Update radon example NB

[AlexAndorra]

Finally got time to finish this PR 🍾
I noticed on Discourse that the radon notebook is often used as an introduction to hierarchical models, but several questions were recurring so I thought it'd useful to:

• Update with new code and versions of PyMC and ArviZ
• Replace flat priors by weakly regularizing ones (both to encourage best practices and to preemptively answer Discourse questions)
• Add prior predictive checks
• Replace the section about varying-slopes only by a model including the covariance between varying slopes and varying intercepts
• Implement non-centered versions of the hierarchical models (this is necessary in this case to get rid of divergences + we often get questions about that, so this will hopefully be a useful ressource)
• Update and add graphs
• Do some miscellaneous changes

First PR on the main repo (🎉 ), so tell me if I forgot anything. And of course I'm available for any change.
Happy new year & PyMCheers 🖖

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Codecov Report

Merging #3765 into master will increase coverage by 0.07%.
The diff coverage is n/a.

@@            Coverage Diff             @@
##           master    #3765      +/-   ##
==========================================
+ Coverage   90.55%   90.63%   +0.07%     
==========================================
  Files         133      133              
  Lines       20316    20328      +12     
==========================================
+ Hits        18397    18424      +27     
+ Misses       1919     1904      -15

Impacted Files	Coverage Δ
pymc3/tests/test_model.py	100% <0%> (ø)	⬆️
pymc3/sampling.py	83.11% <0%> (+0.02%)	⬆️
pymc3/step_methods/compound.py	95.12% <0%> (+0.12%)	⬆️
pymc3/model.py	89.17% <0%> (+1.91%)	⬆️

[fonnesbeck]

Thanks for doing this! -- its definitely an upgrade.

A couple of notes:

• for didactic purposes, it might be best to separate sampling and posterior predictive sampling into tow different cells (at least for the first time it is presented)
• spelling: should be "envelope" (a noun) not "envelop" (a verb)
• its probably best to use cores=2 rather than chains=2 in calls to sample, as for some systems (e.g. Colab, some Chromebooks) the latter will result in sequential sampling of chains.
• add plt.tight_layout() call to multi-panel plots to make the labels more readable (at least the ones showing the county-wise estimates)
• why not use the symbol ρ rather than spelling out "Rho", since we use symbols for the other Greek letters?

[AlexAndorra]

Thanks Chris!
I integrated all your points and I just had a small remaining question: I removed chains=2 and didn't specify a number of cores (since I understood that parallel sampling is not possible on Windows), but now the fit is worse for complex models (starting from varying_intercept_slope): some divergences and the acceptance probability does not match the target.
Do you know what could be the cause? (Running on Mac, 4 chains in 4 job)

[AlexAndorra]

Ok I think I figured it out: it needed a mix of increased tuning and higher target accept. Pushing the changes ;)

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michaelosthege commented on 2020-01-08T18:13:28Z
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The warnings thrown by this cell were fixed in a recent arviz PR: https://github.com/arviz-devs/arviz/pull/989/files

If you update your arviz and re-run, they should go away.

AlexAndorra commented on 2020-01-09T10:22:16Z
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Thanks for the flag Michael, I updated and re-ran ;)

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aloctavodia commented on 2020-01-09T18:21:27Z
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There is no longer need to use %matplotlib inline

It is better to set the style az.style.use("arviz-darkgrid") in a cell other than the one used to import arviz, otherwise not all the setting in the arviz (or matploblib) style will be properly set.

AlexAndorra commented on 2020-01-10T09:13:18Z
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Ow thanks, didn't know that! Just pushed the changes ;)

[aloctavodia]

Good to see you around @AlexAndorra, I just added a couple of super minor comments. I really like what you did with this notebook.

[AlexAndorra]

Ha ha thx Osvaldo, glad to be around too - the neighborhood is pretty cool 😉