<jupyter_start><jupyter_text>Inference: Monomial-Gamma Hamiltonian Monte CarloThis example shows you how to perform Bayesian inference on a Gaussian distribution and a time-series problem, using [Monomial-Gamma HMC](http://pints.readthedocs.io/en/latest/mcmc_samplers/monomial_gamma_hamiltonian_mcmc.html). First, we create a simple normal distribution<jupyter_code>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()<jupyter_output><empty_output><jupyter_text>Now we set up and run a sampling routine using Monomial-Gamma HMC MCMC<jupyter_code># 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!')
# Show traces and histograms
import pints.plot
pints.plot.trace(full_chains)
plt.show()
# 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()<jupyter_output>R-hat:
[1.0006476543920884, 0.99942523141911921]
0.0014302120063
0.00286525604443
0.00735266826099<jupyter_text>Monomial-Gamma HMC on a time-series problemWe now try the same method on a time-series problem<jupyter_code>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!')<jupyter_output>Running...
Using Monomial-Gamma Hamiltonian Monte Carlo
Generating 3 chains.
Running in sequential mode.<jupyter_text>The chains do not take long to reach equilibrium with this method.<jupyter_code># Show traces and histograms
pints.plot.trace(chains)
plt.show()<jupyter_output><empty_output><jupyter_text>Chains have converged!<jupyter_code># Discard warm up
chains = chains[:, 200:]

# Check convergence using rhat criterion
print('R-hat:')
print(pints.rhat_all_params(chains))<jupyter_output>R-hat:
[1.0003255905068089, 1.0008356410410884]<jupyter_text>Extract any divergent iterations -- looks fine as there were none.<jupyter_code>div_iterations = []
for sampler in mcmc.samplers():
    div_iterations.append(sampler.divergent_iterations())
print("There were " + str(np.sum(div_iterations)) + " divergent iterations.")<jupyter_output>There were 0 divergent iterations.
{'fraction_non_alphanumeric': 0.11357031075340934, 'fraction_numerical': 0.046724793203666445, 'mean_word_length': 5.7076461769115445, 'pattern_counts': {'":': 0, '<': 21, '<?xml version=': 0, '>': 21, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '6', 'dir': 'jupyter-structured-clean-dedup', 'id': '664428', 'license': 'RSA-MD', 'n_tokens_mistral': 1807, 'n_tokens_neox': 1610, 'n_words': 412, 'path': 'examples/sampling/monomial-gamma-hmc.ipynb', 'repo_name': 'lisaplag/pints', 'size': '4473'}