# Chapter 16: Bayesian statistics¶

Robert Johansson

Source code listings for Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib (ISBN 978-1-484242-45-2).

In [1]:
import pymc3 as mc

In [2]:
import numpy as np

In [3]:
%matplotlib inline
%config InlineBackend.figure_format='retina'
import matplotlib.pyplot as plt

In [4]:
import seaborn as sns
sns.set()

In [5]:
from scipy import stats

In [6]:
import statsmodels.api as sm

In [7]:
import statsmodels.formula.api as smf

In [8]:
import pandas as pd


## Changelog:¶

• 160828: The keyword argument vars to the functions mc.traceplot and mc.forestplot was changed to varnames.

# Simple example: Normal distributed random variable¶

In [9]:
# try this
# dir(mc.distributions)

In [10]:
np.random.seed(100)

In [11]:
mu = 4.0

In [12]:
sigma = 2.0

In [13]:
model = mc.Model()

In [14]:
with model:
mc.Normal('X', mu, tau=1/sigma**2)

In [15]:
model.vars

Out[15]:
[X]
In [16]:
start = dict(X=2)

In [17]:
with model:
step = mc.Metropolis()
trace = mc.sample(10000, step=step, start=start, jobs=1)

Multiprocess sampling (4 chains in 4 jobs)
Metropolis: [X]
Sampling 4 chains: 100%|██████████| 42000/42000 [00:06<00:00, 6966.26draws/s]
The number of effective samples is smaller than 25% for some parameters.

In [18]:
x = np.linspace(-4, 12, 1000)

In [19]:
y = stats.norm(mu, sigma).pdf(x)

In [20]:
X = trace.get_values("X")

In [21]:
fig, ax = plt.subplots(figsize=(8, 3))

ax.plot(x, y, 'r', lw=2)
sns.distplot(X, ax=ax)
ax.set_xlim(-4, 12)
ax.set_xlabel("x")
ax.set_ylabel("Probability distribution")
fig.tight_layout()
fig.savefig("ch16-normal-distribution-sampled.pdf")

In [22]:
fig, axes = plt.subplots(1, 2, figsize=(12, 4), squeeze=False)
mc.traceplot(trace, ax=axes)
axes[0,0].plot(x, y, 'r', lw=0.5)
fig.tight_layout()
fig.savefig("ch16-normal-sampling-trace.png")
fig.savefig("ch16-normal-sampling-trace.pdf")


## Dependent random variables¶

In [23]:
model = mc.Model()

In [24]:
with model:
mean = mc.Normal('mean', 3.0)
sigma = mc.HalfNormal('sigma', sd=1.0)
X = mc.Normal('X', mean, tau=sigma)

In [25]:
model.vars

Out[25]:
[mean, sigma_log__, X]
In [26]:
with model:
start = mc.find_MAP()

/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/pymc3/tuning/starting.py:61: UserWarning: find_MAP should not be used to initialize the NUTS sampler, simply call pymc3.sample() and it will automatically initialize NUTS in a better way.
warnings.warn('find_MAP should not be used to initialize the NUTS sampler, simply call pymc3.sample() and it will automatically initialize NUTS in a better way.')
logp = -2.4949, ||grad|| = 0.13662: 100%|██████████| 7/7 [00:00<00:00, 1168.42it/s]

In [27]:
start

Out[27]:
{'mean': array(3.),
'sigma_log__': array(-0.34657365),
'X': array(3.),
'sigma': array(0.70710674)}
In [28]:
with model:
step = mc.Metropolis()
trace = mc.sample(10000, start=start, step=step)

Multiprocess sampling (4 chains in 4 jobs)
CompoundStep
>Metropolis: [X]
>Metropolis: [sigma]
>Metropolis: [mean]
Sampling 4 chains: 100%|██████████| 42000/42000 [00:10<00:00, 4053.99draws/s]
The number of effective samples is smaller than 10% for some parameters.

In [29]:
trace.get_values('sigma').mean()

Out[29]:
0.7783082456057299
In [30]:
X = trace.get_values('X')

In [31]:
X.mean()

Out[31]:
3.1083864227035884
In [32]:
trace.get_values('X').std()

Out[32]:
2.843632679264834
In [33]:
fig, axes = plt.subplots(3, 2, figsize=(8, 6), squeeze=False)
mc.traceplot(trace, varnames=['mean', 'sigma', 'X'], ax=axes)
fig.tight_layout()
fig.savefig("ch16-dependent-rv-sample-trace.png")
fig.savefig("ch16-dependent-rv-sample-trace.pdf")


## Posterior distributions¶

In [34]:
mu = 2.5

In [35]:
s = 1.5

In [36]:
data = stats.norm(mu, s).rvs(100)

In [37]:
with mc.Model() as model:

mean = mc.Normal('mean', 4.0, tau=1.0) # true 2.5
sigma = mc.HalfNormal('sigma', tau=3.0 * np.sqrt(np.pi/2)) # true 1.5

