Markov random fields (MRFs) are an interesting class of statistical models that originate in physics. They can be thought of as a collection of random variables where we allow arbitrary joint dependencies between variables. These dependencies are often not all-to-all and can frequently be represented as a graph, where random variables are nodes and edges indicate interactions. These interactions are typically written as potentials (log-likelihood contributions) of the state under the model.

MRFs have strong conceptual overlap with belief networks (or Bayesian networks). However, belief networks are generally more tractable for inference, as their directed structure enables algorithms like belief propagation and variable elimination to compute likelihoods efficiently. Random fields, on the other hand, allow more general dependency structures and therefore often require approximate inference methods such as loopy belief propagation or mean-field approximations.

I am considering working with MRFs in some future work, so I thought I would write a quick introduction to sampling from them using the probabilistic programming library NumPyro. Here, I do not consider inference of model parameters and instead focus on sampling from these models. The visualizations associated with sampling these models can be quite cool, as I show below.

Let’s begin with a basic random field. This is simply a collection of random variables

\[\{X_s : s \in S\}\]

Typically, we choose \(S\) (the index set) to be a grid (see the visualizations below for examples of these grids evolving over time). We obtain a Markov random field by imposing a Markov property with respect to a graph \(G = (V, E)\), such that

\[X_i \perp X_{V \setminus \{i \cup N(i)\}} \mid X_{N(i)}\]

where \(N(i)\) denotes the neighbors (the Markov blanket) of node \(i\). In other words, each node is conditionally independent of the rest of the graph given its neighbors. The joint distribution of the full system can then be written as a product over cliques of the graph:

\[p(x) \propto \prod_{c \in C} \psi_c(x_c)\]

It is worth noting that this definition is extremely general and the set of cliques could, in principle, be the entire graph.

Below, we consider two cases of MRFs: the Ising model (a classic model from statistical physics with local interactions) and a Hopfield network (a fully connected model from classical machine learning).

The Ising Model

The Ising model is the prototypical MRF. This model was originally developed to describe the spin of atoms in a lattice. In this model, each variable only interacts with its local neighbors, resulting in a small Markov blanket. We consider a grid \(S\) of size \(n \times n\), where each variable takes values

\[X_{ij} \in \{-1, +1\}, \quad i,j \in \{1,2,\dots,n\}\]

We introduce two parameters:

  • \(J\): The interaction strength between neighboring spins
  • \(h\): An external field bias
    A large \(J\) encourages neighboring variables to align (take the same sign), while \(h\) introduces a global bias towards one sign or the other. The probability of a realization of the field \(x\) is given by
\[p(x) \propto \exp\left( \sum_{(i,j,k,l) \in E} J x_{ij} x_{kl} + \sum_{i,j \in S} h x_{ij} \right)\]

where \(E\) indexes adjacent cells.

To sample from this model, we need to define its energy (potential) and implement a Metropolis–Hastings sampler. Below is a NumPyro implementation:

import jax
import jax.numpy as jnp
import numpyro.distributions as dist
from numpyro.infer.mcmc import MCMC, MCMCKernel
from numpyro.util import identity
from collections import namedtuple

interaction_strength = 100.0
external_strength = 0.0
def local_ising_energy(state, i, j, beta = 1.0):
    energy = -1.0 * external_strength * state[i,j]

    energy -= (i>0) * interaction_strength * state[i,j] * state[i-1,j]
    energy -= (i < state.shape[0]-1) * interaction_strength * state[i,j] * state[i+1,j]

    energy -= (j > 0) * interaction_strength * state[i,j] * state[i,j-1]
    energy -= (j < state.shape[1]-1) * interaction_strength * state[i,j] * state[i,j+1]

    return beta * energy

MHState = namedtuple("MHState", ["spins", "rng_key"])
class IsingMH(MCMCKernel):
    sample_field = "spins"

    def __init__(self, potential_fn=None, **kwargs):
        super().__init__()
        self.potential_fn = potential_fn

    def init(self, rng_key, num_warmup, init_params, model_args, model_kwargs):
        return MHState(init_params, rng_key)

    # Attempt a simple flip of a random state using the Metropolis-Hastings algorithm
    def sample(self, state, model_args, model_kwargs):
        spins, rng_key = state
        rng_key, key_i, key_j, key_accept = jax.random.split(rng_key, 4)

        random_i = jax.random.randint(key_i, (), 0, spins.shape[0], dtype=jnp.int32)
        random_j = jax.random.randint(key_j, (), 0, spins.shape[1], dtype=jnp.int32)

        new_spins = spins.at[random_i, random_j].set(spins[random_i, random_j] * -1)

        accept_prob = jnp.exp(self.potential_fn(spins, random_i, random_j) - self.potential_fn(new_spins, random_i, random_j))
        new_spins = jnp.where(dist.Uniform().sample(key_accept) < accept_prob, new_spins, spins)

        return MHState(new_spins, rng_key)

    def postprocess_fn(self, model_args, model_kwargs):
        return identity

if __name__ == "__main__":
    key = jax.random.PRNGKey(1)
    random_state = dist.Bernoulli(0.5).sample(key, sample_shape=(64, 64)) * 2 - 1

    kernel = IsingMH(local_ising_energy)
    mcmc = MCMC(kernel, num_warmup=0, num_samples=150000)
    mcmc.run(key, init_params=random_state)
    posterior_samples = mcmc.get_samples()

