Markov Chain Monte Carlo and model fitting

Bayesian Data Analysis in Python

Michal Oleszak

Machine Learning Engineer

Bayesian data analysis in production

Grid approximation: inconvenient with many parameters
Sampling from known posterior: requires conjugate priors
Markov Chain Monte Carlo (MCMC): sampling from unknown posterior!

Monte Carlo

Approximating some quantity by generating random numbers
From the formula, $\pi r^2 \simeq 78.5$

A circle with a radius of 5.

Monte Carlo

Approximating some quantity by generating random numbers
From the formula, $\pi r^2 \simeq 78.5$
Draw a 10x10 square around the circle.

A circle with a radius of 5 with a square defined on it.

Monte Carlo

Approximating some quantity by generating random numbers
From the formula, $\pi r^2 \simeq 78.5$
Draw a 10x10 square around the circle.
Sample 25 random points in the square.
How many are within the circle? $19/25=76\%$
Circle's area approximation: 76% * 100 = 76

A circle with a radius of 5 with a square defined on it and 25 points randomly positioned within the square.

Markov Chains

Models a sequence of states, between which one transitions with given probabilities.

Markov Chains

Models a sequence of states, between which one transitions with given probabilities.

What will the bear do next:

	hunt	eat	sleep
hunt	0.1	0.8	0.1
eat	0.05	0.4	0.55
sleep	0.8	0.15	0.05

After many time periods, transition probabilities become the same no matter where we started.

Markov Chains

Models a sequence of states, between which one transitions with given probabilities.

What will the bear do next:

	hunt	eat	sleep
hunt	0.1	0.8	0.1
eat	0.05	0.4	0.55
sleep	0.8	0.15	0.05

After many time periods, transition probabilities become the same no matter where we started.

What will the bear do in a distant future:

	hunt	eat	sleep
hunt	0.28	0.44	0.28
eat	0.28	0.44	0.28
sleep	0.28	0.44	0.28

Markov Chain Monte Carlo

A single red dot along a numbered axis.

Markov Chain Monte Carlo

Two dots along a numbered axis, one red and one black.

Markov Chain Monte Carlo

Two dots along a numbered axis, one red and one green.

Markov Chain Monte Carlo

Three dots along a numbered axis, one red, one green, and one black.

Markov Chain Monte Carlo

Two dots along a numbered axis, two red, and one green.

Markov Chain Monte Carlo

Four dots along a numbered axis, two red, and two green.

Markov Chain Monte Carlo

Seven dots along a numbered axis, two red, and five green.

Markov Chain Monte Carlo

Many dots along a numbered axis, some of them red, some most green.

Aggregated ads data

print(ads_aggregated)

           date  clothes_banners_shown  sneakers_banners_shown  num_clicks
0    2019-01-01                     20                      18           2
1    2019-01-02                     24                      19           8
2    2019-01-03                     20                      20           5
..          ...                    ...                     ...         ...
148  2019-05-29                     24                      25           8
149  2019-05-30                     26                      27          11
150  2019-05-31                     26                      24           8

[151 rows x 4 columns]

Linear regression with pyMC3

formula = "num_clicks ~ clothes_banners_shown + sneakers_banners_shown"


with pm.Model() as model:
    pm.GLM.from_formula(formula, data=ads_aggregated)

    # Print model specification
    print(model)

    # Sample posterior draws
    trace = pm.sample(draws=1000, tune=500)

Output from pymc3 print function, listing the priors for model parameters.

Let's practice MCMC!

Bayesian Data Analysis in Python