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Download a pdf copy of this help file here |
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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 known
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. This is a more unusual case than where one
does not know the standard deviation but might occur, for example,
if one was making many measurements of the same parameter, but believed
that the measurements had independent, normally distributed errors and
no bias (so the distribution of possible values would be centred about
the true value).
We assign an uniformed (Uniform) prior for m and use a Normal likelihood function for the observed n measurements {xi}. No prior is needed for s since it is known and we arrive at a posterior distribution for m given by:
Taking logs:
Since s is known:
where k is some constant. Differentiating twice we get:
The best estimate m0 of
m
is that value for which 
:
i.e., m0 is the average of the data values x - no surprise there! A Taylor series expansion of this function about m0 gives:
(1)
The second term is missing because it equals zero and there are no other
higher order terms since is
independent of m
and any further differential therefore equals zero.
Consequently, Equation 1 is an exact result.
Taking exponents to convert back to f(m), and rearranging a little, we get:
where K is a normalizing constant. Comparison with the probability
density function for the Normal distribution, shows that this is a Normal
density function with mean x and
standard deviation
. In other words:
which is the same as the classical statistics result.
