classical statistics.

ModelRisk

Monte Carlo simulation in Excel. Learn more

Tamara

Adding risk and uncertainty to your project schedule. Learn more

Navigation

Enterprise Risk Management software (ERM)

Learn more about our enterprise risk analysis management software tool, Pelican

Enterprise risk management software introduction