Hierarchical model

[ramanshah]

For the third box of t-shirts, it's becoming clear that the Dirichlet prior isn't really adequate to come up with the best orders. From a state like this:

	MXS	MS	MM	ML	MXL	M2XL	M3XL	WS	WM	WL	WXL	W2XL	totals
Lifetime received	1	4	19	13	4	3	1	3	9	4	6	3	70
Lifetime queued	0	1	19	12	2	0	0	3	8	1	2	1	49
Lifetime shipped	0	1	18	11	2	0	0	3	8	1	2	1	47
Actual inventory (received - shipped)	1	3	1	2	2	3	1	0	1	3	4	2	23
Inventory once caught up (received - queued)	1	3	0	1	2	3	1	0	1	3	4	2	21

We're getting an order like this:

Optimal order:
MXS : 0
MS  : 0
MM  : 13
ML  : 8
MXL : 2
M2XL: 0
M3XL: 0
WS  : 3
WM  : 6
WL  : 1
WXL : 1
W2XL: 1

Note how the Dirichlet prior is resulting in requests to buy more L, XL, and 2XL women's shirts even though in all cases I already have >=2x as many of each size as has ever been requested. This seems clearly to be waste: the problem is that the model can't use the knowledge that women are generally under-represented to reduce the expected number of orders for any given women's size.

Instead, we want to estimate a hierarchical structure, pooling knowledge of the gender ratio of orders to improve the posterior frequencies of a given gendered size. This would require upgrading from trivial closed-form Dirichlet math to (likely PyMC3) MCMC methods for the hierarchical likelihood.