rejection-sampler 0.1.2

A statistical computing toolkit for validating proposal distributions and computing optimal rejection-sampling constants.

rejection-sampler

A small Python package for validating rejection sampling setups and computing the optimal rejection constant (M). The package supports both callable Python functions and symbolic SymPy expressions for target and proposal probability density functions (PDF).

In rejection sampling, M is the smallest constant such that, for all x in the target support:

target_pdf(x) <= M * proposal_pdf(x)

Installation

pip install rejection-sampler

or

uv add rejection-sampler

Usage

Import the main function:

from rejection_sampler import find_optimal_M

Example 1: Callable input

def target_pdf(x):
    return 2 * x if 0 <= x <= 1 else 0.0

def proposal_pdf(x):
    return 1.0 if 0 <= x <= 1 else 0.0

M = find_optimal_M(
    target_pdf=target_pdf,
    target_support=(0.0, 1.0),
    proposal_pdf=proposal_pdf,
    proposal_support=(0.0, 1.0),
)

print(M)

Example 2: SymPy input

import sympy as sp

x = sp.Symbol("x", real=True)

target_pdf = 2 * x
proposal_pdf = sp.Integer(1)

M = find_optimal_M(
    target_pdf=target_pdf,
    target_support=(0, 1),
    proposal_pdf=proposal_pdf,
    proposal_support=(0, 1),
)

print(M)

Infinite support

For numerical inputs with infinite support, provide finite optimization bounds:

import numpy as np
from rejection_sampler import find_optimal_M

def target_pdf(x):
    return np.exp(-0.5 * x * x) / np.sqrt(2 * np.pi)

def proposal_pdf(x):
    return 1.0 / (np.pi * (1 + x * x))

M = find_optimal_M(
    target_pdf=target_pdf,
    target_support=(-np.inf, np.inf),
    # or use (-float("inf"), float("inf")) for infinite support
    proposal_pdf=proposal_pdf,
    proposal_support=(-np.inf, np.inf),
    bounds=(-10.0, 10.0),
)

print(M)

Parameters

• target_pdf: target probability density function, either callable or SymPy expression
• target_support: support of the target PDF
• proposal_pdf: proposal probability density function, either callable or SymPy expression
• proposal_support: support of the proposal PDF
• error: numerical tolerance for validation
• bounds: search interval for numerical optimization for pdfs with infinite support

Note

• When writing mathematical expressions (eg. exp, log, sqrt, inf), use numpy instead of the built-in math module.
• For infinite-support callable inputs, choose sufficiently large bounds. bounds that are too small may miss the global optimum.

License

This project is licensed under the MIT License.