# Gaussian process for parameters that are strictly positive

Hey there,

I was hoping some of you might give me some reassurance about the way I am modelling a set of parameters. In particular, I have a model that contains a set of N parameters \sigma_i that are strictly positive and for which I have some prior information that I would want to account for. This information comes in the form of a distance matrix D_{ij} that characterises the differences between indices i. To do so, at the moment, I am implementing this as Gaussian process, such as the prior for \sigma_i is modelled as follows:

\log(\sigma_i) \sim \text{MVNormal}\left(\bar{\sigma}, K\right),
\bar{\sigma}\sim \text{Normal}\left(0,1\right),
K_{ij} = \eta\,\exp(- \rho D_{ij}^2) +\delta_{ij}\,s,
s, \eta \sim \text{Exponential}\left(1\right)
\rho\sim \text{Exponential}\left(0.5\right)

This assumes \sigma_i to be lognormally distributed, and it seems to works pretty well with simulated data. That said, I am not sure if I am making any fundamentally flawed assumptions regarding the covariance of \sigma_i. I am surely coding things such that the covariance depends not only on D_{ij}, but also on the value of \sigma_i and \sigma_j. However, is that really bad if it seems to work well with simulated data? I haven’t quite found anything regarding this in the forum; though I’m not quite sure I am using the right search terms. Other distributions such as truncated MVNormal would behave similarly, but I don’t necessarily see the benefit of this over a lognormal distribution. Any ideas or thoughts?

Lognormal will have the consequence that the variability will increase with the mean on the measurement scale. In some data contexts this is an expected relationship in which case you’re fine, but if you’re not explicitly expecting that relationship, having a model that enforces it might be an issue. That said, a lognormal GP might be less compromised by this, but I don’t have much more than a vague intuition to support that. You indicate that you’re testing things using simulation and really that is already the best way to check these things.

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The model looks ok. There is an example of using GP prior for log(sigma) at Gaussian process demonstration with Stan

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Thank you both for your quick replies. The example you pointed out, @avehtari, is the sort of reassurance I was hoping for.

If you need a reference, see, e.g. this paper from 1997 Regression with Input-dependent Noise: A Gaussian Process Treatment

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That’s even better! Thanks a lot.

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