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Maximum Entropy Inference

GitHub tests paper docs PyPI version

This provides a Keras implementation of maximum entropy simulation based inference. The point of this package is to reweight outcomes from a simulator to agree with observations, rather than trying to optimize your simulators input parameters. The simulator must necessarily give multiple outcomes - either because you're trying multiple sets of input parameters or it has intrinsic noise. The assumption of this model is that your simulator is approximately correct. The observations being fit could have come the distribution of outcomes of your simulator.

About maximum entropy

Maximum entropy reweighting is a straightforward black box method that can be applied to arbitrary simulators with few observations. Its runtime is independent of the number of parameters used by the simulator, and it has been shown analytically to minimally change the prior to agree with observations. This method fills a niche in the small-data, high-complexity regime of SBI parameter inference, because it accurately and minimally biases a prior to match observations and does not scale in runtime with the number of model parameters.


pip install maxent-infer

Quick Start

A Pandas Data Frame

Consider a data frame representing outcomes from our prior model/simulator. We would like to regress these outcomes to data.

import pandas as pd
import numpy as np
import maxent

data = pd.read_csv('data.csv')

Perhaps we have a single observation we would like to match. We will reweight our rows (outcomes) to agree with the single observation. This is under-determined, but there is one solution because of the maximum entropy condition. To fit this one observation, we specify two things: a function to compute the observation from outcomes of our prior/simulator and the value of the observation (called target). Let's say our observation is just the 3rd column in a row that came from one outcome:

def observe(single_row):
  return single_row[3]

r = maxent.Restraint(observe, target=1.5)

Do you have uncertainty with your observation? No problem. Here we specify our uncertainty as a Laplace distributed with a variance of 2 (Laplace scale parameter 1):

r = maxent.Restraint(observe, target=1.5, prior=maxent.Laplace(1))

Now we'll fit our outcomes to the single observation.

model = maxent.MaxentModel(r)

We now have a set of weights -- one per row -- that we can use to compute other expressions. For example, here is the most likely outcome (mode)

i = np.argmax(model.traj_weights)
mode = data.iloc[i, :]

Here are the new column averages

col_avg = np.sum(data.values * model.traj_weights[:, np.newaxis], axis=0)

A simulator

Here we show how to take a random walk simulator and use maxent to have reweight the random walk so that the average end is at x = 2, y= 1.

# simulate
def random_walk_simulator(T=10):
    x = [0,0]
    traj = np.empty((T,2))
    for i in range(T):
        traj[i] = x
        x += np.random.normal(size=2)
    return traj

N = 500
trajs = [random_walk_simulator() for _ in range(N)]

# now have N x T x 2 tensor
trajs = np.array(trajs)

# here is a plot of these trajectories


# we want the random walk to have average end of 2,1
rx = maxent.Restraint(lambda traj: traj[-1,0], target=2)
ry = maxent.Restraint(lambda traj: traj[-1,1], target=1)

# create model by passing in restraints
model = maxent.MaxentModel([rx, ry])

# convert model to be differentiable/GPU (if available)
# fit to data
h =

# can now compute other averages properties
# with new weights

# plot showing weights of trajectories:


Further Examples

You can find the examples used in the manuscript, including comparisons with competing methods: here. These examples use the latest package versions, so the figures will not exactly match those in the manuscript. If you would like to reproduce the manuscript exactly, install the packages in paper/requirements.txt and execute the notebooks in paper (this is the output from the paper workflow above).




See paper and the citation:

  title={Simulation-Based Inference with Approximately Correct Parameters via Maximum Entropy},
  author={Barrett, Rainier and Ansari, Mehrad and Ghoshal, Gourab and White, Andrew D},
  journal={Machine Learning: Science and Technology},


License: GPL v2