binned-cdf

A PyTorch-based distribution parametrized by the logits of CDF bins

Background

The Cumulative Distribution Function (CDF) is a fundamental concept in probability theory and statistics that describes the probability that a random variable $X$ takes on a value less than or equal to a given threshold $x$. Formally, the CDF is defined as $F(x) = P(X \leq x)$, where $F(x)$ ranges from 0 to 1 as $x$ varies from negative to positive infinity. The CDF provides a complete characterization of the probability distribution of a random variable: for continuous distributions, it is the integral of the probability density function (PDF), while for discrete distributions, it is the sum of probabilities up to and including $x$. Key properties of any CDF are the monotonicity and the boundary conditions $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$. CDFs are particularly useful for computing probabilities of intervals, quantiles, and for statistical inference.

Application to Machine Learning

This repository uses the CDF to model and learn flexible probability distributions in machine learning tasks. By parameterizing the CDF with binned logits, it enables differentiable training and efficient sampling, making it suitable for uncertainty estimation, probabilistic prediction, and distributional modeling in neural networks.

Implementation

The PiecewiseConstantBinnedCDF, PiecewiseLinearBinnedCDF, and BezierCDF classes inherit directly from torch.distributions.Distribution, implementing all necessary methods plus some convenience functions. They support multi-dimensional batch shapes and CUDA devices. The bins can be initialized linearly or log-spaced.

torch>=2.7 it the only non-dev dependency of this repo.

Getting Started

I recommend using PiecewiseLinearBinnedCDF for most applications.

from binned_cdf import PiecewiseLinearBinnedCDF

# logits = ... some model output
distr = PiecewiseLinearBinnedCDF(
    logits=logits,  # shape: (*batch_shape, num_bins)
    bound_low=-5,  # adapt to your data
    bound_up=7,  # adapt to your data
    log_spacing=True,  # if False, linear spacing is used
    normalization_method="sigmoid",  # "sigmoid" or "softmax"
    validate_args=None,  # None uses PyTorch's default argument validation; or set True/False
)
# ... use it like any other torch.distribution.Distribution

This package also provides the more esoteric BezierCDF.

from binned_cdf import BezierCDF

# logits_y, logits_x = ... model outputs with the same floating dtype/device and broadcastable batch shapes
# Degrees may differ; omit logits_x or pass None for fixed horizontal spacing.
distr = BezierCDF(
    logits_y=logits_y,  # vertical increments; shape: (*batch_shape_y, degree_y)
    logits_x=logits_x,  # horizontal increments; shape: (*batch_shape_x, degree_x)
    bound_low=-2,  # finite lower support bound; adapt to your data
    bound_up=2,  # finite upper support bound; must exceed bound_low
    normalization_method="softmax",  # "softmax" or "sigmoid", applied to both axes
    validate_args=None,  # None uses PyTorch's default argument validation; or set True/False
)
# ... use it like any other torch.distribution.Distribution

👉 Please have a look at the documentation to get started.