Starting from main, we initialize a random grid using a transformed Bernoulli distribution and pass it into our MCMC sampler. The sampler uses a custom kernel implementing Metropolis–Hastings, where each iteration proposes flipping a single spin. Each iteration, this algorithm attempts to flip the state of a random cell from state \(x_{ij}\) to \(x'_{ij}\) and accepts the new flip with probability

\[\min \bigg(1, \frac{\exp-\ell(x'_{ij})}{\exp-\ell(x_{ij})} \bigg)\]

where \(\ell\) is the local energy contribution. This ensures the Markov chain spends time in each configuration proportional to its probability under the model.

We can animate our sampling process using the following code and we can get some interesting visual behavior:

import numpy as np
import imageio

data_np = np.array(posterior_samples)[::100, :, :]
frames = ((data_np + 1) / 2 * 255).astype(np.uint8)
imageio.mimsave("ising.gif", frames, fps=50)

Notice how the interaction strength encourages neighbors of a common state among the cells.

Hopfield Networks

Hopfield networks are an interesting model from classical machine learning. The key idea is to construct an energy function whose local minima correspond to stored patterns (for example, images). The system can then recover these patterns by evolving toward energy minima. These were studied in the context of memory recall, where the model was given only part of an image and asked to recall the unobserved portion. Here, we do not focus on memory recall, but instead treat the Hopfield network as a distribution to sample from. The probability distribution of a Hopfield network takes the form

\[p(x) \propto \exp(-\frac{1}{2} \sum_{i \neq j} W_{ij} x_i x_j)\]

where \(W_{ij}\) is the symmetric weight matrix encoding the stored patterns.

We can apply many of the same ideas from the Ising model. The primary difference is that the graph is fully connected, meaning every node interacts with every other node.

Below, I generated a simple \(64\times 64\) pixel-art cat. This serves as an energy minimum of the Hopfield network:

cat_64x64 = jnp.array([[-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]], dtype=jnp.int8)

patterns = jnp.stack([cat_64x64], axis=0)

We can now implement our potential function and energy. The only difference here is that we can’t simply update based on the local energy, as the Markov blanket of the Hopfield network is the whole graph.

import jax
import jax.numpy as jnp
import numpyro.distributions as dist
from numpyro.infer.mcmc import MCMC, MCMCKernel
from numpyro.util import identity
from collections import namedtuple

# Include the code from that cat image above. I did not include it in this block because it makes this code harder to read.
X = patterns.reshape(patterns.shape[0], -1)
W = (X.T @ X) / X.shape[1]
W = W.at[jnp.diag_indices(W.shape[0])].set(0)
def hopfield_energy(state, beta=100):
    s = state.reshape(-1)
    return -0.5 * s @ W @ s * beta

  
MHState = namedtuple("MHState", ["spins", "rng_key"])
class HopfieldMH(MCMCKernel):
    sample_field = "spins"

    def __init__(self, potential_fn=None, **kwargs):
        super().__init__()
        self.potential_fn = potential_fn

    def init(self, rng_key, num_warmup, init_params, model_args, model_kwargs):
        return MHState(init_params, rng_key)
        
    # Attempt a simple flip of a random state using the Metropolis-Hastings algorithm.
    def sample(self, state, model_args, model_kwargs):
        spins, rng_key = state
        rng_key, key_i, key_j, key_accept = jax.random.split(rng_key, 4)

        random_i = jax.random.randint(key_i, (), 0, spins.shape[0], dtype=jnp.int32)
        random_j = jax.random.randint(key_j, (), 0, spins.shape[1], dtype=jnp.int32)
        
        new_spins = spins.at[random_i, random_j].set(spins[random_i, random_j] * -1)

        accept_prob = jnp.exp(self.potential_fn(spins) - self.potential_fn(new_spins))
        new_spins = jnp.where(dist.Uniform().sample(key_accept) < accept_prob, new_spins, spins)

        return MHState(new_spins, rng_key)

    def postprocess_fn(self, model_args, model_kwargs):
        return identity
  

if __name__ == "__main__":
    key = jax.random.PRNGKey(1)
    random_state = dist.Bernoulli(0.5).sample(key, sample_shape=(64, 64)) * 2 - 1

    kernel = HopfieldMH(hopfield_energy)
    mcmc = MCMC(kernel, num_warmup=0, num_samples=20000)
    mcmc.run(key, init_params=random_state)
    posterior_samples = mcmc.get_samples()

As we can see, we can start the grid from complete noise, and obtain our cat at the end.

Overall, MRFs provide a flexible framework for defining structured distributions over high-dimensional spaces. While exact inference is often intractable, even simple sampling schemes can produce visually interesting behavior.