Implementation:Pyro ppl Pyro InclinedPlane

Property	Value
Implementation Type	Pattern Doc
Source File	examples/inclined_plane.py
Module	examples
Pyro Features	pyro.sample, Importance sampling, EmpiricalMarginal, simulator inversion, Uniform/Normal distributions
Pattern	Bayesian inference to invert a physics simulator

Overview

This file demonstrates a compelling use case of probabilistic programming: inverting a physics simulator using Bayesian inference. The scenario is a student (Samantha) who observes a box sliding down an inclined plane 20 times and wants to infer the coefficient of friction (mu) between the box and the plane.

The approach:

1. A deterministic simulator computes descent time given friction coefficient via numerical integration of equations of motion
2. The model places a Uniform(0, 1) prior on mu, runs the simulator, and adds Gaussian measurement noise
3. Importance sampling is used to infer the posterior distribution of mu given the observed descent times

This example is notable because the simulator contains a while loop with variable iteration count, demonstrating Pyro's ability to handle complex, non-differentiable program structures.

Code Reference

def simulate(mu, length=2.0, phi=np.pi / 6.0, dt=0.005, noise_sigma=None):
    T = torch.zeros(())
    velocity = torch.zeros(())
    displacement = torch.zeros(())
    acceleration = torch.tensor(little_g * np.sin(phi)) - \
                   torch.tensor(little_g * np.cos(phi)) * mu
    while displacement.numpy() < length:
        displacement += velocity * dt
        velocity += acceleration * dt
        T += dt
    if noise_sigma is None:
        return T
    else:
        return T + noise_sigma * torch.randn(())

def model(observed_data):
    mu_prior = Uniform(0.0, 1.0)
    mu = pyro.sample("mu", mu_prior)
    def observe_T(T_obs, obs_name):
        T_simulated = simulate(mu)
        T_obs_dist = Normal(T_simulated, torch.tensor(time_measurement_sigma))
        pyro.sample(obs_name, T_obs_dist, obs=T_obs)
    for i, T_obs in enumerate(observed_data):
        observe_T(T_obs, "obs_%d" % i)
    return mu

I/O Contract

Parameter	Type	Description
observed_data	torch.Tensor	Tensor of N observed descent times (seconds)
-n / --num-samples	int	Number of importance samples (default: 500)

Physical parameters:

• Plane length: 2 meters
• Incline angle: 30 degrees (pi/6)
• Measurement noise: 20 milliseconds
• True mu: 0.12

Output:

• Posterior mean and standard deviation of friction coefficient mu
• Comparison of observed, simulated, and analytic descent times

Usage Examples

from pyro.infer import Importance, EmpiricalMarginal

# Run importance sampling
importance = Importance(model, guide=None, num_samples=500)
emp_marginal = EmpiricalMarginal(importance.run(observed_data))

# Extract posterior statistics
posterior_mean = emp_marginal.mean
posterior_std = emp_marginal.variance.sqrt()

# Run from command line
# python inclined_plane.py -n 1000

Related Pages

• Pyro_ppl_Pyro_MiniPyro_Example - Simple inference example with MiniPyro