choice of priors in stan

[Linas Mockus]

Hi,

What are recommended choice of priors in stan?  I always struggle with variance - should it be half-cauchy or uniform if I know that variance is different from 0?  I gathered that for very small variances it is better to use half-cauchy.

What about neg_binomial_2 with hierarchical scale parameter phi: neg_binomial_2(mu,phi[i])?  I am always confused if phi~gamma(loc_phi,scale_phi) or phi~lognormal(loc_phi,scale_phi).  If I use lognormal I should use  normal prior for loc_phi and half-cauchy (or uniform) for scale_phi?  It would be great having a cookbook with  recommendations on which priors for parameters/hyper-parameters work or don't work in stan since we don't have to worry about conjugateness.

Thank you,
Linas

[Bob Carpenter]

There's a discussion in the regression chapter of the manual.
The recommendations aren't really specific to Stan, either.


We don't discuss negative binomial --- Andrew may be able to help
here.

I believe Daniel created a Wiki page where Andrew was going to
write responses to questions about priors. But I'm not even sure where
it's at --- I don't see any public repos on:

https://github.com/andrewgelman?tab=repositories

Let's see if this inspires the first entry.

- Bob

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[Andrew Gelman]

Lately we’ve been using N+(0,1) priors for sd parameters (N+ indicates normal constrained to be positive).  This is a bit stronger than Cauchy and assumes the model has been scaled so that parameters should rarely be more than 1.

On Jun 30, 2015, at 12:28 PM, Linas Mockus <linasm...@gmail.com> wrote:

[Linas Mockus]

Dumb question - how to scale the model? 

Thank you,
Linas

[Andrew Gelman]

It depends on the context, it just means you need to understand the parameters well enough to get a sense of how large they can be.

[Richard McElreath]

I've found half-Cauchy fine in many cases, but with non-linear models, it is easy for a variance parameter to be weakly identified, due to ceiling/floor effects. In those cases, something like an exponential prior or half-Guassian works much better, because both have much thinner tails than the Cauchy.

I think Daniel Simpson likes exponential priors for scale parameters, based upon a subtle argument having to do with constant penalty with distance from a base model with variance = zero. http://arxiv.org/abs/1403.4630

[Andrew Gelman]

I’ve been doing half-normal but I could be persuaded to switch to exponential.

[Linas Mockus]

If I understand correctly, the best practice for sd prior:

- half-cauchy woks fine for sd

- if we know that most of sd < 2 then we scale the model:

real <lower=0> sd;

y~normal(mu,2*sd);

sd~normal(0,1)

- use exponential

- what if we know that sd is concentrated around 1?  Then half-cauchy may not be most efficient.  I don't know if it is true or not but I found sd~uniform(0,2) to be more efficient than half_cauchy.

- what prior should be used for neg_binomial_2 scale parameter? Is it gamma or lognormal or something else?  What should be the hyperpriors?

Thank you,
Linas

--

Thank you,
Linas

[Andrew Gelman]

Neg_binomial_2 scale parameter . . . hmmm . . . Let me just clarify that we’re talking about the same thing.  neg_binomial has two parameters, alpha and beta, where alpha is the shape parameter and 1/beta is the scale parameter.

neg_binomial_2 has 2 parameters, mu and phi.  mu = alpha/beta and phi = alpha.  At least, I think I’m getting the algebra right.  I suppose we could call mu a scale parameter here.  But I think section 39.2 of the manual is wrong:  there it says that phi is a precision parameters and 1/phi is represents overdispersion, and I think neither of these statements are correct.  I think (if I’m not getting confused) that phi in the neg_binomial_2 distribution is simply alpha, the shape parameter.

The overdispersion (as usually defined) is 1 + 1/beta, hence is always at least 1.

I think this all might be explaining some of the confusion we’ve been having lately, also suggests we should have negative_binomial_3 that is given in terms of mu and the overdispersion.

A

[Linas Mockus]

I always thought that the first parameter of neg_binomial_2 represents mean.  Since mean in GLM context is exp(beta'x) where beta is a parameter and x is independent variable, I never intended to put a prior on mu.  However I wanted to put a hierarchical prior on the second parameter, phi.  My expectation was that variance is expressed as mu+mu^2/phi so by giving a hierarchical prior on phi I implicitly attributed different variability to each sample (in my case particle count).  So what is correct?  

Anyway, what hierarchical prior (lognormal/gamma/etc) should phi have (i.e. phi~lognormal(loc,scale) as well as what hyper priors (i.e. loc~normal(0,5) and scale~half-cauchy(0,2)) should be used?   

Thank you,
Linas

--

Thank you,
Linas

[Bob Carpenter]

Negative binomial is implicitly confusing given that there
are four standard variants in play in the stats literature.

neg_binomial(y | alpha, beta) is exactly as defined in BDA,
with shape alpha and inverse scale beta, so I'm assuming that
one's OK.

I added an issue for neg_binomial_2():

https://github.com/stan-dev/stan/pull/1523#issuecomment-118544690

Any suggestions on what to call the neg_binomial_2 parameters and
what Greek letters to use for them?

Note there's also neg_binomial_2_log, which takes a parameter on
the log scale.

- Bob

[Andrew Gelman]

Hi, I’m not sure.  If you set phi < infinity you have overdispersion.  The lower phi, the more overdispersion.  It’s hard for me to get a handle on this in the abstract; I think it depends on context.

A

[Michael Betancourt]

Just think about the variance — the over dispersion is

significant when mu^{2} / phi is on the same order of mu,

so

mu^{2} / phi ~ mu

mu / phi ~ 1

mu ~ phi

Consequently the absolute amount of over dispersion 

depends on the mean so you have to think about your prior 

in terms of relatively over dispersion. 

[Michael Betancourt]

This book, http://www.amazon.com/Negative-Binomial-Regression-Joseph-Hilbe/dp/0521198151,
has something like 20 parameterizations. Don’t ever trust someone when they say “negative binomial”
without providing a formula.

As far as I’ve seen the mu, phi names as used in the manual are pretty standard for the negative binomial
GLM parameterization.

[Luc Coffeng]

In infectious disease epidemiology, the NB2 parameters are referred to as mean \mu and aggregration k (amount of clumping of infectious particles within hosts :) ).