Manifold Markov chain Monte Carlo methods in Python
Mici is a Python package providing implementations of Markov chain Monte Carlo (MCMC) methods for approximate inference in probabilistic models, with a particular focus on MCMC methods based on simulating Hamiltonian dynamics on a manifold.
Key features include
To install and use Mici the minimal requirements are a Python 3.6+ environment with NumPy and SciPy installed. The latest Mici release on PyPI (and its dependencies) can be installed in the current Python environment by running
pip install mici
To instead install the latest development version from the
masterbranch on Github run
pip install git+https://github.com/matt-graham/mici
If available in the installed Python environment the following additional packages provide extra functionality and features
autograd.numpyand
autograd.scipyinterfaces). To sample chains in parallel using
autogradfunctions you also need to install multiprocess. This will cause
multiprocess.Poolto be used in preference to the in-built
mutiprocessing.Poolfor parallelisation as multiprocess supports serialisation (via dill) of a much wider range of types, including of Autograd generated functions. Both Autograd and multiprocess can be installed alongside Mici by running
pip install mici[autodiff].
arviz.InferenceDatacontainer object using
arviz.convert_to_inference_dataor implicitly converted by passing the traces dictionary as the
dataargument to ArviZ API functions, allowing straightforward use of the ArviZ's extensive visualisation and diagnostic functions.
Mici is named for Augusta 'Mici' Teller, who along with Arianna Rosenbluth developed the code for the MANIAC I computer used in the seminal paper Equations of State Calculations by Fast Computing Machines which introduced the first example of a Markov chain Monte Carlo method.
Other Python packages for performing MCMC inference include PyMC3, PyStan (the Python interface to Stan), Pyro / NumPyro, TensorFlow Probability, emcee and Sampyl.
Unlike PyMC3, PyStan, (Num)Pyro and TensorFlow Probability which are complete probabilistic programming frameworks including functionality for definining a probabilistic model / program, but like emcee and Sampyl, Mici is solely focussed on providing implementations of inference algorithms, with the user expected to be able to define at a minimum a function specifying the negative log (unnormalized) density of the distribution of interest.
Further while PyStan, (Num)Pyro and TensorFlow Probability all push the sampling loop into external compiled non-Python code, in Mici the sampling loop is run directly within Python. This has the consequence that for small models in which the negative log density of the target distribution and other model functions are cheap to evaluate, the interpreter overhead in iterating over the chains in Python can dominate the computational cost, making sampling much slower than packages which outsource the sampling loop to a efficient compiled implementation.
## Overview of package
API documentation for the package is available here. The three main user-facing modules within the
micipackage are the
systems,
integratorsand
samplersmodules and you will generally need to create an instance of one class from each module.
mici.systems- Hamiltonian systems encapsulating model functions and their derivatives
EuclideanMetricSystem- systems with a metric on the position space with a constant matrix representation,
GaussianEuclideanMetricSystem- systems in which the target distribution is defined by a density with respect to the standard Gaussian measure on the position space allowing analytically solving for flow corresponding to the quadratic components of Hamiltonian (Shahbaba et al., 2014),
RiemannianMetricSystem- systems with a metric on the position space with a position-dependent matrix representation (Girolami and Calderhead, 2011),
SoftAbsRiemannianMetricSystem- system with SoftAbs eigenvalue-regularized Hessian of negative log target density as metric matrix representation (Betancourt, 2013),
DenseConstrainedEuclideanMetricSystem- Euclidean-metric system subject to holonomic constraints (Hartmann and Schütte, 2005; Brubaker, Salzmann and Urtasun, 2012; Lelièvre, Rousset and Stoltz, 2019) with a dense constraint function Jacobian matrix,
mici.integrators- symplectic integrators for Hamiltonian dynamics
LeapfrogIntegrator- explicit leapfrog (Störmer-Verlet) integrator for separable Hamiltonian systems (Leimkulher and Reich, 2004),
ImplicitLeapfrogIntegrator- implicit (or generalized) leapfrog integrator for non-separable Hamiltonian systems (Leimkulher and Reich, 2004),
ConstrainedLeapfrogIntegrator- constrained leapfrog integrator for Hamiltonian systems subject to holonomic constraints (Andersen, 1983; Leimkuhler and Reich, 1994).
mici.samplers- MCMC samplers for peforming inference
StaticMetropolisHMC- static integration time Hamiltonian Monte Carlo with Metropolis accept step (Duane et al., 1987),
RandomMetropolisHMC- random integration time Hamiltonian Monte Carlo with Metropolis accept step (Mackenzie, 1989),
DynamicSliceHMC- dynamic integration time Hamiltonian Monte Carlo with slice sampling from trajectory, equivalent to the original 'NUTS' algorithm (Hoffman and Gelman, 2014).
