This example shows you how to perform Bayesian inference on a Normal distribution and a time-series problem, using MALA for MCMC.
First, we create a simple normal distribution
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 MALA MCMC
# Choose starting points for 3 mcmc chains
xs = [
[2, 1],
[3, 3],
[5, 4],
]
# Create mcmc routine
mcmc = pints.MCMCController(log_pdf, 3, xs, method=pints.MALAMCMC)
# Add stopping criterion
mcmc.set_max_iterations(2000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(100)
# # Update step sizes used by individual samplers (which is then scaled by sigma0)
for sampler in mcmc.samplers():
sampler.set_epsilon([1.5, 1.5])
# Run!
print('Running...')
full_chains = mcmc.run()
print('Done!')
Running... Using Metropolis-Adjusted Langevin Algorithm (MALA) Generating 3 chains. Running in sequential mode. Iter. Eval. Accept. Accept. Accept. Time m:s 0 3 0 0 0 0:00.0 1 6 0 0.5 0.5 0:00.0 2 9 0 0.667 0.667 0:00.0 3 12 0.25 0.5 0.75 0:00.0 100 303 0.723 0.673 0.832 0:00.3 200 603 0.751 0.741 0.796 0:00.5 300 903 0.767 0.751 0.797 0:00.8 400 1203 0.736 0.743 0.771 0:01.1 500 1503 0.750499 0.748503 0.752495 0:01.4 600 1803 0.757 0.76 0.747 0:01.6 700 2103 0.753 0.755 0.743224 0:01.9 800 2403 0.75 0.757 0.735 0:02.2 900 2703 0.75 0.757 0.73 0:02.4 1000 3003 0.751 0.756 0.733 0:02.7 1100 3303 0.751 0.752 0.736 0:03.0 1200 3603 0.749 0.746045 0.729 0:03.2 1300 3903 0.743 0.745 0.73 0:03.5 1400 4203 0.747 0.747 0.726 0:03.8 1500 4503 0.744 0.749 0.729 0:04.0 1600 4803 0.747 0.751 0.723 0:04.3 1700 5103 0.743 0.748 0.718 0:04.6 1800 5403 0.745 0.751 0.723487 0:04.8 1900 5703 0.742767 0.75 0.725 0:05.1 2000 6000 0.7435 0.7485 0.7285 0:05.4 Halting: Maximum number of iterations (2000) reached. Done!
# Show traces and histograms
import pints.plot
pints.plot.trace(full_chains)
plt.show()
# Discard warm up
chains = full_chains[:, 1000:]
# 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()
R-hat: [1.0011357940509646, 1.0017031271797681] 0.0103631865061 0.00530188143198 0.00250108715373
We now try the same method on a time-series problem
First, we try it in 1d, using a wrapper around the LogisticModel to make it one-dimensional.
import pints.toy as toy
# Create a wrapper around the logistic model, turning it into a 1d model
class Model(pints.ForwardModel):
def __init__(self):
self.model = toy.LogisticModel()
def simulate(self, x, times):
return self.model.simulate([x[0], 500], times)
def simulateS1(self, x, times):
values, gradient = self.model.simulateS1([x[0], 500], times)
gradient = gradient[:, 0]
return values, gradient
def n_parameters(self):
return 1
# Load a forward model
model = Model()
# Create some toy data
real_parameters = np.array([0.015])
times = np.linspace(0, 1000, 50)
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)
plt.figure()
plt.plot(times, values)
plt.plot(times, org_values)
plt.show()
# 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],
[0.02]
)
log_posterior = pints.LogPosterior(log_likelihood, log_prior)
# Choose starting points for mcmc chains
xs = [
real_parameters * 1.01,
real_parameters * 0.9,
real_parameters * 1.15,
]
# Create mcmc routine
mcmc = pints.MCMCController(log_likelihood, len(xs), xs, method=pints.MALAMCMC)
# Add stopping criterion
mcmc.set_max_iterations(2000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(100)
# Run!
print('Running...')
chains = mcmc.run()
print('Done!')
