Best practices for handling very small numbers?

Thanks Duncan,
Indeed I have found a problem in my likelihood, and I'll work on the log
approach again. Regarding your comment; do you think it would make sense to
tackle a data set of few millions for parameter estimation using MCMC? In my
case we are talking about Griddy Gibbs sampling.

One relevant idea is to break the observations into smaller chunks and use
sequential updating by taking the posterior of one step as the prior of the
other, however, I don't feel confident that this would work, since it is not
the same situation with conjugate priors, where you have analytic form of
the posterior.

Any pointers (papers, books) towards large scale data processing with MCMC,
and Gibbs sampling in particular would be much appreciated.

Kind regards
Seref


On Tue, Oct 18, 2011 at 11:12 AM, Duncan Murdoch
<murdoch.duncan at gmail.com>wrote:

 On 11-10-18 4:30 AM, Seref Arikan wrote:

 Hi Dan,
 I've tried the log likelihood, but it reaches zero again, if I work with
 say
 1000 samples.
 I need an approach that would scale to quite large sample sizes. Surely I
 can't be the first one to encounter this problem, and I'm sure I'm missing
 an option that is embarrassingly obvious.

 I think you are calculating the log likelihood incorrectly.  Don't
 calculate the likelihood and take the log; work out the formula for the log
 of the likelihood, and calculate that.  (If the likelihood contains a sum of
 terms, as in a mixture model, this takes some thinking, but it is still
 worthwhile.)

 With most models, it is just about impossible to cause the log likelihood
 to underflow if it is calculated carefully.

 Duncan Murdoch

 Regards
 Seref

 On Mon, Oct 17, 2011 at 6:15 PM, Nordlund, Dan (DSHS/RDA)<
 NordlDJ at dshs.wa.gov>   wrote:

  -----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

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