This example shows you how to perform Bayesian inference on a time series, using emcee hammer MCMC as introduced in [1].
It follows on from the first sampling example.
[1] "emcee: The MCMC Hammer", Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, Jonathan Goodman, 2013, arXiv, https://arxiv.org/pdf/1202.3665.pdf
In [7]:
import pints
import pints.toy as toy
import pints.plot
import numpy as np
import matplotlib.pyplot as plt
# Load a forward model
model = toy.LogisticModel()
# Create some toy data
real_parameters = [0.015, 500]
times = np.linspace(0, 1000, 1000)
org_values = model.simulate(real_parameters, times)
# Add noise
noise = 10
values = org_values + np.random.normal(0, noise, org_values.shape)
real_parameters = np.array(real_parameters + [noise])
# Get properties of the noise sample
noise_sample_mean = np.mean(values - org_values)
noise_sample_std = np.std(values - org_values)
# Create an object with links to the model and time series
problem = pints.SingleOutputProblem(model, times, values)
# Create a log-likelihood function (adds an extra parameter!)
log_likelihood = pints.GaussianLogLikelihood(problem)
# Create a uniform prior over both the parameters and the new noise variable
log_prior = pints.UniformLogPrior(
[0.01, 400, noise * 0.1],
[0.02, 600, noise * 100],
)
# Create a posterior log-likelihood (log(likelihood * prior))
log_posterior = pints.LogPosterior(log_likelihood, log_prior)
# Choose starting points for 3 mcmc chains
num_chains = 3
xs = [real_parameters * (1 + 0.1 * np.random.rand()) for i in range(num_chains)]
# Create mcmc routine
mcmc = pints.MCMCController(
log_posterior, num_chains, xs, method=pints.EmceeHammerMCMC)
# Add stopping criterion
mcmc.set_max_iterations(6000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(500)
# Run!
print('Running...')
chains = mcmc.run()
print('Done!')
In [8]:
# Check convergence using rhat criterion
print('R-hat:')
print(pints.rhat_all_params(chains))
# Show traces and histograms
pints.plot.trace(chains)
# Discard warm up
chains = chains[:, 500:, :]
# Apply thinning
chains = chains[:, ::10]
# Look at distribution in chain 0
pints.plot.pairwise(chains[0], kde=True)
# Show graphs
plt.show()