A Decision Tree (Conditional) Independence Test (DTIT).

Introduction

Let x, y, z be random variables. Then deciding whether P(y | x, z) = P(y | z) can be difficult, especially if the variables are continuous. This package implements a simple yet efficient and effective conditional independence test, described in [link to arXiv when we write it up!]. Important features that differentiate this test from competition:

• It is fast. Worst-case speed scales as O(n_data * log(n_data) * dim), where dim is max(x_dim + z_dim, y_dim). However, amortized speed is O(n_data * log(n_data) * log(dim)).
• It applies to cases where some of x, y, z are continuous and some are discrete, or categorical (one-hot-encoded).
• It is very simple to understand and modify.
• It can be used for unconditional independence testing with almost no changes to the procedure.

We have applied this test to tens of thousands of samples of thousand-dimensional datapoints in seconds. For smaller dimensionalities and sample sizes, it takes a fraction of a second. The algorithm is described in [arXiv link coming], where we also provide detailed experimental results and comparison with other methods. However for now, you should be able to just look through the code to understand what's going on -- it's only 90 lines of Python, including detailed comments!

Usage

Basic usage is simple, and the default settings should work in most cases. To perform an unconditional test, use dtit.test(x, y):

import numpy as np
from dtit import dtit

x = np.random.rand(1000, 1)
y = np.random.randn(1000, 1)

pval_i = dtit.test(x, y) # p-value should be uniform on [0, 1].
pval_d = dtit.test(x, x + y) # p-value should be very small.

To perform a conditional test, just add the third variable z to the inputs:

import numpy as np
from dtit import dtit

# Generate some data such that x is indpendent of y given z.
n_samples = 1000
z = np.random.dirichlet(alpha=np.ones(2), size=n_samples)
x = np.vstack([np.random.multinomial(20, p) for p in z]).astype(float)
y = np.vstack([np.random.multinomial(20, p) for p in z]).astype(float)

# Check that x and y are dependent (p-value should be uniform on [0, 1]).
pval_d = dtit.test(x, y)
# Check that z d-separates x and y (the p-value should be small).
pval_i = dtit.test(x, y, z)

Installation

pip install dtit

Requirements

Tested with Python 3.6 and

• joblib >= 0.11
• numpy >= 1.12
• scikit-learn >= 0.18.1
• scipy >= 0.16.1