Markov Chain

Monte Carlo and

model fitting
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N

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!

BAYESIAN DATA ANALYSIS IN PYTHON

Monte Carlo
Approximating some quantity by
generating random numbers

From the formula, πr2 ≃ 78.5

BAYESIAN DATA ANALYSIS IN PYTHON

Monte Carlo
Approximating some quantity by
generating random numbers

From the formula, πr2 ≃ 78.5

Draw a 10x10 square around the circle.

BAYESIAN DATA ANALYSIS IN PYTHON

Monte Carlo
Approximating some quantity by
generating random numbers

From the formula, πr2 ≃ 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

BAYESIAN DATA ANALYSIS IN PYTHON

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

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chains
Models a sequence of states, between What will the bear do next:
which one transitions with given
probabilities. hunt eat sleep
hunt 0.1 0.8 0.1
eat 0.05 0.4 0.55
sleep 0.8 0.15 0.05
A er many time periods, transition
probabilities become the same no ma er
where we started.

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chains
Models a sequence of states, between What will the bear do next:
which one transitions with given
probabilities. hunt eat sleep
hunt 0.1 0.8 0.1
eat 0.05 0.4 0.55
sleep 0.8 0.15 0.05
A er many time periods, transition What will the bear do in a distant future:
probabilities become the same no ma er
where we started. hunt eat sleep
hunt 0.28 0.44 0.28
eat 0.28 0.44 0.28
sleep 0.28 0.44 0.28

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

Markov Chain Monte Carlo

BAYESIAN DATA ANALYSIS IN PYTHON

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]

BAYESIAN DATA ANALYSIS IN PYTHON

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)

BAYESIAN DATA ANALYSIS IN PYTHON

 Let's practice
MCMC!
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N
 Interpreting results
and comparing
models
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N

Michal Oleszak
Machine Learning Engineer
Running the model revisited
formula = "num_clicks ~ clothes_banners_shown + sneakers_banners_shown"

with pm.Model() as model_1:

pm.GLM.from_formula(formula, data=ads_aggregated)
trace_1 = pm.sample(draws=1000, tune=500)

BAYESIAN DATA ANALYSIS IN PYTHON

Running the model revisited
formula = "num_clicks ~ clothes_banners_shown + sneakers_banners_shown"

with pm.Model() as model_1:

pm.GLM.from_formula(formula, data=ads_aggregated)
trace_1 = pm.sample(draws=1000, tune=500, chains=4)

Number of parameters: 4

Number of draws for each parameter: 1000 × 4 = 4000

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Trace plot
pm.traceplot(trace_1)

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Trace plot: zoom in on one parameter

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Forest plot
pm.forestplot(trace_1)

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Trace summary
pm.summary(trace_1)

mean sd hdi_3% hdi_97% mcse_mean mcse_sd \

Intercept 1.307 0.886 -0.305 2.962 0.018 0.013
clothes_banners_shown 0.103 0.031 0.043 0.160 0.001 0.000
sneakers_banners_shown 0.104 0.032 0.045 0.163 0.001 0.001
sd 2.654 0.157 2.382 2.970 0.003 0.002

ess_mean ess_sd ess_bulk ess_tail r_hat

Intercept 2346.0 2318.0 2351.0 2083.0 1.0
clothes_banners_shown 2085.0 2085.0 2089.0 1868.0 1.0
sneakers_banners_shown 2105.0 1953.0 2122.0 1869.0 1.0
sd 2615.0 2590.0 2646.0 1834.0 1.0

BAYESIAN DATA ANALYSIS IN PYTHON

Fitting another model
formula = "num_clicks ~ clothes_banners_shown + sneakers_banners_shown + weekend"

with pm.Model() as model_2:

pm.GLM.from_formula(formula, data=ads_aggregated)
trace_2 = pm.sample(draws=1000, tune=500)

BAYESIAN DATA ANALYSIS IN PYTHON

Widely Applicable Information Criterion (WAIC)
comparison = pm.compare({"trace_1": trace_1, "trace_2": trace_2},
ic="waic", scale="deviance")
print(comparison)

rank waic p_waic d_waic weight se dse warning \

trace_2 0 -362.8 5.1576 0 0.513792 9.37269 0 True
trace_1 1 -362.926 4.13318 0.126236 0.486208 9.48352 1.50682 True

waic_scale
trace_2 log
trace_1 log

BAYESIAN DATA ANALYSIS IN PYTHON

Compare plot
pm.compareplot(comparison)

BAYESIAN DATA ANALYSIS IN PYTHON

 Let's practice
comparing models!
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N
 Making predictions
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N

Michal Oleszak
Machine Learning Engineer
Number-of-clicks model again
formula = "num_clicks ~ clothes_banners_shown + sneakers_banners_shown + weekend"

with pm.Model() as model_2:

pm.GLM.from_formula(formula, data=ads_aggregated)
trace_2 = pm.sample(draws=1000, tune=500)

BAYESIAN DATA ANALYSIS IN PYTHON

Ads test data
print(ads_test)

clothes_banners_shown sneakers_banners_shown num_clicks weekend

0 40 36 7 True
1 42 47 8 False
2 45 37 11 False
3 22 15 4 False
4 20 18 2 False

BAYESIAN DATA ANALYSIS IN PYTHON

Sampling predictive draws
with pm.Model() as model:
pm.GLM.from_formula(formula, data=ads_test)
posterior_predictive = pm.fast_sample_posterior_predictive(trace_2)