X = mc.Normal('X', mean, tau=1/sigma**2, observed=data)

In [38]:
model.vars

Out[38]:
[mean, sigma_log__]
In [39]:
with model:
start = mc.find_MAP()
step = mc.Metropolis()
trace = mc.sample(10000, start=start, step=step)
#step = mc.NUTS()
#trace = mc.sample(10000, start=start, step=step)

/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/pymc3/tuning/starting.py:61: UserWarning: find_MAP should not be used to initialize the NUTS sampler, simply call pymc3.sample() and it will automatically initialize NUTS in a better way.
warnings.warn('find_MAP should not be used to initialize the NUTS sampler, simply call pymc3.sample() and it will automatically initialize NUTS in a better way.')
logp = -189.07, ||grad|| = 0.052014: 100%|██████████| 14/14 [00:00<00:00, 1754.83it/s]
Multiprocess sampling (4 chains in 4 jobs)
CompoundStep
>Metropolis: [sigma]
>Metropolis: [mean]
Sampling 4 chains: 100%|██████████| 42000/42000 [00:07<00:00, 5311.96draws/s]
The number of effective samples is smaller than 25% for some parameters.

In [40]:
start

Out[40]:
{'mean': array(2.30048494),
'sigma_log__': array(0.3732095),
'sigma': array(1.45238858)}
In [41]:
model.vars

Out[41]:
[mean, sigma_log__]
In [42]:
fig, axes = plt.subplots(2, 2, figsize=(8, 4), squeeze=False)
mc.traceplot(trace, varnames=['mean', 'sigma'], ax=axes)
fig.tight_layout()
fig.savefig("ch16-posterior-sample-trace.png")
fig.savefig("ch16-posterior-sample-trace.pdf")

In [43]:
mu, trace.get_values('mean').mean()

Out[43]:
(2.5, 2.300439672735639)
In [44]:
s, trace.get_values('sigma').mean()

Out[44]:
(1.5, 1.4773833088778436)
In [45]:
gs = mc.forestplot(trace, varnames=['mean', 'sigma'])
plt.savefig("ch16-forestplot.pdf")

In [46]:
help(mc.summary)

Help on function summary in module pymc3.stats:

summary(trace, varnames=None, transform=<function <lambda> at 0x1c1d6e9ae8>, stat_funcs=None, extend=False, include_transformed=False, alpha=0.05, start=0, batches=None)
Create a data frame with summary statistics.

Parameters
----------
trace : MultiTrace instance
varnames : list
Names of variables to include in summary
transform : callable
Function to transform data (defaults to identity)
stat_funcs : None or list
A list of functions used to calculate statistics. By default,
the mean, standard deviation, simulation standard error, and
highest posterior density intervals are included.

The functions will be given one argument, the samples for a
variable as a 2 dimensional array, where the first axis
corresponds to sampling iterations and the second axis
represents the flattened variable (e.g., x__0, x__1,...). Each
function should return either

1) A pandas.Series instance containing the result of
calculating the statistic along the first axis. The name
attribute will be taken as the name of the statistic.
2) A pandas.DataFrame where each column contains the
result of calculating the statistic along the first axis.
The column names will be taken as the names of the
statistics.
extend : boolean
If True, use the statistics returned by stat_funcs in
addition to, rather than in place of, the default statistics.
This is only meaningful when stat_funcs is not None.
include_transformed : bool
Flag for reporting automatically transformed variables in addition
to original variables (defaults to False).
alpha : float
The alpha level for generating posterior intervals. Defaults
to 0.05. This is only meaningful when stat_funcs is None.
start : int
The starting index from which to summarize (each) chain. Defaults
to zero.
batches : None or int
Batch size for calculating standard deviation for non-independent
samples. Defaults to the smaller of 100 or the number of samples.
This is only meaningful when stat_funcs is None.

Returns
-------
pandas.DataFrame with summary statistics for each variable Defaults one
are: mean, sd, mc_error, hpd_2.5, hpd_97.5, n_eff and Rhat.
Last two are only computed for traces with 2 or more chains.

Examples
--------
.. code:: ipython

>>> import pymc3 as pm
>>> trace.mu.shape
(1000, 2)
>>> pm.summary(trace, ['mu'])
mean        sd  mc_error     hpd_5    hpd_95
mu__0  0.106897  0.066473  0.001818 -0.020612  0.231626
mu__1 -0.046597  0.067513  0.002048 -0.174753  0.081924

n_eff      Rhat
mu__0     487.0   1.00001
mu__1     379.0   1.00203

Other statistics can be calculated by passing a list of functions.

