Re: Question about Bayesian analysis


Stephen J. Herschkorn wrote:
Lou Thraki wrote:

Suppose I have a sample of data {x}={x1,x2,...,xN} and
a family of pdf's f(x;a) parametrized with parameter "a".
I assume conditional independence for the data and put

P( {x} | a ) = f(x1;a)*f(x2;a)*...*f(xN;a)

Furthermore, I take a suitable prior pdf Prior(a) for the
parameter "a". The posterior pdf Post(a;{x}) for the parameter
"a" is given by

Post(a;{x}) = f(x1;a)*f(x2;a)*...*f(xN;a)*Prior(a)
/ integral[ f(x1;b)*f(x2;b)*...*f(xN;b)*Prior(b)*db ]

Question: does the pdf (in "y") given by

p(y;{x}) = integral[ f(y;a)*Post(a;{x})*da ]

have any meaning? Is it an estimate of the density the
data seem to have (independently) been drawn from?


Yes, your p(y;{x}) is the posterior density for your population. That
is, having observed {x}, your next observation comes from this
density. You could have intergated out a in your prior as well.

Note that, probabilistically speaking, your sample data {x} are
*conditionally* independent given a. The random variables x1, x2,
x3,..., x_n are not unconditionally independent. This is usually the
case in Bayesian analysis. Indeed, each observation gives you more
information about the underlying unknown distribution.

Thanks for the answer.
Now suppose I want to simulate "new data".
What else could I do but generating them
*independently* following p(y;{x})?

I think my question finally is wether Bayesian analysis can
be used for density estimation.

.

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