# Introduction to Probabilistic Programming¶

## Probabilistic Programming¶

• Programming with random variables
• User specifies a generative model for the observed data
• "Tell the story" of how the data were created
• The language runtime/software library automatically performs inference

### What do we mean by inference?¶

• Maximum likelihood inference
• Maxumum a posteriori inference
• Full posterior inference

## Disease Testing¶

A certain rare disease affects one in 10,000 people. A test for the disease gives the correct result 99.9% of the time.

What is the probability that a given test is positive?

In [8]:
import pymc3 as pm

with pm.Model() as pretest_disease_model:
# This is our prior belief about whether or not the subject has the disease,
# given its prevalence of the disease in the general population
has_disease = pm.Bernoulli('has_disease', 1e-4)

# This is the probability of a positive test given the presence or lack
# of the disease
p_positive = pm.Deterministic('p_positive',
T.switch(T.eq(has_disease, 1),
# If the subject has the disease,
# this is the probability that the
# test is positive
0.999,
# If the subject does not have the
# disease, this is the probability
# that the test is negative
1 - 0.999))

# This is the observed test result
test_positive = pm.Bernoulli('test_positive', p_positive)

In [9]:
samples = 40000

In [10]:
with pretest_disease_model:
step = pm.Metropolis()
pretest_disease_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 40000 of 40000 complete in 6.1 sec
In [11]:
pretest_disease_trace['test_positive']

Out[11]:
array([0, 0, 0, ..., 0, 0, 0])
In [12]:
pretest_disease_trace['test_positive'].size

Out[12]:
40000

The probability that a given test is positive is

In [13]:
pretest_disease_trace['test_positive'].mean()

Out[13]:
0.001575
\begin{align*} P(\textrm{Test } +) = &\ P(\textrm{Test } +\ |\ \textrm{Disease}) P(\textrm{Disease}) \\ & + P(\textrm{Test } +\ |\ \textrm{No disease}) P(\textrm{No disease}) \end{align*}
In [14]:
0.999 * 1e-4  + (1 - 0.999) * (1 - 1e-4)

Out[14]:
0.0010998000000000008

If a subject tests positive, what is the probability they have the disease?

In [15]:
with pm.Model() as disease_model:
# This is our prior belief about whether or not the subject has the disease,
# given its prevalence of the disease in the general population
has_disease = pm.Bernoulli('has_disease', 1e-4)

# This is the probability of a positive test given the presence or lack
# of the disease
p_positive = pm.Deterministic('p_positive',
T.switch(T.eq(has_disease, 1),
# If the subject has the disease,
# this is the probability that the
# test is positive
0.999,
# If the subject does not have the
# disease, this is the probability
# that the test is negative
1 - 0.999))

# This is the observed positive test
test_positive = pm.Bernoulli('test_positive', p_positive, observed=1)

In [16]:
with disease_model:
step = pm.Metropolis()
disease_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 40000 of 40000 complete in 3.8 sec

The probability that someone who tests positive for the disease has it is

In [17]:
disease_trace['has_disease'].mean()

Out[17]:
0.096775
\begin{align*} P(\textrm{Disease}\ |\ \textrm{Test }+) & = \frac{P(\textrm{Test }+\ |\ \textrm{Disease}) P(\textrm{Disease})}{P(\textrm{Test }+)} \end{align*}
In [18]:
0.999 * 1e-4 / (0.999 * 1e-4 + (1 - 0.999) * (1 - 1e-4))

Out[18]:
0.09083469721767587

## The Monty Hall Problem¶

If we select the first door and Monty opens the third to reveal a goat, should we change doors?

In [19]:
with pm.Model() as monty_model:
# We have no idea where the prize is at the beginning
prize = pm.DiscreteUniform('prize', 0, 2)

# The probability that Monty opens each door
p_open = pm.Deterministic('p_open',
T.switch(T.eq(prize, 0),
# If the prize is behind the first door,
# he chooses to open one of the others
# at random
np.array([0., 0.5, 0.5]),
T.switch(T.eq(prize, 1),
# If it is behind the second door,
# he must open the third door
np.array([0., 0., 1.]),
# If it is behind the third door,
# he must open the second door
np.array([0., 1., 0.]))))

# Monty opened the third door, revealing a goat
opened = pm.Categorical('opened', p_open, observed=2)

In [20]:
samples = 10000

In [21]:
with monty_model:
step = pm.Metropolis()
monty_trace = pm.sample(samples, step, random_seed=SEED)

monty_trace_df = pm.trace_to_dataframe(monty_trace)

 [-----------------100%-----------------] 10000 of 10000 complete in 1.4 sec
In [22]:
monty_trace_df.head()

Out[22]:
p_open__0 p_open__1 p_open__2 prize
0 0.0 0.0 1.0 1
1 0.0 0.5 0.5 0
2 0.0 0.0 1.0 1
3 0.0 0.0 1.0 1
4 0.0 0.0 1.0 1
In [23]:
(monty_trace_df.groupby('prize')
.size()
.div(monty_trace_df.shape[0]))

Out[23]:
prize
0    0.3196
1    0.6804
dtype: float64

## PyMC3¶

PyMC3 is a python module for Bayesian statistical modeling and model fitting which focuses on advanced Markov chain Monte Carlo fitting algorithms. Its flexibility and extensibility make it applicable to a large suite of problems.

Built on top of theano

## A/B Testing¶

Suppose variant A sees 510 visitors and 40 purchases, but variant B sees 505 visitors and 50 conversions.

