Hi,
It is good to hear that the phi() function is nice and robust for
probit models. I have a few responses.
1. Credit where credit is due: The sampling from truncated normals
approach is from James H. Albert and Siddhartha Chib. 1993. "Bayesian
Analysis of Binary and Polychotomous Response Data." Journal of the
American Statistical Association 88:669--79.
2. Other remedies? I was initially attracted to this approach after
getting frustrated by the "linear predictor in probit too large" error
with the probit() link. Good start values and tighter priors can
remedy the problem, as can switching to logit()! I note that in my
part of the world (political science, sociology etc), there is almost
always no theory as to logit vs probit for Bernoulli data (indeed, the
Albert and Chib piece has a setup where the link function is estimated
from the data, by specifying a t distribution with unknown degrees of
freedom, but thats another story...)
3. Beware the I(,) construct: The truncated normals approach relies on
the I(,) construct. Consider the following code fragment:
for(i in 1:N){
mu[i] <- b[1] + b[2]*x[i]
ystar[i] ~ dnorm(mu[i],1)I(lo[y[i]+1],up[y[i]+1])
}
lo[1] <- -10 lo[2] <- 0
up[1] <- 0 up[2] <- 10
i.e., the observed y supply the truncation points.
This use of the I(,) construct is warned against in the WinBUGS
documentation (i.e., you don't learn about the parameters of the
truncated distribution -- mu and its parents -- the right way), but in
large-ish data sets, I've never found it to make a difference (in the
sense that with vague priors on b, the posterior modes for b from the
WinBUGS output are the MLEs one obtains fitting a glm in S/R etc). But
caveat emptor. And a political science colleague at UCLA, Jeff Lewis,
has devised a delightfully simple example where things go wrong with
the I(,) construct, and in subtle ways. I hope to put a ODC file up on
my web pages soon.
-- Simon Jackman
On Wednesday, October 16, 2002, at 07:04 AM, Eugene Hahn wrote:
> Hello all,
>
> A number of solutions have been proposed for the probit trap involving
> the linear predictor in probit being too large. One of my favorites
> is Jackman's truncated normal sampling approach for Bernoulli data.
>
> It turns out that a little testing shows an interesting result. For
> the construction
>
> probit(p) <- mu
>
> the "critical value" inducing a trap is somewhere in the vicinity of
> ABS(8.5) to ABS(8.6). That is, if mu is +-8.5, no crash occurs; if mu
> is +-8.6, a crash occurs.
>
> Interestingly, the phi function seems to be much more forgiving and so
> the construction
>
> p <- phi(mu)
>
> will happily tolerate much greater values of mu (say 20) without
> returning a trap.
>
> As for why this is the case, I'm not certain but at a pragmatic level
> this construction seems to solve the probit trap problem.
>
> One nice thing about phi() is that it can be used with both binomial
> and Bernoulli data. Those so-inclined can try out the probit() vs.
> phi() constructions with the binomial data in the Seeds example. If
> the probit(p) construction is used and the model is run out for a
> little while (say 50,000 iterations), a trap will occur. Phi() runs
> to completion with no traps.
>
> Best,
> -Gene Hahn
>
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>
Simon Jackman, Assoc Prof and Victoria Schuck Faculty Scholar,
Director, Political Science Computational Laboratory,
Department of Political Science, ph: +1 (650) 723-4760
Encina Hall, fax: +1 (650) 723-1808
Stanford University, [log in to unmask]
Stanford, CA 94305-6044, USA. http://jackman.stanford.edu
see Josephine at http://josephinejackman.stanford.edu
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