BAYESIAN DATA ANALYSIS IN PYTHON

Predictive draws
posterior_predictive["y"].shape

(4000, 5)

print(posterior_predictive["y"])

array([[12.83527253, 10.22454815, 11.20386868, 7.50227286, 6.85458594],

[ 3.1015655 , 6.1253004 , 11.38324931, 2.1844722 , 4.21451756],
[ 3.40141276, 9.10157964, 6.57689421, 8.26669814, 4.23812161],
...,
[10.97303606, 9.0772305 , 10.6877039 , 1.78448969, 6.75663075],
[ 8.53734584, 12.14079593, 11.00969881, 4.69875055, 8.317338 ],
[16.44713387, 17.35163824, 19.59359831, 2.84058536, 4.21108186]])

BAYESIAN DATA ANALYSIS IN PYTHON

How good is the prediction?
clothes_banners_shown sneakers_banners_shown num_clicks weekend
0 40 36 7 True

pm.plot_posterior(posterior_predictive["y"][:, 0])

BAYESIAN DATA ANALYSIS IN PYTHON

Test error distribution
errors = []

for index, test_example in ads_test.iterrows():

error = posterior_predictive["y"][:, index] - test_example["num_clicks"]
errors.append(error)

error_distribution = np.array(errors).reshape(-1)
error_distribution.shape

(20000,)

pm.plot_posterior(error_distribution)

BAYESIAN DATA ANALYSIS IN PYTHON

Test error distribution

BAYESIAN DATA ANALYSIS IN PYTHON

 Let's make
predictions!
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N
 How much is an
avocado?
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N

Michal Oleszak
Machine Learning Engineer
The Avocado, Inc.

BAYESIAN DATA ANALYSIS IN PYTHON

Case study: estimating price elasticity
Goal: estimate price elasticity of avocados and optimize the price

(price elasticity = impact of the change in price on the sales volume)

1. Fit a Bayesian regression model.

2. Inspect the model to verify its correctness.

3. Predict sales volume for di erent prices.

4. Propose the pro t-maximizing price and the associated uncertainty.

BAYESIAN DATA ANALYSIS IN PYTHON

 Avocado data
print(avocado)

date price volume type_organic

0 2015-01-04 0.95 313.242777 0
1 2015-01-11 1.01 290.635427 0
2 2015-01-18 1.03 290.434588 0
3 2015-01-25 1.04 284.703108 0
.. ... ... ... ...
334 2018-03-04 1.52 16.344308 1
335 2018-03-11 1.52 16.642349 1
336 2018-03-18 1.54 16.758042 1
337 2018-03-25 1.55 15.599672 1

1 Data source: h ps://www.kaggle.com/neuromusic/avocado-prices

BAYESIAN DATA ANALYSIS IN PYTHON

Priors in pymc3
formula = "num_bikes ~ temp + work_day + wind_speed"

with pm.Model() as model:

pm.GLM.from_formula(formula, data=bikes)
trace = pm.sample(draws=1000, tune=500)

BAYESIAN DATA ANALYSIS IN PYTHON

Priors in pymc3
formula = "num_bikes ~ temp + work_day + wind_speed"

with pm.Model() as model:

priors = {"wind_speed": pm.Normal.dist(mu=-5)}
pm.GLM.from_formula(formula, data=bikes, priors=priors)
trace = pm.sample(draws=1000, tune=500)

BAYESIAN DATA ANALYSIS IN PYTHON

Extracting draws from trace
temp_draws = trace.get_values("temp")
print(temp_draws)

array([6.8705346, 6.7421152, 6.7393061, ..., 5.966574 , 6.1274128, 6.7149277])

BAYESIAN DATA ANALYSIS IN PYTHON

What you will need
Model ing: Visualization:

pm.Model() pm.forestplot()

pm.GLM.from_formula() pm.traceplot()

pm.sample()

pm.Normal()

Making predictions: Inference:

pm.fast_sample_posterior_predictive() pm.hpd()

BAYESIAN DATA ANALYSIS IN PYTHON

 Let's put what
you've learned to
practice!
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N
 Final remarks
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N

Michal Oleszak
Machine Learning Engineer
What you know
Chapter 1: The Bayesian Way Chapter 3: Bayesian Inference

Bayesian vs. frequentist approach A/B testing

Probability theory & distributions Decision analysis

Updating beliefs with more data Forecasting & regression

Chapter 2: Bayesian Estimation Chapter 4: Bayesian Linear Regression

Grid approximation Markov Chain Monte Carlo (MCMC)

Prior distributions Fi ing and interpreting models with pymc3

Reporting Bayesian results Bayesian data analysis: a case study

BAYESIAN DATA ANALYSIS IN PYTHON

More Bayes
Hierarchical models: PyMC3 docs:
h ps://pymc3.readthedocs.io/en/latest
y = β0 + β1 x1 + β2 x2

β2 = β20 + β21 x3
Think Bayes by Allen Downey
More regression (logistic, Poisson, ...)
h p://allendowney.github.io/ThinkBayes2
Bayesian machine learning

BAYESIAN DATA ANALYSIS IN PYTHON

Congratulations and
good luck!
B AY E S I A N D ATA A N A LY S I S I N P Y T H O N