Added Rat Tumour Example

[Dpananos]

Added notebook for rat tumor example in chapter 5 of BDA.

I was looking for some resources in python for Gelman's first example of hierarchical models in BDA but couldn't find any. I decided to write one myself.

I'd appreciate any feedback re: this example.

[junpenglao]

Looks nice! A few comments:

• Please index the notebook under https://github.com/pymc-devs/pymc3/blob/master/docs/source/examples.rst#glm
• There are some guideline about unifying the style of the docs, please have a look at this issue: Update notebooks and examples #2834
• you can put ; at the end of plotting command suppresses the text output: pm.traceplot(trace);
• you should use the default initialization, also shorter trace is sufficient:
  trace = pm.sample(50_000, step = pm.NUTS()) --> trace = pm.sample(1000, tune=2000)
• Transformed is needed for the Densitydist as the parameter is defined on (0, inf), which is why on the discourse answer I supplied transform=tr.log https://discourse.pymc.io/t/reparametizing-gelmans-rat-tumour-example/1417/3?u=junpenglao. Otherwise you will get a divergence when a proposal went out of bound.
• use import theano.tensor as tt, tt is mostly used in our docs
• please write p = pm.Binomial('y', n=n_rat, p=theta, observed=n_tumour) which is more conventional.

[Dpananos]

I've omitted a discussion about transforms (as per your request on discourse) because the example no longer uses DensityDist. If you still feel an explanation is required, I'd be happy to include it. I just need to know what you'd like me to write.

Alternatively, we could use DensityDist in the example, but I would need a solution such that the estimates are not biased.

[junpenglao]

Could you please move this to the last session of GLM, also change the title of the notebook:
rat tumor--> Hierarchical Binominal Model: Rat Tumor Example

[junpenglao]

The math rendering looks a bit odd, i believe \\>. are not needed:

[Dpananos]

Changes made.

[junpenglao]

a few \\> still need to be removed.

[Dpananos]

Got em

[junpenglao]

Awesome, thanks!