The HMC Revolution is Open Source

Probabilistic Programming with PyMC3

@AustinRochfordTom Tom AMLC • Charlottesville • 4/11/19

Who am I?

PyMC3 developer • Chief Data Scientist @ Monetate


About This Talk

About Monetate

  • Founded 2008, web optimization and personalization SaaS
  • Observed 5B impressions and $4.1B in revenue during Cyber Week 2017

Modern Bayesian Inference

Motivating Examples

2017 UK General Election

NBA Foul Calls

Player skills(?)

Probabilistic Programming

Data Science — inference enables storytelling

Probabilistic Programming — storytelling enables inference

The Monty Hall Problem

Initially, we have no information about which door the prize is behind.

In [8]:
import pymc3 as pm

with pm.Model() as monty_model:
    prize = pm.DiscreteUniform('prize', 0, 2)

If we choose door one:

Monty can open
Prize behind Door 1 Door 2 Door 3
Door 1 No Yes Yes
Door 2 No No Yes
Door 3 No Yes No
In [9]:
from theano import tensor as tt

with monty_model:
    p_open = pm.Deterministic(
        tt.switch(tt.eq(prize, 0),
            np.array([0., 0.5, 0.5]), # it is behind the first door
        tt.switch(tt.eq(prize, 1),
            np.array([0., 0., 1.]),   # it is behind the second door
            np.array([0., 1., 0.])))  # it is behind the third door

Monty opened the third door, revealing a goat.

In [10]:
with monty_model:
    opened = pm.Categorical('opened', p_open, observed=2)

Should we switch our choice of door?

In [13]:
with monty_model:
    monty_trace = pm.sample(1000, **MONTY_SAMPLE_KWARGS)
monty_df = pm.trace_to_dataframe(monty_trace)
Multiprocess sampling (3 chains in 3 jobs)
Metropolis: [prize]
Sampling 3 chains: 100%|██████████| 4500/4500 [00:01<00:00, 4173.25draws/s]
In [14]:
0    0
1    0
2    0
3    0
4    1
Name: prize, dtype: int64
In [15]:
ax = (monty_df['prize']
              .value_counts(normalize=True, ascending=True)
              .plot(kind='bar', color='C0'))

ax.set_ylabel("Probability of prize");

From the PyMC3 documentation:

PyMC3 is a Python package for Bayesian statistical modeling and Probabilistic Machine Learning which focuses on advanced Markov chain Monte Carlo and variational fitting algorithms. Its flexibility and extensibility make it applicable to a large suite of problems.


Case Study: NBA Foul Calls

Question: Is (not) committing and/or drawing fouls a measurable player skill?

In [19]:
play_id 20150301CLEHOU-0 20150301CLEHOU-1
period Q4 Q4
seconds_left 112 103
call_type Foul: Shooting Foul: Shooting
committing_player Josh Smith J.R. Smith
disadvantaged_player Kevin Love James Harden
review_decision CNC CC
away CLE CLE
home HOU HOU
date 2015-03-01 00:00:00 2015-03-01 00:00:00
score_away 103 103
score_home 105 105
disadvantaged_team CLE HOU
committing_team HOU CLE

Model outline

$$ \operatorname{log-odds}(\textrm{Foul}) \ \sim \textrm{Season factor} + \left(\textrm{Disadvantaged skill} - \textrm{Committing skill}\right) $$

In [29]:
with pm.Model() as irt_model:
    β_season = pm.Normal('β_season', 0., 2.5, shape=n_season)
    θ = hierarchical_normal('θ', n_player) # disadvantaged skill
    b = hierarchical_normal('b', n_player) # committing skill

    p = pm.math.sigmoid(
        β_season[season] + θ[player_disadvantaged] - b[player_committing]
    obs = pm.Bernoulli(
        'obs', p,

Metropolis-Hastings Inference

In [30]:
with irt_model: 
    step = pm.Metropolis()
    met_trace = pm.sample(5000, step=step, **SAMPLE_KWARGS)
Multiprocess sampling (3 chains in 3 jobs)
>Metropolis: [σ_b]
>Metropolis: [Δ_b]
>Metropolis: [σ_θ]
>Metropolis: [Δ_θ]
>Metropolis: [β_season]
Sampling 3 chains: 100%|██████████| 16500/16500 [01:48<00:00, 152.44draws/s]
The gelman-rubin statistic is larger than 1.4 for some parameters. The sampler did not converge.
The estimated number of effective samples is smaller than 200 for some parameters.
In [33]:
In [34]:
max(np.max(var_stats) for var_stats in pm.gelman_rubin(met_trace).values())

The Curse of Dimensionality

This model has

In [35]:
n_param = n_season + 2 * n_player


In [38]:

Hamiltonian Monte Carlo Inference

Bayesian inference ⇔ Differential geometry

Case Study Continued: NBA Foul Calls

In [39]:
with irt_model:
    nuts_trace = pm.sample(500, **SAMPLE_KWARGS)
Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (3 chains in 3 jobs)
NUTS: [σ_b, Δ_b, σ_θ, Δ_θ, β_season]
Sampling 3 chains: 100%|██████████| 3000/3000 [02:14<00:00, 22.23draws/s]
In [40]:
nuts_gr_stats = pm.gelman_rubin(nuts_trace)
max(np.max(var_stats) for var_stats in nuts_gr_stats.values())
In [43]:

Basketball strategy leads to more complexity

In [48]:

Next Steps with Probabilistic Programming


Probabilistic Programming Ecosystem