 |
Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
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.
