Jupyter Snippet NP ch16-code-listing
Jupyter Snippet NP ch16-code-listing
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).
import pymc3 as mc
import numpy as np
%matplotlib inline
%config InlineBackend.figure_format='retina'
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()
from scipy import stats
import statsmodels.api as sm
import statsmodels.formula.api as smf
import pandas as pd
Changelog:
- 160828: The keyword argument
varsto the functionsmc.traceplotandmc.forestplotwas changed tovarnames.
Simple example: Normal distributed random variable
# try this
# dir(mc.distributions)
np.random.seed(100)
mu = 4.0
sigma = 2.0
model = mc.Model()
with model:
mc.Normal('X', mu, tau=1/sigma**2)
model.vars
[X]
start = dict(X=2)
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.
x = np.linspace(-4, 12, 1000)
y = stats.norm(mu, sigma).pdf(x)
X = trace.get_values("X")
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")
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
model = mc.Model()
with model:
mean = mc.Normal('mean', 3.0)
sigma = mc.HalfNormal('sigma', sd=1.0)
X = mc.Normal('X', mean, tau=sigma)
model.vars
[mean, sigma_log__, X]
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]
start
{'mean': array(3.),
'sigma_log__': array(-0.34657365),
'X': array(3.),
'sigma': array(0.70710674)}
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.
trace.get_values('sigma').mean()
0.7783082456057299
X = trace.get_values('X')
X.mean()
3.1083864227035884
trace.get_values('X').std()
2.843632679264834
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
mu = 2.5
s = 1.5
data = stats.norm(mu, s).rvs(100)
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)
model.vars
[mean, sigma_log__]
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.
start
{'mean': array(2.30048494),
'sigma_log__': array(0.3732095),
'sigma': array(1.45238858)}
model.vars
[mean, sigma_log__]
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")
mu, trace.get_values('mean').mean()
(2.5, 2.300439672735639)
s, trace.get_values('sigma').mean()
(1.5, 1.4773833088778436)
gs = mc.forestplot(trace, varnames=['mean', 'sigma'])
plt.savefig("ch16-forestplot.pdf")
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
mc.summary(trace, varnames=['mean', 'sigma'])
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Linear regression
dataset = sm.datasets.get_rdataset("Davis", "carData")
data = dataset.data[dataset.data.sex == 'M']
data = data[data.weight < 110]
data.head(3)
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model = smf.ols("height ~ weight", data=data)
result = model.fit()
print(result.summary())
OLS Regression Results
==============================================================================
Dep. Variable: height R-squared: 0.327
Model: OLS Adj. R-squared: 0.319
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.
x = np.linspace(50, 110, 25)
y = result.predict({"weight": x})
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")
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))
model.vars
[sigma_interval__, intercept, beta, predict_height]
# help(mc.NUTS)
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.
model.vars
[sigma_interval__, intercept, beta, predict_height]
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")
intercept = trace.get_values("intercept").mean()
intercept
152.15912056403442
beta = trace.get_values("beta").mean()
beta
0.3424401905691073
result.params
Intercept 152.617348
weight 0.336477
dtype: float64
result.predict({"weight": 90})
0 182.9003
dtype: float64
weight_index = np.where(x == 90)[0][0]
trace.get_values("predict_height")[:, weight_index].mean()
182.97564659506045
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")
fig, ax = plt.subplots(1, 1, figsize=(12, 4))
for n in range(500, 2000, 1):
intercept = trace.get_values("intercept")[n]
beta = trace.get_values("beta")[n]
ax.plot(x, intercept + beta * x, color='red', lw=0.25, alpha=0.05)
intercept = trace.get_values("intercept").mean()
beta = trace.get_values("beta").mean()
ax.plot(x, intercept + beta * x, color='k', label="Mean Bayesian prediction")
ax.plot(data.weight, data.height, 'o')
ax.plot(x, y, '--', color="blue", label="OLS prediction")
ax.set_xlabel("weight")
ax.set_ylabel("height")
ax.legend(loc=0)
fig.tight_layout()
fig.savefig("ch16-linear-model-fit.pdf")
fig.savefig("ch16-linear-model-fit.png")
with mc.Model() as model:
mc.glm.GLM.from_formula('height ~ weight', data)
step = mc.NUTS()
trace = mc.sample(2000, step)
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sd, weight, Intercept]
Sampling 4 chains: 100%|██████████| 10000/10000 [00:15<00:00, 634.51draws/s]
The acceptance probability does not match the target. It is 0.8788452420871716, 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 0.9413562668375693, 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 0.9289299829864369, but should be close to 0.8. Try to increase the number of tuning steps.
