R help - Oct 2011 - Best practices for handling very small numbers?

Greetings
I have been experimenting with sampling from posterior distributions using
R. Assume that I have the following observations from a normal distribution,
with an unscaled joint likelihood function:

normsamples = rnorm(1000,8,3)

joint_likelihood = function(observations, mean, sigma){
    return((sigma ^ (-1 * length(observations))) * exp(-0.5 * sum(
((observations - mean ) ^ 2)) / (sigma ^ 2) ));
}

the joint likelihood omits the constant (1/(2Pi)^n), which is what I want,
because I've been experimenting with some crude sampling methods. The
problem is, with the samples above, the joint likelihood becomes 0 very
quickly.
I wanted to experiment with tens of thousands of observations, but without
an implementation of a transformation that can handle very small values, my
sampling algorithms would not work.

This is an attempt to use some sampling algorithms for Bayesian parameter
estimation. I do not want to resort to conjugacy, since I am developing this
to handle non conjugate scenarios, I just wanted to test it on a conjugate
scenario, but I've quickly realized that I'm in trouble due to
likelihood
reaching 0 quickly.

Your feedback would be appreciated. It makes me wonder how JAGS/BUGS handles
this problem

Best regards
Seref

	[[alternative HTML version deleted]]

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
> project.org] On Behalf Of Seref Arikan
> Sent: Monday, October 17, 2011 9:11 AM
> To: r-help at r-project.org
> Subject: [R] Best practices for handling very small numbers?
> 
> Greetings
> I have been experimenting with sampling from posterior distributions
> using
> R. Assume that I have the following observations from a normal
> distribution,
> with an unscaled joint likelihood function:
> 
> normsamples = rnorm(1000,8,3)
> 
> joint_likelihood = function(observations, mean, sigma){
>     return((sigma ^ (-1 * length(observations))) * exp(-0.5 * sum(
> ((observations - mean ) ^ 2)) / (sigma ^ 2) ));
> }
> 
> the joint likelihood omits the constant (1/(2Pi)^n), which is what I
> want,
> because I've been experimenting with some crude sampling methods. The
> problem is, with the samples above, the joint likelihood becomes 0 very
> quickly.
> I wanted to experiment with tens of thousands of observations, but
> without
> an implementation of a transformation that can handle very small
> values, my
> sampling algorithms would not work.
> 
> This is an attempt to use some sampling algorithms for Bayesian
> parameter
> estimation. I do not want to resort to conjugacy, since I am developing
> this
> to handle non conjugate scenarios, I just wanted to test it on a
> conjugate
> scenario, but I've quickly realized that I'm in trouble due to
> likelihood
> reaching 0 quickly.
> 
> Your feedback would be appreciated. It makes me wonder how JAGS/BUGS
> handles
> this problem
> 
> Best regards
> Seref
> 
Maybe you should work with the log-likelihood?


Hope this is helpful,

Dan

Daniel J. Nordlund
Washington State Department of Social and Health Services
Planning, Performance, and Accountability
Research and Data Analysis Division
Olympia, WA 98504-5204