symlik

Symbolic Likelihood Models for Statistical Inference

symlik lets you define statistical models symbolically and automatically derives everything needed for inference: score functions, Hessians, Fisher information, standard errors, and maximum likelihood estimates.

Why symlik?

Traditional statistical computing requires manually deriving score functions and information matrices, or relying on numerical approximations. symlik takes a different approach: write the log-likelihood symbolically, and let the computer handle the calculus.

from symlik.distributions import exponential

model = exponential()
data = {'x': [1.2, 0.8, 2.1, 1.5]}

mle, _ = model.mle(data=data, init={'lambda': 1.0})
se = model.se(mle, data)

print(f"Rate: {mle['lambda']:.3f} ± {se['lambda']:.3f}")
# Rate: 0.714 ± 0.357

Installation

pip install symlik

Requirements: Python 3.8+, NumPy, and rerum (symbolic rewriting engine).

Key Features

• Symbolic differentiation - Automatic score functions and Hessians
• Heterogeneous data - Handle censoring, masking, and mixed observation types
• Series systems - Reliability analysis for multi-component systems
• DataFrame support - Works with pandas, polars, or plain dicts
• Extensible - Custom distributions, operators, and hazard functions

Quick Start

Pre-built Distributions

from symlik.distributions import normal, poisson, exponential

# Normal distribution
model = normal()
mle, _ = model.mle(
    data={'x': [4.2, 5.1, 4.8, 5.3, 4.9]},
    init={'mu': 0, 'sigma2': 1},
    bounds={'sigma2': (0.01, None)}
)

Custom Models

Define log-likelihoods using s-expressions:

from symlik import LikelihoodModel

# Exponential: ℓ(λ) = Σ[log(λ) - λxᵢ]
log_lik = ['sum', 'i', ['len', 'x'],
           ['+', ['log', 'lambda'],
            ['*', -1, ['*', 'lambda', ['@', 'x', 'i']]]]]

model = LikelihoodModel(log_lik, params=['lambda'])

# Symbolic derivatives available
score = model.score()       # Gradient
hess = model.hessian()      # Hessian matrix
info = model.information()  # Fisher information

Heterogeneous Data (Censoring)

Handle mixed observation types with ContributionModel:

from symlik import ContributionModel
from symlik.contributions import complete_exponential, right_censored_exponential

model = ContributionModel(
    params=["lambda"],
    type_column="status",
    contributions={
        "observed": complete_exponential(),
        "censored": right_censored_exponential(),
    }
)

data = {
    "status": ["observed", "censored", "observed", "observed", "censored"],
    "t": [1.2, 3.0, 0.8, 2.1, 4.5],
}

mle, _ = model.mle(data=data, init={"lambda": 1.0})

Series System Reliability

Model multi-component systems with known cause, masked cause, or censored observations:

from symlik import ContributionModel
from symlik.series import build_exponential_series_contributions

# 3-component series system
contribs = build_exponential_series_contributions(m=3)

model = ContributionModel(
    params=["lambda1", "lambda2", "lambda3"],
    type_column="obs_type",
    contributions=contribs,
)

# Handles: known_1, known_2, known_3, masked_12, masked_13,
#          masked_23, masked_123, right_censored

Masked cause observations satisfy C1, C2, C3 conditions:

• C1: True cause always in candidate set
• C2: Candidate set probability independent of which component failed
• C3: Masking probabilities independent of parameters

DataFrame Support

Works seamlessly with pandas and polars:

import pandas as pd

df = pd.DataFrame({
    "status": ["observed", "censored", "observed"],
    "t": [1.2, 3.0, 0.8],
})

mle, _ = model.mle(data=df, init={"lambda": 1.0})  # Just works!

S-Expression Syntax

symlik uses s-expressions to represent mathematical formulas:

Available Distributions

Contribution Types

Series System Components

Build custom reliability models:

from symlik.series import (
    exponential_component,
    weibull_component,
    exponential_ph_component,  # Proportional hazards
    custom_component,          # Any hazard function
    series_known_cause,
    series_masked_cause,
)

# Mixed system: exponential + Weibull
components = [
    exponential_component("lambda"),
    weibull_component("k", "theta"),
]

# Covariate-dependent hazards
comp = exponential_ph_component(
    baseline_rate="lambda0",
    coefficients=["beta"],
    covariates=["temperature"],
)

Direct Calculus Operations

Use symlik's calculus module for standalone symbolic math:

from symlik import diff, gradient, hessian, simplify

# Differentiate x³ + 2x
expr = ['+', ['^', 'x', 3], ['*', 2, 'x']]
deriv = diff(expr, 'x')  # 3x² + 2

# Gradient of f(x,y) = x² + xy
expr = ['+', ['^', 'x', 2], ['*', 'x', 'y']]
grad = gradient(expr, ['x', 'y'])  # [2x + y, x]

Documentation

Full documentation: queelius.github.io/symlik

• Getting Started
• Contribution Models
• Series Systems
• API Reference

License

MIT License. See LICENSE for details.

Contributing

Contributions welcome! Please open an issue or pull request on GitHub.