fig, axes = plt.subplots(3, 2, figsize=(8, 6), squeeze=False)
mc.traceplot(trace, varnames=['Intercept', 'weight', 'sd'], ax=axes)
fig.tight_layout()
fig.savefig("ch16-glm-sample-trace.pdf")
fig.savefig("ch16-glm-sample-trace.png")
Multilevel model
dataset = sm.datasets.get_rdataset("Davis", "carData")
data = dataset.data.copy()
data = data[data.weight < 110]
data["sex"] = data["sex"].apply(lambda x: 1 if x == "F" else 0)
data.head()
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with mc.Model() as model:
# heirarchical model: hyper priors
#intercept_mu = mc.Normal("intercept_mu", 125)
#intercept_sigma = 30.0 #mc.Uniform('intercept_sigma', lower=0, upper=50)
#beta_mu = mc.Normal("beta_mu", 0.0)
#beta_sigma = 5.0 #mc.Uniform('beta_sigma', lower=0, upper=10)
# multilevel model: prior parameters
intercept_mu, intercept_sigma = 125, 30
beta_mu, beta_sigma = 0.0, 5.0
# priors
intercept = mc.Normal('intercept', intercept_mu, sd=intercept_sigma, shape=2)
beta = mc.Normal('beta', beta_mu, sd=beta_sigma, shape=2)
error = mc.Uniform('error', 0, 10)
# model equation
sex_idx = data.sex.values
height_mu = intercept[sex_idx] + beta[sex_idx] * data.weight
mc.Normal('height', mu=height_mu, sd=error, observed=data.height)
model.vars
[intercept, beta, error_interval__]
with model:
start = mc.find_MAP()
# hessian = mc.find_hessian(start)
step = mc.NUTS()
trace = mc.sample(5000, step, start=start)
/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 = -618.03, ||grad|| = 8.8049: 100%|██████████| 51/51 [00:00<00:00, 1715.39it/s]
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [error, beta, intercept]
Sampling 4 chains: 100%|██████████| 22000/22000 [00:51<00:00, 428.65draws/s]
fig, axes = plt.subplots(3, 2, figsize=(8, 6), squeeze=False)
mc.traceplot(trace, varnames=['intercept', 'beta', 'error'], ax=axes)
fig.tight_layout()
fig.savefig("ch16-multilevel-sample-trace.pdf")
fig.savefig("ch16-multilevel-sample-trace.png")
intercept_m, intercept_f = trace.get_values('intercept').mean(axis=0)
intercept = trace.get_values('intercept').mean()
beta_m, beta_f = trace.get_values('beta').mean(axis=0)
beta = trace.get_values('beta').mean()
fig, ax = plt.subplots(1, 1, figsize=(8, 3))
mask_m = data.sex == 0
mask_f = data.sex == 1
ax.plot(data.weight[mask_m], data.height[mask_m], 'o', color="steelblue", label="male", alpha=0.5)
ax.plot(data.weight[mask_f], data.height[mask_f], 'o', color="green", label="female", alpha=0.5)
x = np.linspace(35, 110, 50)
ax.plot(x, intercept_m + x * beta_m, color="steelblue", label="model male group")
ax.plot(x, intercept_f + x * beta_f, color="green", label="model female group")
ax.plot(x, intercept + x * beta, color="black", label="model both groups")
ax.set_xlabel("weight")
ax.set_ylabel("height")
ax.legend(loc=0)
fig.tight_layout()
fig.savefig("ch16-multilevel-linear-model-fit.pdf")
fig.savefig("ch16-multilevel-linear-model-fit.png")
trace.get_values('error').mean()
5.131501939157903
Version
%reload_ext version_information
%version_information numpy, pandas, matplotlib, statsmodels, pymc3, theano