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Manifold Markov chain Monte Carlo methods in Python

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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

  • a modular design allowing use of a wide range of inference algorithms by mixing and matching different components, and making it easy to extend the package,
  • a pure Python code base with minimal dependencies, allowing easy integration within other code,
  • implementations of MCMC methods for sampling from distributions on embedded manifolds implicitly-defined by a constraint equation and distributions on Riemannian manifolds with a user-specified metric,
  • computationally efficient inference via transparent caching of the results of expensive operations and intermediate results calculated in derivative computations allowing later reuse without recalculation,
  • memory efficient inference for large models by memory-mapping chains to disk, allowing long runs on large models without hitting memory issues.


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

branch on Github run
pip install git+

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

  • Autograd: if available Autograd will be used to automatically compute the required derivatives of the model functions (providing they are specified using functions from the
    interfaces). To sample chains in parallel using
    functions you also need to install multiprocess. This will cause
    to be used in preference to the in-built
    for 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
    install mici[autodiff]
  • ArviZ: if ArviZ is available the traces (dictionary) output of a sampling run can be directly converted to an
    container object using
    or implicitly converted by passing the traces dictionary as the
    argument to ArviZ API functions, allowing straightforward use of the ArviZ's extensive visualisation and diagnostic functions.

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

package are the
modules and you will generally need to create an instance of one class from each module.
- 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,

- symplectic integrators for Hamiltonian dynamics

- 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
Open non-interactive version with nbviewer Render with nbviewer
Open interactive version with Binder Launch with Binder
Open interactive version with Google Colab Open in Colab
Manifold MCMC methods for inference in diffusion models
Open non-interactive version with nbviewer Render with nbviewer
Open interactive version with Binder Launch with Binder
Open interactive version with Google Colab Open in Colab

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

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
, use
to calculate diagnostics and use
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
import arviz

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([((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


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)


  1. Andersen, H.C., 1983. RATTLE: A “velocity” version of the SHAKE algorithm for molecular dynamics calculations. Journal of Computational Physics, 52(1), pp.24-34. DOI:10.1016/0021-9991(83)90014-1
  2. Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D., 1987. Hybrid Monte Carlo. Physics letters B, 195(2), pp.216-222. DOI:10.1016/0370-2693(87)91197-X
  3. Mackenzie, P.B., 1989. An improved hybrid Monte Carlo method. Physics Letters B, 226(3-4), pp.369-371. DOI:10.1016/0370-2693(89)91212-4
  4. Horowitz, A.M., 1991. A generalized guided Monte Carlo algorithm. Physics Letters B, 268(CERN-TH-6172-91), pp.247-252. DOI:10.1016/0370-2693(91)90812-5
  5. Leimkuhler, B. and Reich, S., 1994. Symplectic integration of constrained Hamiltonian systems. Mathematics of Computation, 63(208), pp.589-605. DOI:10.2307/2153284
  6. Leimkuhler, B. and Reich, S., 2004. Simulating Hamiltonian dynamics (Vol. 14). Cambridge University Press. DOI:10.1017/CBO9780511614118
  7. Hartmann, C. and Schütte, C., 2005. A constrained hybrid Monte‐Carlo algorithm and the problem of calculating the free energy in several variables. ZAMM ‐ Journal of Applied Mathematics and Mechanics, 85(10), pp.700-710. DOI:10.1002/zamm.200410218
  8. Girolami, M. and Calderhead, B., 2011. Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2), pp.123-214. DOI:10.1111/j.1467-9868.2010.00765.x
  9. Brubaker, M., Salzmann, M. and Urtasun, R., 2012. A family of MCMC methods on implicitly defined manifolds. In Artificial intelligence and statistics (pp. 161-172). CiteSeerX:
    1. Betancourt, M., 2013. A general metric for Riemannian manifold Hamiltonian Monte Carlo. In Geometric science of information (pp. 327-334). DOI:10.1007/978-3-642-40020-9_35 arXiv:1212.4693
    2. Hoffman, M.D. and Gelman, A., 2014. The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), pp.1593-1623. CiteSeerX: arXiv:1111.4246
    3. Shahbaba, B., Lan, S., Johnson, W.O. and Neal, R.M., 2014. Split Hamiltonian Monte Carlo. Statistics and Computing, 24(3), pp.339-349. DOI:10.1007/s11222-012-9373-1 arXiv:1106.5941
    4. Betancourt, M., 2017. A conceptual introduction to Hamiltonian Monte Carlo. arXiv:1701.02434
    5. Lelièvre, T., Rousset, M. and Stoltz, G.,
      1. Hybrid Monte Carlo methods for sampling probability measures on submanifolds. In Numerische Mathematik, 143(2), (pp.379-421). DOI:10.1007/s00211-019-01056-4 arXiv:1807.02356

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