Posterior Distribution of The Parameters
Assume that x is distributed according to a normal distribution with unknown mean and precision .
and that the prior distribution on and, has a normal-gamma distribution
for which the density π satisfies
Given a dataset, consisting of independent and identically distributed random_variables (i.i.d), the posterior distribution of and given this dataset can be analytically determined by Bayes' theorem. Explicitly,
- ,
where is the likelihood of the data given the parameters.
Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples:
This expression can be simplified as follows:
where, the mean of the data samples, and, the sample variance.
The posterior distribution of the parameters is proportional to the prior times the likelihood.
The final exponential term is simplified by completing the square.
On inserting this back into the expression above,
This final expression is in exactly the same form as a Normal-Gamma distribution, i.e.,
Read more about this topic: Normal-gamma Distribution
Famous quotes containing the words distribution and/or parameters:
“The question for the country now is how to secure a more equal distribution of property among the people. There can be no republican institutions with vast masses of property permanently in a few hands, and large masses of voters without property.... Let no man get by inheritance, or by will, more than will produce at four per cent interest an income ... of fifteen thousand dollars] per year, or an estate of five hundred thousand dollars.”
—Rutherford Birchard Hayes (18221893)
“What our children have to fear is not the cars on the highways of tomorrow but our own pleasure in calculating the most elegant parameters of their deaths.”
—J.G. (James Graham)