Add matrix factorization example back to pymc3

[zaxtax]

Reintroducing the probabilistic matrix factorization example originally from @macks22 with the Movielens dataset instead of Jester. This could still use some polish, but the notebook works albeit slowly.

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[fonnesbeck]

This is great, and is already pretty polished, actually. Runs in about 30 min on my MBP.

[codecov]

Codecov Report

Merging #3709 into master will not change coverage.
The diff coverage is n/a.

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[ColCarroll]

If you run this again, can you bump to PyMC3 v3.8?

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[ColCarroll]

could be just a single bar plot for this, right? the density plot is a little weird

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[ColCarroll]

I'd use

print(f"Users: {num_users}\nMovies:", {num_items}\nSparsity: {sparsity}")

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[ColCarroll]

I would replace this with a function:

def make_pmf_model(train, dim, alpha, std):

    bounds = (1, 5)

    data = train.copy()

    n, m = data.shape

    # Perform mean value imputation

    nan_mask = np.isnan(data)

    data[nan_mask] = data[~nan_mask].mean()

    # Low precision reflects uncertainty; prevents overfitting.

    # Set to the mean variance across users and items.

    alpha_u = 1 / data.var(axis=1).mean()

    alpha_v = 1 / data.var(axis=0).mean()

    # Specify the model.

    logging.info('building the PMF model')

    with pm.Model() as pmf:

      U = pm.MvNormal(

        'U', mu=0, tau=alpha_u * np.eye(dim),

        shape=(n, dim), testval=np.random.randn(n, dim) * std)

      V = pm.MvNormal(

        'V', mu=0, tau=alpha_v * np.eye(dim),

        shape=(m, dim), testval=np.random.randn(m, dim) * std)

      R = pm.Normal(

        'R', mu=(U @ V.T)[~nan_mask], tau=alpha, observed=data[~nan_mask])

return pmf

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[ColCarroll]

I'd replace the self in these with a pmf instance.

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[ColCarroll]

i'd update this to use just pm.sample(draws=500, tune=500) (or similar!)

This should automatically use 1 chain per physical core and run diagnostics.

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[ColCarroll]

this self.std is defined as 1 / alpha in the constructor, and is maybe only used here

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[ColCarroll]

pmf = make_pmf_model(train, dim=10, alpha=2, std=0.05)

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[ColCarroll]

with pmf:

map_estimate = pm.find_MAP()

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[ColCarroll]

This is great @zaxtax -- thanks for getting this example back up and running! I can regen the docs so that it is up on the examples site, too.