# [R] Analytical Optimization (Stat question)

francogrex francogrex at mail.com
Wed Jan 16 12:06:37 CET 2008

```Dear Experts, this is more a general stat question, I tried to ask in other
places but had no luck with answers (expect one that suggested numerical
The likelihood below is a mixture of two negative binomial
distributions:
P*f(n;x1,x2,E) + (1-P)*f(n;x3,x4,E)
N and E are vectors of same length. I would like to find the fist
derivatives with respect to x1,x2,x3,x4 and P (so that I can set them
to zero and calculate a bayesian MLE by iterations according to  David
Draper's
"Bayesian Modeling, Inference and Prediction").
Does anyone have a "quick and dirty" way? To avoid drawning in rather
complicated math...
Here's the transcription of the likelihood in S language:
P*exp((lgamma(x1+N)))-(((lgamma(x1)+lfactorial(N))+log((1+(E/
x2))^x1)+log((1+(x2/E))^N)))+
(1-P)*exp((lgamma(x3+N)))-(((lgamma(x3)+lfactorial(N))+log((1+(E/
x4))^x3)+log((1+(x4/E))^N)))
--
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```