This example shows you how to perform Bayesian inference on a Gaussian distribution and a time-series problem, using Monomial-Gamma HMC.
First, we create a simple normal distribution
In [1]:
import pints
import pints.toy
import numpy as np
import matplotlib.pyplot as plt
# Create log pdf
log_pdf = pints.toy.GaussianLogPDF([2, 4], [[1, 0], [0, 3]])
# Contour plot of pdf
levels = np.linspace(-3,12,20)
num_points = 100
x = np.linspace(-1, 5, num_points)
y = np.linspace(-0, 8, num_points)
X, Y = np.meshgrid(x, y)
Z = np.zeros(X.shape)
Z = np.exp([[log_pdf([i, j]) for i in x] for j in y])
plt.contour(X, Y, Z)
plt.xlabel('x')
plt.ylabel('y')
plt.show()
Now we set up and run a sampling routine using Monomial-Gamma HMC MCMC
In [2]:
# Choose starting points for 3 mcmc chains
xs = [
[2, 1],
[3, 3],
[5, 4],
]
# Create mcmc routine
sigma = [1, 1]
mcmc = pints.MCMCController(log_pdf, 3, xs, method=pints.MonomialGammaHamiltonianMCMC, sigma0=sigma)
# Add stopping criterion
mcmc.set_max_iterations(1000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(100)
# change 'a' parameter in kinetic energy function used by individual samplers
for sampler in mcmc.samplers():
sampler.set_a(0.5)
# Run!
print('Running...')
full_chains = mcmc.run()
print('Done!')
In [3]:
# Show traces and histograms
import pints.plot
pints.plot.trace(full_chains)
plt.show()
In [4]:
# Discard warm up
chains = full_chains[:, 200:]
# Check convergence using rhat criterion
print('R-hat:')
print(pints.rhat_all_params(chains))
# Check Kullback-Leibler divergence of chains
print(log_pdf.kl_divergence(chains[0]))
print(log_pdf.kl_divergence(chains[1]))
print(log_pdf.kl_divergence(chains[2]))
# Look at distribution in chain 0
pints.plot.pairwise(chains[0], kde=True)
plt.show()
In [5]:
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
times = np.linspace(0, 1000, 50)
real_parameters = np.array([0.015, 500])
org_values = model.simulate(real_parameters, times)
# Add noise
np.random.seed(1)
noise = 10
values = org_values + np.random.normal(0, noise, org_values.shape)
# Create an object with links to the model and time series
problem = pints.SingleOutputProblem(model, times, values)
# Create a log-likelihood function
log_likelihood = pints.GaussianKnownSigmaLogLikelihood(problem, noise)
# Create a uniform prior over the parameters
log_prior = pints.UniformLogPrior(
[0.01, 400],
[0.02, 600]
)
# Create a posterior log-likelihood (log(likelihood * prior))
log_posterior = pints.LogPosterior(log_likelihood, log_prior)
# Choose starting points for 3 mcmc chains
xs = [
real_parameters * 1.01,
real_parameters * 0.9,
real_parameters * 1.1,
]
# Create mcmc routine
mcmc = pints.MCMCController(log_posterior, len(xs), xs, method=pints.MonomialGammaHamiltonianMCMC)
# Add stopping criterion
mcmc.set_max_iterations(1000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(100)
# Run!
print('Running...')
chains = mcmc.run()
print('Done!')
The chains do not take long to reach equilibrium with this method.
In [6]:
# Show traces and histograms
pints.plot.trace(chains)
plt.show()
Chains have converged!
In [7]:
# Discard warm up
chains = chains[:, 200:]
# Check convergence using rhat criterion
print('R-hat:')
print(pints.rhat_all_params(chains))
Extract any divergent iterations -- looks fine as there were none.
In [8]:
div_iterations = []
for sampler in mcmc.samplers():
div_iterations.append(sampler.divergent_iterations())
print("There were " + str(np.sum(div_iterations)) + " divergent iterations.")