Importance sampling performs probabilistic inference by generating outcomes from a joint distribution, weighting them according to their probability in a target distribution, and averaging the weighted outcomes with Monte Carlo estimation. It is useful when directly conditioning would discard many generated samples. section 2.2.5
In a Pyro causal model, condition the model on transportation='train', draw 5,000 importance samples, extract the education marginal, and convert sample counts into estimated probabilities. The resulting distribution approximates P(E | T='train'), though probabilistic inference alone is not yet causal inference. section 3.1.11
Estimating education for train riders
A conditioned Pyro model samples the education variable after fixing transportation to train. An empirical marginal and histogram summarize the sampled probabilities for high-school and university education. section 3.1.11
Listing 3.9 Inference on the causal model in Pyro
import numpy as np
import pyro
from pyro.distributions import Categorical
from pyro.infer import Importance, EmpiricalMarginal #1
import matplotlib.pyplot as plt
conditioned_model = pyro.condition( #2
model, #3
data={'transportation':torch.tensor(1.)} #3
)
m = 5000 #4
posterior = pyro.infer.Importance( #5
conditioned_model, #5
num_samples=m
).run() #6
E_marginal = EmpiricalMarginal(posterior, "education") #7
E_samples = [E_marginal().item() for _ in range(m)] #7
E_unique, E_counts = np.unique(E_samples, return_counts=True) #8
E_probs = E_counts / m #8
plt.bar(E_unique, E_probs, align='center', alpha=0.5) #9
plt.xticks(E_unique, E_alias) #9
plt.ylabel('probability') #9
plt.xlabel('education') #9
plt.title('P(E | T = "train") - Importance Sampling') #9
Inspect the Pyro implementation that conditions the model, runs 5,000 importance samples, extracts an empirical marginal, and plots estimated probabilities. section 3.1.11
