Re: Full conditional probability out of product of Gaussians?

From: Ian Jermyn (Ian.Jermyn_at_sophia.inria.fr)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 10:51:50 +0100

If I understand your question, the exponent in your Gaussian distribution is
given by

E(X) = a1 (x1 - m1)^{2} + \sum_{i} ai (xi - x1 - mi)^{2} ,

where i runs from 2 to n; the a's are the inverse variances; and the m's are
the means. Define new variables as

z1 = x1 - m1

zi = xi - x1 - mi .

The Jacobian of this transformation is 1. So you can write the exponent in
your pdf as

E(Z) = a1 z1^{2} + \sum_{i} ai zi^{2} ,

where now the variables are all independent. You can sample from each one,
and then transform back to the x variables.

Ian.

<beliavsky@aol.com> a écrit dans le message de news:
1108922212.799392.91720@z14g2000cwz.googlegroups.com...
>
> sigu4wa02@sneakemail.com wrote:
>> Hello,
>> I need to generate samples from a n dimentional pdf which is a
>> product of Gaussians:
>> p(x1, x2, ..., xn) = f1(x1) * f2(x2-x1) * ... * fn(xn-x1)
>> where each fi(x) is a Gaussian pdf with known means and variances.
>> I'm considering using Gibbs sampler. So I need to compute the full
>> conditional probability p(xi | *). i.e. the probability of xi
>> conditioned on all other n-1 variables. Since the joint pdf is known,
> I
>> try to compute p(xi | *) by:
>> p(xi | *) = p(x1, x2, ..., xn)
>> -----------------
>> p(*)
>> where p(*) is the marginal pdf of the rest n-1 variables obtained
> by
>> integrating out xi from the joint pdf.
>>
>> I find, even in the simplest case of two gaussians, the resulting
>> p(xi | *) can be horriably complex.
>> Is there a better way of computing the full conditional
> probability
>> out of joint p.d.f. in my case? Or is it possible to do Gibbs
> sampling
>> from the unnormalized full conditional p.d.f. p'(xi | *) =
>> fi(xi)*fi(xi-x_{i-1}), which can be easily computed.
>
> Are the Gaussian variates in the product uncorrelated? In that case,
> analytic expressions for the density in the case of 2 or 3 variates are
> at http://mathworld.wolfram.com/NormalProductDistribution.html . There
> is Fortran 90 code for the PDF of two correlated normal variates at
> http://users.bigpond.net.au/amiller/fnprod.f90 .
>

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