# Data augmentation and latent variable

**URL:** https://discourse.pymc.io/t/data-augmentation-and-latent-variable/17398
**Category:** v5
**Tags:** modeling
**Created:** [October 23, 2025, 4:43pm UTC](https://discourse.pymc.io/t/data-augmentation-and-latent-variable/17398 "2025-10-23T16:43:05Z")
**Posts on this page:** 1
**Page:** 1

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### Author: ![tgx](https://avatars.discourse-cdn.com/v4/letter/t/ecae2f/32.png) [@tgx](https://discourse.pymc.io/u/tgx)
#### Post date: [October 23, 2025, 4:43pm UTC](https://discourse.pymc.io/t/data-augmentation-and-latent-variable/17398/1 "2025-10-23T16:43:05Z")

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Hi everyone,

I am a Bayesian beginner, and I am having some difficulty with a high-dimensional problem.

My goal is to fit a multivariate distribution (the r largest order statistics (GEVr) model) called f to an observational dataset drawn from this distribution + some Gaussian random noise. This distribution depends on 3 free parameters (noted \Theta) and takes as input a vector \mathbf{u} = (u\_1,u\_2,...,u\_r) of size r whose components are ranked i.e., u\_1\>u\_2\>...\>u\_r.

I have already managed to fit it to an idealistic observational dataset (without random noise), but now I’d like to move on to a more realistic case by adding some random Gaussian noise. In this scenario, the likelihood is obtained by convolving the GEVr f with a Gaussian distribution G, which is very costly and does not seem feasible in a reasonable amount of time. In that case the likelihood is given by;  
\mathcal{L}(\mathbf{U}^{obs}|\Theta) = \prod\_{i = 1}^{N}\int G(\mathbf{u}^{obs}\_i|\mathbf{u}) f(\mathbf{u}|\Theta)\rm{d}\mathbf{u}

I have read that a Bayesian way to solve this problem would be to avoid doing the integral and treat the vector \mathbf{u} as a latent variable and that one need to sample \mathbf{u} in addition to the rGEV free parameters. In that case the posterior would just be the product of this rGEV with the Gaussian distribution and the prior i.e.  
P(\theta|\mathbf{u}) \propto \prod\_{i = 1}^{N}G(\mathbf{u}^{obs}\_i|\mathbf{u}) f(\mathbf{u}|\Theta)\Pi(\Theta)

Given that I have N = 60 measurements of size r = 5, it means that the dimension of the problem is = 303. Can PyMC handle this dimensionality?

And my other difficulty is to take into account the fact that when sampling \mathbf{u}, I must take into account that u\_1\>u\_2\>...\>u\_r, , is there a way to efficiently do that with PyMC?

Thank you very much for your time.