# Re: st: Multinomial logit-type probability in log likelihood for ML model

 From Maarten buis To statalist@hsphsun2.harvard.edu Subject Re: st: Multinomial logit-type probability in log likelihood for ML model Date Tue, 4 Sep 2007 14:18:32 +0100 (BST)

```Three remarks:
1) I don't entirely understand what you want to do, but it sounds like
some sort of mixture model. Maybe the likelihood becomes more tractable
if you formulate it as such.

2) If I understand you correctly you want to sum over observations
within the likelihood function. That does not sound right to me. -ml-
will do the summing for you.

3a) If you really want to sum over observations than a can be stored as
a scalar, as it will be constant over observations. Storing a single
number _N times is a waste of time and space.

3b) Also I would use the -tempname- command to ensure you don't leave
it hanging around after it has finished.

3c) Finally I would not call it a but denom (for denominator). Such
names will greatly improve the readablility of your code. (You will be
really really grateful when you have to debug your program 6 months
from now...)

Hope this helps,
Maarten

--- Bob Hammond <robert.g.hammond@vanderbilt.edu> wrote:

> Hey Maarten,
>
> Thanks for your reply.  The objective in the example you sent is to
> estimate a multinomial logit (MNL) model, but my objective is
> different.  I want to estimate a binary choice model where the
> probability of Y=1 is governed by a log likelihood function that
> contains a MNL-type probability.  I can't simply use a MNL model
> because
> I do not observe the outcome of the MN process.  If the MN process is
>
> described by Binomial(n, j, q), then I observe n, but not j,
> preventing
> me from using Stata's built-in MNL.  My simplification of the log
> likelihood function made this less clear than it should have been;
> sorry
> for the confusion.  My approach is to model q as a function of
> observed
> covariates, allowing me to use the distribution of the unknown j.
>
> After working more on this, I wonder if I can use the following
> (maintaining my simplification from the initial email that ignores
> the
> fact that j is unobserved):
>
> gen a = .
> ...
> egen double `tempvar1' = sum(exp(`theta'))
> replace a = 1 + `tempvar1'
> replace `lnf' = (Binomial(n,j,exp(`theta')/a)  ///
>     - Binomial(n,j+1,exp(`theta')/a) * f(x) if \$ML_y1 == 1
>
> That is, can I simply sum up the `theta' for the entire data set into
> a
> new variable, then use this variable in the "replace `lnf'"
> statement?
> Thanks,
>
> Bob
> > I posted a -ml- program that does a multinomial logit on statalist
> some
> > time ago:
> > http://www.stata.com/statalist/archive/2007-05/msg00449.html
> >
> > Hope this helps,
> > Maarten
> >
> >
> > --- Bob Hammond <robert.g.hammond@vanderbilt.edu> wrote:
> >
> > > All,
> > >
> > > I am having trouble coding an ML program for my log likelihood
> > > function.  I'll simplify several aspects of the log likelihood
> that
> > > seem
> > > unrelated to my question.  Define `lnf' for an individual
> observation
> > > as
> > > follows:
> > >
> > > replace `lnf' = (Binomial(n,j,q) - Binomial(n,j+1,q)) * f(x)  if
> > > \$ML_y1 == 1
> > >
> > > In words, the probability that Y is 1 is the probability of
> observing
> > >
> > > exactly j successes out of n (where q is the probability of a
> success
> > > on
> > > an individual trial) times some function f(x).  The probability q
> > > takes
> > > a multinomial logit form:
> > >
> > > q_i = exp(X_i * `theta') / (1 + sum(exp(X_h * `theta')))
> > >
> > > where the sum goes from h=1, ..., J, so it sums the product of
> the
> > > covariate vector X times the parameter vector `theta' for all
> > > observations in the data set.
> > >
> > > It seems that I need some way to construct the summation in the
> > > denominator of q first, but my confusion is that this denominator
> > > contains the `theta' parameter vector.  Basically, for every
> > > observation
> > > in the data set, I need to construct a scalar that multiplies the
> 1 x
> > > k
> > > covariate vector X by the k x 1 parameter vector `theta' that
> needs
> > > to
> > > be estimated.  Then I need to sum these J scalars up (call this
> a,
> > > which
> > > is a function of `theta').  If I could do that, then, after
> > > constructing
> > > a, I would write:
> > >
> > > replace `lnf' = (Binomial(n,j, exp(`theta') / (1 + a) -
> Binomial(n,j,
> > >
> > > exp(`theta') / (1 + a)) * f(x)
> > >             if \$ML_y1 == 1
> > >
> > > I don't know how to construct "a" because it contains the
> parameter
> > > vector `theta'.  I've tried taking a look at the ado file for
> mlogit,
> > >
> > >
> > > Bob
>
> --
>
------------------------------------------------------------------------
> Bob Hammond
> Department of Economics
> Vanderbilt University
> http://people.vanderbilt.edu/~robert.g.hammond/
> *
> *   For searches and help try:
> *   http://www.stata.com/support/faqs/res/findit.html
> *   http://www.stata.com/support/statalist/faq
> *   http://www.ats.ucla.edu/stat/stata/
>

-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands

Buitenveldertselaan 3 (Metropolitan), room Z434

+31 20 5986715

http://home.fsw.vu.nl/m.buis/
-----------------------------------------

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