In [24]:
with pm.Model() as ab_model:
# A priori, we have no idea what the purchase rate for variant A will be
a_rate = pm.Uniform('a_rate')
# We observed 40 purchases from 510 visitors who saw variant A
a_purchases = pm.Binomial('a_purchases', 510, a_rate, observed=40)

# A priori, we have no idea what the purchase rate for variant B will be
b_rate = pm.Uniform('b_rate')
# We observed 40 purchases from 510 visitors who saw variant A
b_purchases = pm.Binomial('b_purchases', 505, b_rate, observed=45)

Applied interval-transform to a_rate and added transformed a_rate_interval to model.
Applied interval-transform to b_rate and added transformed b_rate_interval to model.

In [25]:
samples = 10000

In [26]:
with ab_model:
step = pm.NUTS()
ab_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 10000 of 10000 complete in 3.2 sec
In [28]:
fig

Out[28]:

The probability that variant B is better than variant A is

In [29]:
(ab_trace['a_rate'] < ab_trace['b_rate']).mean()

Out[29]:
0.7218

## Robust Regression¶

In [32]:
fig

Out[32]:
In [36]:
fig

Out[36]:
In [37]:
with pm.Model() as ols_model:
# alpha is the slope and beta is the intercept
alpha = pm.Uniform('alpha', -100, 100)
beta = pm.Uniform('beta', -100, 100)

# Ordinary least squares assumes that the noise is normally distributed
y_obs = pm.Normal('y_obs', alpha + beta * x, 1., observed=y)

Applied interval-transform to alpha and added transformed alpha_interval to model.
Applied interval-transform to beta and added transformed beta_interval to model.

In [38]:
samples = 10000

In [39]:
with ols_model:
step = pm.Metropolis()
ols_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 10000 of 10000 complete in 2.2 sec
In [40]:
post_y = ols_trace['alpha'] + np.outer(plot_X[:, 1], ols_trace['beta'])

In [42]:
fig

Out[42]:
In [44]:
fig

Out[44]:
In [45]:
with pm.Model() as robust_model:
# alpha is the slope and beta is the intercept
alpha = pm.Uniform('alpha', -100, 100)
beta = pm.Uniform('beta', -100, 100)

# t-distributed residuals are less sensitive to outliers
y_obs = pm.StudentT('y_obs', nu=3., mu=alpha + beta * x, observed=y)

Applied interval-transform to alpha and added transformed alpha_interval to model.
Applied interval-transform to beta and added transformed beta_interval to model.

In [46]:
with robust_model:
step = pm.Metropolis()

robust_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 10000 of 10000 complete in 2.3 sec
In [47]:
post_y_robust = robust_trace['alpha'] + np.outer(plot_X[:, 1], robust_trace['beta'])

In [49]:
fig

Out[49]:

## Congressional Ideal Point Model¶

In [52]:
df.head()

Out[52]:
party 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
rep
0 0 0 1 0 1 1 1 0 0 0 1 NaN 1 1 1 0 1
1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 0 NaN
2 1 NaN 1 1 NaN 1 1 0 0 0 0 1 0 1 1 0 0
3 1 0 1 1 0 NaN 1 0 0 0 0 1 0 1 0 0 1
4 1 1 1 1 0 1 1 0 0 0 0 1 NaN 1 1 1 1
In [53]:
parties

Out[53]:
Index(['republican', 'democrat'], dtype='object')
In [56]:
grid.fig

Out[56]:

$p_{i, j}$ is the probability that representative $i$ votes for bill $j$.

\begin{align*} \log \left(\frac{p_{i, j}}{1 - p_{i, j}}\right) & = \gamma_j \left(\alpha_i - \beta_j\right) \\ & = \textrm{ability of bill } j \textrm{ to discriminate} \times (\textrm{conservativity of represetative } i\ - \textrm{conservativity of bill } j) \end{align*}
In [57]:
with pm.Model() as ideal_point_model:
# The conservativity of representative i
alpha = pm.Normal('alpha', 0, 1., shape=N_REPS)

# The conservativity of bill j
mu_beta = pm.Normal('mu_beta', 0, 1e-3)
sigma_beta = pm.Uniform('sigma_beta', 0, 1e2)
beta = pm.Normal('beta', mu_beta, sd=sigma_beta, shape=N_BILLS)

# The ability of bill j to discriminate
mu_gamma = pm.Normal('mu_gamma', 0, 1e-3)
sigma_gamma = pm.Uniform('sigma_gamma', 0, 1e2)
gamma = pm.Normal('gamma', mu_gamma, sd=sigma_gamma, shape=N_BILLS)

# The probability that representative i votes for bill j
p = pm.Deterministic('p', T.nnet.sigmoid(gamma[vote_df.bill] * (alpha[vote_df.rep] - beta[vote_df.bill])))

vote = pm.Bernoulli('vote', p, observed=vote_df.vote)

Applied interval-transform to sigma_beta and added transformed sigma_beta_interval to model.
Applied interval-transform to sigma_gamma and added transformed sigma_gamma_interval to model.

In [58]:
samples = 20000

In [59]:
with ideal_point_model:
step = pm.Metropolis()
ideal_point_trace = pm.sample(samples, step, random_seed=SEED)

 [-----------------100%-----------------] 20000 of 20000 complete in 107.5 sec
In [60]:
rep_alpha = ideal_point_trace['alpha'].mean(axis=0)

In [62]:
fig

Out[62]:
In [63]:
pm.forestplot(ideal_point_trace, varnames=['gamma']);


## Probabilistic Programming Ecosystem¶

Probabilistic Programming System Language Discrete Variable Support Automatic Differentiation/Hamiltonian Monte Carlo Variational Inference
PyMC3 Python 2/3
PyMC2 Python 2/3
Stan C++, R, Python 2/3
BUGS Standalone program, R
JAGS Standalone program, R

## References¶

Think Bayes (Available freely online, calculations in Python)

## Thank You!¶

The Jupyter notebook these slides were generated from is available here