Learning maximum likelihood tree structure with the Chow-Liu algorithm

Jan 5, 2024 · 2 min read
blog coursework

Write a function ChowLiu(X) -> A where X is a D by N data matrix containing a multivariate data point on each column that returns a Chow-Liu maximum likelihood tree for X. The tree structure is to be returned in the sparse matrix A.

The file ChowLiuData.mat contains a data matrix for 10 variables. Use your function to find the maximum-likelihood Chow-Liu tree, and draw a picture of the resulting DAG with edges oriented away from variable 1.

Solution

from scipy.io import loadmat

data = loadmat('ChowLiuData.mat')['X'].T - 1
print(data.shape)
data

(1000, 10)

array([[0, 1, 1, ..., 2, 1, 0],
       [1, 1, 1, ..., 0, 2, 0],
       [0, 0, 1, ..., 2, 1, 1],
       ...,
       [1, 1, 1, ..., 0, 2, 0],
       [1, 1, 0, ..., 0, 0, 0],
       [1, 0, 1, ..., 2, 1, 0]], dtype=uint8)

The Chow-Liu algorithm simply computes the mutual information $I(X_i,X_j)$ between each pair of variables and constructs a maximum spanning tree where the edge weights are the mutual information.

import numpy as np
import networkx as nx
import matplotlib.pyplot as plt

def chow_liu(X):
    M, N = X.shape
    K = X.max() + 1  # num discrete values each X can take
    # compute empirical p(x_i, x_j)
    p_pair = np.zeros((N, K, N, K))
    # and empirical p(x_i)
    p_single = np.zeros((N, K))
    for row in X:
        for i in range(N):
            p_single[i, row[i]] += 1
            for j in range(i, N):
                p_pair[i, row[i], j, row[j]] += 1
                if i != j:
                    p_pair[j, row[j], i, row[i]] += 1
    p_pair /= M
    p_single /= M
    assert np.allclose(p_pair.sum(axis=(1, 3)), 1)
    assert p_single.shape == (N, K)
    assert np.allclose(p_single.sum(axis=1), 1)
    # compute all I_ij = I(X_i, X_j)
    with np.errstate(divide='ignore', invalid='ignore'):
        logterm = (np.log(p_pair) \
            - np.log(p_single[..., np.newaxis, np.newaxis]) \
            - np.log(p_single))
    # set these to constants, will be nullified assuming they come from
    # instances when p_pair=0
    assert np.all(p_pair[np.isnan(logterm)] == 0)
    logterm[np.isnan(logterm)] = 1  # from log0 - log0
    assert np.all(p_pair[np.isinf(logterm)] == 0)
    logterm[np.isinf(logterm)] = 1  # from log0 - ...
    I = (p_pair * logterm).sum(axis=(1, 3))
    assert I.shape == (N, N)

    # compute max spanning tree
    G = nx.Graph()
    G.add_nodes_from(range(1,N+1))
    for i in range(1, N+1):
        for j in range(i, N+1):
            G.add_edge(i, j, weight=I[i-1, j-1])
    T = nx.maximum_spanning_tree(G)
    # get directions from breadth-first search
    A = nx.bfs_tree(T, 1)
    return A
A = chow_liu(data)
d = nx.draw_planar(A, with_labels=True)

The directions are coincidental from the breadth-first search—the result is an undirected tree (Markov random field).

Probabilistic Graphical Models
Kyle Johnsen
Authors
Kyle Johnsen
PhD Candidate
Biomedical Engineering

I’m a PhD student advised by Chris Rozell at Georgia Tech, developing advanced closed-loop optogenetic control techniques for neuroscience. Specifically, I am applying optimal, model-based control to precisely perturb population dynamics.

I’m fascinated by the biological computational principles and how we can exploit them for next-generation AI/ML technologies. And vice-versa: how we can apply modern ML to neurotech/neuroscience, especially for building life- and time-saving brain-computer interfaces. I am looking for an industry research scientist role post-graduation.