# Hierarchical model with bounded Normals?

**URL:** <https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369>\
**Category:** Questions\
**Created:** [January 16, 2020, 9:36am UTC](https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369 "2020-01-16T09:36:21Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![jeti](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/jeti/32/7206_2.png) [@jeti](https://discourse.pymc.io/u/jeti)\
**Post date:** [January 16, 2020, 9:36am UTC](https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369/1 "2020-01-16T09:36:21Z")

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Dear all,

quite often when I set up a hierarchical model in in pymc3 I encounter the following problem: My lower-level parameters (e.g., subject-level parameters) are drawn from higher-level distributions (e.g. group-level parameters). Now quite often I have parameters on the lower level that are not supposed to take negative values. In a non-hierarchical version, I can use direct priors that establish this bound. But how do I proceed in the hierarchical version, where I typically use Normal distributions between the group and subject level? Here are a few more detailed questions/ideas:

a) Can I use bounded Normals instead? I often use a non-central parametrization ([https://twiecki.io/blog/2017/02/08/bayesian-hierchical-non-centered/](https://twiecki.io/blog/2017/02/08/bayesian-hierchical-non-centered/)). Can I simply add a min() operation to the deterministic calculation of the normally distributed values?

b) Should I rather use a log-Normal to link the subject and group level? How would I have to transform the parameters?

c) Or should I (log-?) transform the data?

I know the question is not pymc3 specific per se. But I believe there are some expert here who hopefully could help and I’d be very happy about some pymc3 specific solutions.

Many thanks in advance!  
Jan

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**Author:** ![dpananos](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/dpananos/32/925_2.png) [@dpananos](https://discourse.pymc.io/u/dpananos)\
**Post date:** [January 18, 2020, 6:36am UTC](https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369/2 "2020-01-18T06:36:16Z")

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I run into this quite often; subject level parameters are only interpretable if they are positive.

One strategy I use is to model the logarithm of the population mean for these random variables and then have them come from a log-normal distribution. Here is an example from my own work on pharmacokinetics.

I usually need to model the volume of blood in a patient’s body. Obviously, this must be a positive number and can vary between patients, so a random effect is suitable. I may specify my generative model as follows

```
log_mu = pm.Normal('log_mu', 0, 1)
z = pm.Normal('z', 0, 1, shape = N_patients)
sigma = 0.25 #Alternatively, specify a prior
volume = pm.math.exp(log_mu + z*sigma)

```

Since `z` is normal, then `log_mu+z*sigma` is normal, which means taking the expoenenital of this is expression yields a log-normal random variable. The exponential forces the parameters to be positive without having to resort to constraining priors via `pm.Bound`.

Is this helpful?

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**Author:** ![jeti](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/jeti/32/7206_2.png) [@jeti](https://discourse.pymc.io/u/jeti)\
**Post date:** [January 20, 2020, 12:23pm UTC](https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369/3 "2020-01-20T12:23:23Z")

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Yes, thank you, I think it might help. Just trying it out. I’m not yet sure about some practical aspects. e.g. do I have to exp(log\_mu) to get my group-level mean? How does the transformation affect the prior I put on sigma …? Any hints highly welcome…

Thanks again  
Jan

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<div class="post-metadata">

**Author:** ![dpananos](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/dpananos/32/925_2.png) [@dpananos](https://discourse.pymc.io/u/dpananos)\
**Post date:** [January 20, 2020, 5:26pm UTC](https://discourse.pymc.io/t/hierarchical-model-with-bounded-normals/4369/4 "2020-01-20T17:26:22Z")

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Under this parameterization, the random variable is log normal. You can leverage all the information about log normal random variables to find things like the expectation. See here [https://en.wikipedia.org/wiki/Log-normal\_distribution](https://en.wikipedia.org/wiki/Log-normal_distribution)

So in this case, the population level mean would be `exp(mu + sigma/2)`

As for the prior on sigma, obviously it can’t be too large because the exponential will just make it worse. I would recommend you do some prior predictive checks.