PyPI version API documentation DOI

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 master branch on Github run

pip install git+

If available in the installed Python environment the following additional packages provide extra functionality and features

Why Mici?

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.

Related projects

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 mici package are the systems, integrators and samplers modules and you will generally need to create an instance of one class from each module. - Hamiltonian systems encapsulating model functions and their derivatives

mici.integrators - symplectic integrators for Hamiltonian dynamics

mici.samplers - MCMC samplers for peforming inference

Example: sampling on a torus

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 autograd to 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 and use matplotlib to 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

# Define 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([((x**2 + y**2)**0.5 - R)**2 + z**2 - 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, 

# 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}")

# 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)
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)


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