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Var(y) = V (E(y)) = V (g −1 (Xβ)) (5.7)
In the frequentist setting, as with ordinary linear regression above, the unknown β coefficients
are estimated via a maximum likelihood approach.
I’m not going to discuss GLMs in depth here as they are not the focus of the chapter or the
book. We are interested in them because we will be using the glm module from PyMC3, which
was written by Thomas Wiecki[102] and others, in order to easily specify our Bayesian linear
regression.
5.3.2 Simulating Data and Fitting the Model with PyMC3
Before we utilise PyMC3 to specify and sample a Bayesian model, we need to simulate some
noisy linear data. The following snippet carries this out, which is modified and extended from
Jonathan Sedar’s post[90]:
import numpy as np
import pandas as pd
import seaborn as sns
sns.set(style="darkgrid", palette="muted")
def simulate_linear_data(N, beta_0, beta_1, eps_sigma_sq):
"""
Simulate a random dataset using a noisy
linear process.
N: Number of data points to simulate
beta_0: Intercept
beta_1: Slope of univariate predictor, X
"""
# Create a pandas DataFrame with column ’x’ containing
# N uniformly sampled values between 0.0 and 1.0
df = pd.DataFrame(
{"x":
np.random.RandomState(42).choice(
map(
lambda x: float(x)/100.0,
np.arange(100)
), N, replace=False
)
}
)