Posterior samples from multi-level model when choosing the index values

AidanMar · July 3, 2021, 4:13pm

I have a simple multi-level model without any parameter pooling in the model defined in the following way without any pooling of parameters:

coords = {"Country":country_list, "obs_id":np.arange(yearly.shape[0])}
with pm.Model(coords=coords) as unpooled_model:
    country_idx = pm.Data("country_idx", yearly.country_idx, dims="obs_id")
    a = pm.Exponential("a", 1/10, dims="Country")
    counts = pm.Poisson("counts", a[yearly.country_idx], observed = yearly["Gold"], dims="obs_id")

I can obtain posterior predictive samples from this model straightforwardly and visualise it with arviz. However, each country index (country_idx) value appears a different number of times within this dataset, and this results in getting imbalanced posterior predictive samples, coming from each country. What I would really like to do is only get posterior predictive samples for certain country_idx values. Whenever I have tried to do this using pm.set_data({“country_idx”:[1.4]}) for instance, I get errors regarding conflicts with the shape of the obs_id dimension in the original model. Is there a smart/easy way to do this?

I have looked through the tutorials on multi-level models and this isn’t done there either. For instance, in the linked tutorial, imagine I wanted posterior predictive samples, but only for the aitkin basement, and not the others. What is the best way to do this in pymc3?

Thanks in advance

drbenvincent · July 7, 2021, 9:51am

Does my answer on this thread help? There’s a link to a new forthcoming example notebook.

AidanMar · July 29, 2021, 5:28pm

I don’t think the link worked. Could you re-attach it?

drbenvincent · July 29, 2021, 5:50pm

Ah yes - The example has been merged into pymc-examples so I deleted by branch. It’s not yet live on the website, but you can access the notebook from the pymc-examples repo…

github.com

pymc-devs/pymc-examples/blob/main/examples/generalized_linear_models/GLM-simpsons-paradox.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simpson's paradox and mixed models\n",
    "Author: [Benjamin T. Vincent](https://github.com/drbenvincent)\n",
    "\n",
    "This notebook covers:\n",
    "- [Simpson's Paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) and its resolution through mixed or hierarchical models. This is a situation where there might be a negative relationship between two variables within a group, but when data from multiple groups are combined, that relationship may disappear or even reverse sign. The gif below (from the [Simpson's Paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox) Wikipedia page) demonstrates this very nicely.\n",
    "- How to build linear regression models, starting with linear regression, moving up to hierarchical linear regression. Simpon's paradox is a nice motivation for why we might want to do this - but of course we should aim to build models which incorporate as much as our knowledge about the structure of the data (e.g. it's nested nature) as possible.\n",
    "- Use of `pm.Data` containers to facilitate posterior prediction at different $x$ values with the same model.\n",
    "- Providing array dimensions (see `coords`) to models to help with shape problems. This involves the use of [xarray](http://xarray.pydata.org/) and is quite helpful in multi-level / hierarchical models.\n",
    "- Differences between posteriors and posterior predictive distributions.\n",
    "- How to visualise models in data space and parameter space, using a mixture of [ArviZ](https://arviz-devs.github.io/arviz/) and [matplotlib](https://matplotlib.org/)."
   ]
  },
  {
   "cell_type": "markdown",

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Posterior samples from multi-level model when choosing the index values

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