# Robust positive continuous likelihood

**URL:** <https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395>\
**Category:** General\
**Tags:** modeling\
**Created:** [May 7, 2024, 5:56am UTC](https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395 "2024-05-07T05:56:57Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![opherdonchin](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/opherdonchin/32/5948_2.png) [@opherdonchin](https://discourse.pymc.io/u/opherdonchin)\
**Post date:** [May 7, 2024, 5:56am UTC](https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395/1 "2024-05-07T05:56:57Z")

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This may be the wrong place to ask this, so feel free to redirect.

I have data that is positive and continuous but has outliers. I normally use the Gamma distribution to model positive continuous data or parameters. Is there some equivalent robust version the way that the Student’s T is used as a robust likelihood in place of the Normal distribution? If not based on Gamma, is there any standard way to do a robust likelihood for positive continuous variables?

(Added in edit:) There is a [generalized Gamma distribution](https://en.wikipedia.org/wiki/Generalized_gamma_distribution) which has an extra shape parameter `p`. I also thought that the easiest thing to do (programmatically) is just to use a truncated Student’s t.

Opher

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**Author:** ![jessegrabowski](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/jessegrabowski/32/5010_2.png) [@jessegrabowski](https://discourse.pymc.io/u/jessegrabowski)\
**Post date:** [May 7, 2024, 6:10am UTC](https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395/2 "2024-05-07T06:10:48Z")

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You try a LogStudentT distribution. The generative graph is just the exp of student T, so it’s easy to implement using a custom dist:

```auto
import pymc as pm

def log_student_t(mu, sigma, nu, size=None):
    return pm.math.exp(pm.StudentT.dist(mu=mu, sigma=sigma, nu=nu, size=size))

with pm.Model() as m:
    # priors for mu, sigma, nu
    y_hat = pm.CustomDist(mu, sigma, nu, dist=log_student_t, observed=y_data)

```

For the prior on `nu`, @bwengals was doing some work on a model like this with a special prior for `nu`, but the specifics escape me. Hopefully he can chime in himself.

We also recently experimented with a mixture of LogNormals, so each datapoint was classified via latent variable into either a low sigma or high sigma distribution, each with the same mean. The idea was an analogy to the student T as a mixture of infinite normals.

But as to the question of whether there’s a “standard” approach here, I haven’t seen one if there is.

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**Author:** ![opherdonchin](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/opherdonchin/32/5948_2.png) [@opherdonchin](https://discourse.pymc.io/u/opherdonchin)\
**Post date:** [May 7, 2024, 7:29am UTC](https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395/3 "2024-05-07T07:29:12Z")

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Thanks this is helpful. I added some ideas I’d had before above. If there’s nothing standard, we’ll just pick something and go with it.

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**Author:** ![bwengals](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/bwengals/32/1237_2.png) [@bwengals](https://discourse.pymc.io/u/bwengals)\
**Post date:** [May 14, 2024, 6:37am UTC](https://discourse.pymc.io/t/robust-positive-continuous-likelihood/14395/4 "2024-05-14T06:37:51Z")

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RE the prior on `nu`, the PC prior is a good choice. There’s a PR for it in PyMC experimental if you’re brave and not on windows. Or, to keep it simple, you can use Gamma(alpha=2, beta=0.1). That’s very close to the PC prior that says, “I think there’s a 50% chance that `nu` is greater than 30”, where 30 is the sort of recognized place where student t’s look very normal.