Running... Using Metropolis-Adjusted Langevin Algorithm (MALA) Generating 3 chains. Running in sequential mode. Iter. Eval. Accept. Accept. Accept. Time m:s 0 3 0 0 0 0:00.0 1 6 0.5 0.5 0.5 0:00.0 2 9 0.667 0.667 0.667 0:00.0 3 12 0.75 0.75 0.75 0:00.0 100 303 0.990099 0.990099 0.990099 0:00.3 200 603 0.995 0.995 0.99 0:00.6 300 903 0.993 0.993 0.99 0:00.9 400 1203 0.995 0.993 0.993 0:01.2 500 1503 0.992016 0.988024 0.99002 0:01.5 600 1803 0.992 0.99 0.99 0:01.8 700 2103 0.993 0.991 0.991 0:02.1 800 2403 0.99 0.993 0.993 0:02.4 900 2703 0.988 0.993 0.992 0:02.6 1000 3003 0.988012 0.994006 0.993007 0:02.9 1100 3303 0.988 0.995 0.993 0:03.2 1200 3603 0.989 0.994 0.993 0:03.5 1300 3903 0.99 0.995 0.992 0:03.8 1400 4203 0.99 0.995 0.992 0:04.1 1500 4503 0.99 0.995 0.991 0:04.4 1600 4803 0.991 0.995 0.992 0:04.7 1700 5103 0.991 0.995 0.991 0:05.0 1800 5403 0.992 0.995 0.991116 0:05.3 1900 5703 0.992 0.995 0.992 0:05.5 2000 6000 0.992 0.9955 0.992 0:05.8 Halting: Maximum number of iterations (2000) reached. Done!
# Show trace and histogram
pints.plot.trace(chains)
plt.show()
# Show predicted time series for the first chain
pints.plot.series(chains[0, 200:], problem, real_parameters)
plt.show()
Finally, we try MALA MCMC on a 2d logistic model problem.
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 = 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.MALAMCMC)
# Add stopping criterion
mcmc.set_max_iterations(4000)
# Set up modest logging
mcmc.set_log_to_screen(True)
mcmc.set_log_interval(100)
# Run!
print('Running...')
chains = mcmc.run()
print('Done!')
Running... Using Metropolis-Adjusted Langevin Algorithm (MALA) Generating 3 chains. Running in sequential mode. Iter. Eval. Accept. Accept. Accept. Time m:s 0 3 0 0 0 0:00.0 1 6 0.5 0.5 0.5 0:00.0 2 9 0.333 0.667 0.667 0:00.0 3 12 0.5 0.75 0.75 0:00.0 100 303 0.970297 0.970297 0.980198 0:00.3 200 603 0.975 0.98 0.99 0:00.6 300 903 0.98 0.983 0.986711 0:00.9 400 1203 0.975 0.985 0.985 0:01.2 500 1503 0.976 0.984 0.986 0:01.5 600 1803 0.975 0.987 0.985025 0:01.8 700 2103 0.977 0.989 0.984 0:02.1 800 2403 0.974 0.99 0.985 0:02.4 900 2703 0.977 0.99 0.984 0:02.7 1000 3003 0.979021 0.991009 0.983017 0:03.0 1100 3303 0.98 0.992 0.981 0:03.3 1200 3603 0.981 0.992 0.981 0:03.6 1300 3903 0.982 0.991545 0.982 0:03.9 1400 4203 0.983 0.991 0.982 0:04.2 1500 4503 0.983 0.99 0.983 0:04.5 1600 4803 0.984 0.989 0.983 0:04.9 1700 5103 0.985 0.99 0.984 0:05.2 1800 5403 0.985 0.991 0.984 0:05.5 1900 5703 0.986 0.991 0.983 0:05.8 2000 6003 0.986 0.99 0.983 0:06.1 2100 6303 0.985 0.989 0.982 0:06.4 2200 6603 0.985 0.989 0.982 0:06.7 2300 6903 0.985 0.989 0.982 0:07.0 2400 7203 0.985 0.99 0.982 0:07.3 2500 7503 0.984 0.990004 0.981 0:07.6 2600 7803 0.984 0.99 0.981 0:07.9 2700 8103 0.984 0.99 0.981 0:08.2 2800 8403 0.984 0.99 0.981 0:08.5 2900 8703 0.984 0.991 0.981 0:08.8 3000 9003 0.984 0.991 0.981 0:09.1 3100 9303 0.984 0.99 0.981 0:09.4 3200 9603 0.984 0.991 0.982 0:09.7 3300 9903 0.985 0.991 0.982 0:10.0 3400 10203 0.985 0.990885 0.983 0:10.3 3500 10503 0.985 0.991 0.982862 0:10.6 3600 10803 0.985 0.991 0.983 0:10.9 3700 11103 0.985 0.991 0.982167 0:11.2 3800 11403 0.986 0.991 0.982 0:11.5 3900 11703 0.986 0.991 0.982 0:11.8 4000 12000 0.986 0.99125 0.98225 0:12.1 Halting: Maximum number of iterations (4000) reached. Done!
# Show traces and histograms
pints.plot.trace(chains)
plt.show()
Chains have converged!
# Discard warm up
chains = chains[:, 1000:]
# Check convergence using rhat criterion
print('R-hat:')
print(pints.rhat_all_params(chains))
R-hat: [1.0035953753279669, 0.99995026418452926]