How is the posterior mean related to the prior mean?

How is the posterior mean related to the prior mean?

It turns out that the posterior mean can be algebraically re-arranged into a weighted average of the prior mean, a / ( a + b ), and the data proportion, z / N, as follows: (6.9) z + a N + a + b ︸ posterior = z N ︸ data N N + a + b ︸ weight + a a + b ︸ prior a + b N + a + b ︸ weight.

How does mixing weight affect the posterior mean?

The mixing weight on the data proportion increases as N increases. Thus, the more data we have, the less is the influence of the prior, and the posterior mean gets closer to the proportion in the data.

How is numerical approximation used in posterior distribution?

Instead of analytical mathematical approaches, another class of methods involves numerical approximation of the integral. When the parameter space is small, one numerical approximation method is to cover the space with a comb or grid of points and compute the integral by exhaustively summing across that grid.

Which is the posterior mean of n flips?

Suppose we observe z heads in N flips, which is a proportion of z/N heads in the data. The posterior mean is ( z + a )/ [ ( z + a) + ( N ‒ z + b )] = ( z + a )/ ( N + a + b ). It turns out that the posterior mean can be algebraically re-arranged into a weighted average of the prior mean, a / ( a + b ), and the data proportion, z / N, as follows:

How is the posterior mean of a data set arranged?

The posterior mean is ( z + a )/ [ ( z + a) + ( N ‒ z + b )] = ( z + a )/ ( N + a + b ). It turns out that the posterior mean can be algebraically re-arranged into a weighted average of the prior mean, a / ( a + b ), and the data proportion, z / N, as follows:

Which is the correct equation for the posterior distribution?

(6.9) z + a N + a + b ︸ posterior = z N ︸ data N N + a + b ︸ weight + a a + b ︸ prior a + b N + a + b ︸ weight. Equation 6.9 indicates that the posterior mean is always somewhere between the prior mean and the proportion in the data.