Hierarchical model with bounded Normals?

Questions
1

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/). 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

2

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?

2 Likes
3

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

4

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

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.