概率编程库Pymc3案例之鲁棒线性回归

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参考：https://twiecki.io/blog/2013/08/27/bayesian-glms-2/

https://twiecki.io/blog/2014/03/17/bayesian-glms-3/

https://twiecki.github.com/blog/2013/08/12/bayesian-glms-1/

针对线性回归中异常点，利用t分布来替换正态分布构建贝叶斯模型。

%matplotlib inline
import pymc3 as pm
import matplotlib.pyplot as plt
import numpy as np
import theano
size = 100
true_intercept = 1
true_slope = 2

x = np.linspace(0, 1, size)
# y = a + b*x
true_regression_line = true_intercept + true_slope * x
# add noise
y = true_regression_line + np.random.normal(scale=.5, size=size)
# Add outliers
x_out = np.append(x, [.1, .15, .2])
y_out = np.append(y, [8, 6, 9])
data = dict(x=x_out, y=y_out)
fig = plt.figure(figsize=(12, 12))
ax = fig.add_subplot(111, xlabel='x', ylabel='y', title='Generated data and underlying model')
ax.plot(x_out, y_out, 'x', label='sampled data')
ax.plot(x, true_regression_line, label='true regression line', lw=2.)
plt.legend(loc=0);

with pm.Model() as model_robust:
    family = pm.glm.families.StudentT()
    pm.GLM.from_formula('y ~ x', data, family=family)
    trace_robust = pm.sample(progressbar=False, tune=1000)

plt.figure(figsize=(12, 12))
plt.plot(x_out, y_out, 'x')
pm.plots.plot_posterior_predictive_glm(trace_robust,
                                       label='posterior predictive regression lines')
plt.plot(x, true_regression_line, 
         label='true regression line', lw=3., c='y')
plt.legend();

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [lam, x, Intercept]

PyMC3's glm() function allows you to pass in a family object that contains information about the likelihood.

• By changing the likelihood from a Normal distribution to a Student T distribution -- which has more mass in the tails -- we can perform Robust Regression.