Monte Carlo simulation estimates an expectation by repeatedly generating samples from a distribution, applying a function to each sample, and averaging the results. It works for discrete and continuous random variables. section 2.2.4
A practical workflow is:
- Generate samples with a probabilistic model or random process. section 2.2.4
- Apply the desired function to every sample, then average the transformed values. section 2.2.4
- Represent probabilities with indicator values: average 1 when a condition holds and 0 otherwise. section 2.2.4
- Expect slightly different estimates across runs because random samples vary. section 2.2.4
Estimating a conditional expectation
A Pyro process generates z from a Gamma distribution, x from a Poisson distribution, and y from repeated Bernoulli draws. After generating 1,000 outcomes, averaging z estimates E(Z). Filtering to outcomes with y = 3 and averaging the retained z values estimates E(Z | Y = 3). This accept/reject approach becomes inefficient when the condition is rare. section 2.2.5
Listing 2.9 Monte Carlo estimation in Pyro
import torch
import pyro
from pyro.distributions import Bernoulli, Gamma, Poisson
def random_process():
z = pyro.sample("z", Gamma(7.5, 1.0))
x = pyro.sample("x", Poisson(z))
y = torch.tensor(0.0)
for i in range(int(x)):
y += pyro.sample(f"{i}", Bernoulli(.5))
return z, y #1
generated_samples = [random_process() for _ in range(1000)] #2
z_mean = torch.stack([z for z, _ in generated_samples]).mean() #3
print(z_mean)
Use the Pyro listing to trace random-process generation, tensor stacking, and Monte Carlo averaging. section 2.2.5
