truncation with *_log syntax?

[Seth Flaxman]

If I do:

y ~ normal(mu, sigma) T[0,ymax];

Can I do the equivalent with normal_log (I have a mixture model so can't use the above syntax), e.g.:

lp <- normal_log(y,mu,sigma) T[0,ymax];

And if this isn't supported at the moment, is there a hack or reparameterization for just a positively constrained normal distribution?

parameters {

   real<lower=0> y;

}

model {

   // ...

   lp <- normal_log(y,mu,sigma) T[0,];

}

In case you're interested in the model, I'm using a Gaussian mixture model for bivariate density estimation. Theoretically my data covers a really large range (time from 0 to 365 days, space from 0 to 20 km) but I'm only interested in a short-range model so I truncate my observations to 7 days and 2 km.

Thanks,

Seth

[Daniel Lee]

Hi Seth,

On Feb 20, 2016, at 6:20 AM, Seth Flaxman <fla...@gmail.com> wrote:

If I do:

y ~ normal(mu, sigma) T[0,ymax];

Can I do the equivalent with normal_log (I have a mixture model so can't use the above syntax), e.g.:

lp <- normal_log(y,mu,sigma) T[0,ymax];

No. See "Truncated and Censored Data" and the subsection on "Truncated Distributions" (pg 280). 

You have to subtract the normalizing term: 

lp <- normal_log(y,mu,sigma) - log(normal_cdf(ymax, mu, sigma) - 0.5);

(What is lo declared as? You might want to use increment_log_prob() directly). 

And if this isn't supported at the moment, is there a hack or reparameterization for just a positively constrained normal distribution?

parameters {

   real<lower=0> y;

}

model {

   // ...

   lp <- normal_log(y,mu,sigma) T[0,];

}

If y is a scalar parameter with constraints, then there's no need to add T[lower, upper] (if the bounds match). Stan will behave as though you e provided the truncation. 

If you need to use it in a mixture model with different constraints, you may need to include the normalizing constants on your own. 

In case you're interested in the model, I'm using a Gaussian mixture model for bivariate density estimation. Theoretically my data covers a really large range (time from 0 to 365 days, space from 0 to 20 km) but I'm only interested in a short-range model so I truncate my observations to 7 days and 2 km.

Providing the full Stan model would be useful. 

Thanks,

Seth

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[Bob Carpenter]

Just add in the appropriate log CDFs to normalize.
They're all spelled out in the manual, or you can
just take a look at how the C++ is generated for
the truncation.

We still don't have particularly numerically robust log CDFs or
log CCDFs yet --- now they're just doing the log of the CDF.

- Bob

> On Feb 20, 2016, at 6:20 AM, Seth Flaxman <fla...@gmail.com> wrote:
>

[Bob Carpenter]

Ack. Sorry for repeating Daniel's answer.
I only saw it in my mailer after I responded.

- Bob