.. code:: ipython

>>> import pandas as pd
>>> def trace_sd(x):
...     return pd.Series(np.std(x, 0), name='sd')
...
>>> def trace_quantiles(x):
...     return pd.DataFrame(pm.quantiles(x, [5, 50, 95]))
...
>>> pm.summary(trace, ['mu'], stat_funcs=[trace_sd, trace_quantiles])
sd         5        50        95
mu__0  0.066473  0.000312  0.105039  0.214242
mu__1  0.067513 -0.159097 -0.045637  0.062912


In [47]:
mc.summary(trace, varnames=['mean', 'sigma'])

Out[47]:
mean sd mc_error hpd_2.5 hpd_97.5 n_eff Rhat
mean 2.300440 0.147201 0.001956 2.002971 2.577492 6580.255931 1.000826
sigma 1.477383 0.098052 0.001563 1.300465 1.677372 4046.655011 1.000122

## Linear regression¶

In [48]:
dataset = sm.datasets.get_rdataset("Davis", "carData")

In [49]:
data = dataset.data[dataset.data.sex == 'M']

In [50]:
data = data[data.weight < 110]

In [51]:
data.head(3)

Out[51]:
sex weight height repwt repht
0 M 77 182 77.0 180.0
3 M 68 177 70.0 175.0
5 M 76 170 76.0 165.0
In [52]:
model = smf.ols("height ~ weight", data=data)

In [53]:
result = model.fit()

In [54]:
print(result.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                 height   R-squared:                       0.327
Method:                 Least Squares   F-statistic:                     41.35
Date:                Mon, 06 May 2019   Prob (F-statistic):           7.11e-09
Time:                        15:52:50   Log-Likelihood:                -268.20
No. Observations:                  87   AIC:                             540.4
Df Residuals:                      85   BIC:                             545.3
Df Model:                           1
Covariance Type:            nonrobust
==============================================================================
coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    152.6173      3.987     38.281      0.000     144.691     160.544
weight         0.3365      0.052      6.431      0.000       0.232       0.441
==============================================================================
Omnibus:                        5.734   Durbin-Watson:                   2.039
Prob(Omnibus):                  0.057   Jarque-Bera (JB):                5.660
Skew:                           0.397   Prob(JB):                       0.0590
Kurtosis:                       3.965   Cond. No.                         531.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

In [55]:
x = np.linspace(50, 110, 25)

In [56]:
y = result.predict({"weight": x})

In [57]:
fig, ax = plt.subplots(1, 1, figsize=(8, 3))
ax.plot(data.weight, data.height, 'o')
ax.plot(x, y, color="blue")
ax.set_xlabel("weight")
ax.set_ylabel("height")
fig.tight_layout()
fig.savefig("ch16-linear-ols-fit.pdf")

In [85]:
with mc.Model() as model:
sigma = mc.Uniform('sigma', 0, 10)
intercept = mc.Normal('intercept', 125, sd=30)
beta = mc.Normal('beta', 0, sd=5)

height_mu = intercept + beta * data.weight

# likelihood function
mc.Normal('height', mu=height_mu, sd=sigma, observed=data.height)

# predict
predict_height = mc.Normal('predict_height', mu=intercept + beta * x, sd=sigma, shape=len(x))

In [86]:
model.vars

Out[86]:
[sigma_interval__, intercept, beta, predict_height]
In [87]:
# help(mc.NUTS)

In [88]:
with model:
# start = mc.find_MAP()
step = mc.NUTS()
trace = mc.sample(10000, step) # , start=start)

Multiprocess sampling (4 chains in 4 jobs)
NUTS: [predict_height, beta, intercept, sigma]
Sampling 4 chains: 100%|██████████| 42000/42000 [01:24<00:00, 495.44draws/s]
The acceptance probability does not match the target. It is 0.8815345011357728, but should be close to 0.8. Try to increase the number of tuning steps.
The acceptance probability does not match the target. It is 0.8914826083396686, but should be close to 0.8. Try to increase the number of tuning steps.

In [89]:
model.vars

Out[89]:
[sigma_interval__, intercept, beta, predict_height]
In [91]:
fig, axes = plt.subplots(2, 2, figsize=(8, 4), squeeze=False)
mc.traceplot(trace, varnames=['intercept', 'beta'], ax=axes)
fig.savefig("ch16-linear-model-sample-trace.pdf")
fig.savefig("ch16-linear-model-sample-trace.png")

In [92]:
intercept = trace.get_values("intercept").mean()
intercept

Out[92]:
152.15912056403442
In [93]:
beta = trace.get_values("beta").mean()
beta

Out[93]:
0.3424401905691073
In [94]:
result.params

Out[94]:
Intercept    152.617348
weight         0.336477
dtype: float64
In [95]:
result.predict({"weight": 90})

Out[95]:
0    182.9003
dtype: float64
In [96]:
weight_index = np.where(x == 90)[0][0]

In [97]:
trace.get_values("predict_height")[:, weight_index].mean()

Out[97]:
182.97564659506045
In [98]:
fig, ax = plt.subplots(figsize=(8, 3))

sns.distplot(trace.get_values("predict_height")[:, weight_index], ax=ax)
ax.set_xlim(150, 210)
ax.set_xlabel("height")
ax.set_ylabel("Probability distribution")
fig.tight_layout()
fig.savefig("ch16-linear-model-predict-cut.pdf")