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What is pyINLA?

pyINLA is a Python interface to INLA (Integrated Nested Laplace Approximations), a high-performance engine written in C/C++ for fast, approximate Bayesian inference.

INLA is designed for Latent Gaussian Models (LGMs), a flexible class that includes generalized linear models, mixed effects models, spatial models, and time series. In these models:

• Observations follow a distribution from the exponential family (Gaussian, Poisson, Binomial, etc.)
• A latent Gaussian field captures fixed effects, random effects, and structured components
• Hyperparameters control variance, correlation, and smoothness

Instead of sampling (like MCMC), INLA uses Laplace approximations to analytically compute posterior marginals, making it orders of magnitude faster for this model class. pyINLA joins R-INLA as the two official wrappers for the INLA engine.

Learn how INLA works →

Installation

We recommend using a virtual environment (uv, conda, or venv) to keep dependencies isolated.

Model Specification

In pyINLA, you write the statistical model as a Python dictionary. Here is how the math maps to code.

The Mathematical Model

Consider a mixed-effects regression with Gaussian observations:

From Math to Code

Each part of the equation maps directly to a key in the model dictionary:

Runnable Example

Copy-paste this into Python, no data files needed:

import pandas as pd
from pyinla import pyinla

# 10 observations, 3 groups
df = pd.DataFrame({
    'y':     [2.1, 3.5, 2.8, 5.2, 4.9, 6.1, 1.9, 3.3, 5.7, 4.4],
    'x1':    [0.3, 0.7, 0.5, 1.2, 1.0, 1.5, 0.2, 0.6, 1.4, 1.1],
    'x2':    [1.0, 1.2, 0.8, 2.1, 1.9, 2.5, 0.9, 1.1, 2.3, 1.8],
    'group': [  1,   1,   1,   2,   2,   2,   3,   3,   3,   3],
})

# Math:  y_i = beta_0 + beta_1*x1 + beta_2*x2 + u_group(i) + e_i
model = {
    'response': 'y',
    'fixed': [
        '1',                       # beta_0  (intercept)
        'x1',                      # beta_1  (covariate)
        'x2'                       # beta_2  (covariate)
    ],
    'random': [
        {'id': 'group', 'model': 'iid'}  # u_j ~ N(0, sigma_u^2)
    ]
}

result = pyinla(model=model, family='gaussian', data=df)

print(result.summary_fixed)     # Fixed effect estimates
print(result.summary_hyperpar)  # Hyperparameter estimates

Learn More

Explore these topics to master pyINLA:

Next Steps