DynamicMultinomialHMC- dynamic integration time Hamiltonian Monte Carlo with multinomial sampling from trajectory, equivalent to the current default MCMC algorithm in Stan (Hoffman and Gelman, 2014; Betancourt, 2017).
The manifold MCMC methods implemented in Mici have been used in several research projects. Below links are provided to a selection of Jupyter notebooks associated with these projects as demonstrations of how to use Mici and to illustrate some of the settings in which manifold MCMC methods can be computationally advantageous.
Manifold lifting: MCMC in the vanishing noise regime | |
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Open non-interactive version with nbviewer | |
Open interactive version with Binder | |
Open interactive version with Google Colab |
Manifold MCMC methods for inference in diffusion models | |
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Open non-interactive version with nbviewer | |
Open interactive version with Binder | |
Open interactive version with Google Colab |
A simple complete example of using the package to compute approximate samples from a distribution on a two-dimensional torus embedded in a three-dimensional space is given below. The computed samples are visualized in the animation above. Here we use
autogradto automatically construct functions to calculate the required derivatives (gradient of negative log density of target distribution and Jacobian of constraint function), sample four chains in parallel using
multiprocess, use
arvizto calculate diagnostics and use
matplotlibto plot the samples.
from mici import systems, integrators, samplers import autograd.numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D import matplotlib.animation as animation import arvizDefine fixed model parameters
R = 1.0 # toroidal radius ∈ (0, ∞) r = 0.5 # poloidal radius ∈ (0, R) α = 0.9 # density fluctuation amplitude ∈ [0, 1)
Define constraint function such that the set {q : constr(q) == 0} is a torus
def constr(q): x, y, z = q.T return np.stack([((x2 + y2)0.5 - R)**2 + z2 - r**2], -1)
Define negative log density for the target distribution on torus
(with respect to 2D 'area' measure for torus)
def neg_log_dens(q): x, y, z = q.T θ = np.arctan2(y, x) ϕ = np.arctan2(z, x / np.cos(θ) - R) return np.log1p(r * np.cos(ϕ) / R) - np.log1p(np.sin(4*θ) * np.cos(ϕ) * α)
Specify constrained Hamiltonian system with default identity metric
system = systems.DenseConstrainedEuclideanMetricSystem(neg_log_dens, constr)
System is constrained therefore use constrained leapfrog integrator
integrator = integrators.ConstrainedLeapfrogIntegrator(system)
Seed a random number generator
rng = np.random.default_rng(seed=1234)
Use dynamic integration-time HMC implementation as MCMC sampler
sampler = samplers.DynamicMultinomialHMC(system, integrator, rng)
Sample initial positions on torus using parameterisation (θ, ϕ) ∈ [0, 2π)²
x, y, z = (R + r * cos(ϕ)) * cos(θ), (R + r * cos(ϕ)) * sin(θ), r * sin(ϕ)
n_chain = 4 θ_init, ϕ_init = rng.uniform(0, 2 * np.pi, size=(2, n_chain)) q_init = np.stack([ (R + r * np.cos(ϕ_init)) * np.cos(θ_init), (R + r * np.cos(ϕ_init)) * np.sin(θ_init), r * np.sin(ϕ_init)], -1)
Define function to extract variables to trace during sampling
def trace_func(state): x, y, z = state.pos return {'x': x, 'y': y, 'z': z}
Sample 4 chains in parallel with 500 adaptive warm up iterations in which the
integrator step size is tuned, followed by 2000 non-adaptive iterations
final_states, traces, stats = sampler.sample_chains_with_adaptive_warm_up( n_warm_up_iter=500, n_main_iter=2000, init_states=q_init, n_process=4, trace_funcs=[trace_func])
Print average accept probability and number of integrator steps per chain
for c in range(n_chain): print(f"Chain {c}:") print(f" Average accept prob. = {stats['accept_stat'][c].mean():.2f}") print(f" Average number steps = {stats['n_step'][c].mean():.1f}")
Print summary statistics and diagnostics computed using ArviZ
print(arviz.summary(traces))
Visualize concatentated chain samples as animated 3D scatter plot
fig = plt.figure(figsize=(4, 4)) ax = Axes3D(fig, [0., 0., 1., 1.], proj_type='ortho') points_3d, = ax.plot(*(np.concatenate(traces[k]) for k in 'xyz'), '.', ms=0.5) ax.axis('off') for set_lim in [ax.set_xlim, ax.set_ylim, ax.set_zlim]: set_lim((-1, 1))
def update(i): angle = 45 * (np.sin(2 * np.pi * i / 60) + 1) ax.view_init(elev=angle, azim=angle) return (points_3d,)
anim = animation.FuncAnimation(fig, update, frames=60, interval=100, blit=True)