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.
Features
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 implicitlydefined by a constraint equation and distributions on Riemannian manifolds with a userspecified 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 memorymapping chains to disk, allowing long runs on large models without hitting memory issues.
Installation
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+https://github.com/mattgraham/mici
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
autograd.numpy
andautograd.scipy
interfaces). To sample chains in parallel usingautograd
functions you also need to install multiprocess. This will causemultiprocess.Pool
to be used in preference to the inbuiltmutiprocessing.Pool
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 runningpip install mici[autodiff]
.  ArviZ: if ArviZ is
available the traces (dictionary) output of a sampling run can be directly
converted to an
arviz.InferenceData
container object usingarviz.convert_to_inference_data
or implicitly converted by passing the traces dictionary as thedata
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 nonPython 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 userfacing
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.
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 positiondependent matrix representation (Girolami and Calderhead, 2011),SoftAbsRiemannianMetricSystem
 system with SoftAbs eigenvalueregularized Hessian of negative log target density as metric matrix representation (Betancourt, 2013),DenseConstrainedEuclideanMetricSystem
 Euclideanmetric 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 (StörmerVerlet) integrator for separable Hamiltonian systems (Leimkulher and Reich, 2004),ImplicitLeapfrogIntegrator
 implicit (or generalized) leapfrog integrator for nonseparable 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).Notebooks
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 noninteractive version with nbviewer  
Open interactive version with Binder  
Open interactive version with Google Colab 
Manifold MCMC methods for inference in diffusion models  

Open noninteractive version with nbviewer  
Open interactive version with Binder  
Open interactive version with Google Colab 
Example: sampling on a torus
A simple complete example of using the package to compute approximate samples
from a distribution on a twodimensional torus embedded in a threedimensional
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
, use arviz
to calculate diagnostics 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
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([((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 integrationtime 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 nonadaptive 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)
References
 Andersen, H.C., 1983. RATTLE: A “velocity” version of the SHAKE algorithm for molecular dynamics calculations. Journal of Computational Physics, 52(1), pp.2434.
 Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D., 1987. Hybrid Monte Carlo. Physics letters B, 195(2), pp.216222.
 Mackenzie, P.B., 1989. An improved hybrid Monte Carlo method. Physics Letters B, 226(34), pp.369371.
 Horowitz, A.M., 1991. A generalized guided Monte Carlo algorithm. Physics Letters B, 268(CERNTH617291), pp.247252.
 Leimkuhler, B. and Reich, S., 1994. Symplectic integration of constrained Hamiltonian systems. Mathematics of Computation, 63(208), pp.589605.
 Leimkuhler, B. and Reich, S., 2004. Simulating Hamiltonian dynamics (Vol. 14). Cambridge University Press.
 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.700710.
 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.123214.
 Brubaker, M., Salzmann, M. and Urtasun, R., 2012. A family of MCMC methods on implicitly defined manifolds. In Artificial intelligence and statistics (pp. 161172).
 Betancourt, M., 2013. A general metric for Riemannian manifold Hamiltonian Monte Carlo. In Geometric science of information (pp. 327334).
 Hoffman, M.D. and Gelman, A., 2014. The NoUturn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), pp.15931623.
 Shahbaba, B., Lan, S., Johnson, W.O. and Neal, R.M., 2014. Split Hamiltonian Monte Carlo. Statistics and Computing, 24(3), pp.339349.
 Betancourt, M., 2017. A conceptual introduction to Hamiltonian Monte Carlo.
 Lelièvre, T., Rousset, M. and Stoltz, G.,