參考：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/

針對線性回歸中異常點，利用t分布來替換正態分布構建貝葉斯模型。

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

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