Fixed Effects

Linear: Latent Single‑Slope

INLA's model='linear' treats a single slope as a latent component with a Gaussian prior. pyINLA exposes the same helper via the model['random'] block.

Parametrization

The latent linear model is an alternative interface for a single fixed effect. It adds a one-column latent component whose contribution is

$$\eta_i = \beta\,z_i + \cdots$$

The slope $\beta$ follows a Gaussian prior

$$\beta \sim \mathcal{N}(\mu,\; \tau^{-1})$$

with $\mu$ and $\tau$ supplied via mean.linear and prec.linear. Conceptually, this is identical to a fixed effect with an informative prior: only the interface differs.

Hyperparameters

No additional hyperparameters are introduced beyond the Gaussian prior on $\beta$; the model does not expose a hyper block.

Allowed Parameters

Parameter	Required	Description
`model`	Yes	Must be `'linear'`.
`covariate`	Yes	Column name or 1D vector of covariate values.
`id`	No	Label used in the section name and in `summary_random`. Defaults to the value of `covariate` when it is a column name.
`mean.linear`	No	Prior mean $\mu$ of the slope (default: $0$).
`prec.linear`	No	Prior precision $\tau$ of the slope (default: $0.001$).
`diagonal`	No	Small constant added to the precision-matrix diagonal for numerical stability (default: $0$).

pyINLA Usage

Declare the covariate as a linear latent component inside model['random']. Supply the covariate column (or vector) with covariate=... and set the prior via mean.linear / prec.linear.

import numpy as np
import pandas as pd
from pyinla import pyinla

n = 100
rng = np.random.default_rng(123)
z = rng.uniform(size=n)
y = 1.0 + z + rng.normal(size=n)

df = pd.DataFrame({"y": y, "z": z})

model = {
    "response": "y",
    "fixed": [],
    "random": [{
        "model":       "linear",
        "covariate":   "z",
        "mean.linear": 1.0,
        "prec.linear": 1.0,
    }],
}

result = pyinla(model=model, family="gaussian", data=df)
print(result.summary_fixed)

The model fits $y_i = \beta_0 + \beta_z\, z_i + \varepsilon_i$ with $\beta_z \sim \mathcal{N}(1,\; 1^{-1})$. pyINLA writes a dedicated linear section so INLA treats the slope as a latent component with the specified prior.

Notes on usage

covariate accepts a column name or an explicit vector. Missing values are replaced with $0$.
mean.linear and prec.linear must be scalars; omit them to use the engine defaults ($\mu = 0$, $\tau = 0.001$).

Alternative via control['fixed']

You can obtain the same prior by keeping the slope in the fixed-effects block and setting label-specific entries in control['fixed']. This remains useful when you want to stay entirely within the fixed-effect design.

formula = {
    "response": "y",
    "fixed":    ["1", "z"],
}

control = {
    "fixed": {
        "mean": {"z": 1.0},
        "prec": {"z": 1.0},
    },
}

result = pyinla(formula=formula, family="gaussian", data=df, control=control)

Notes

Combine model='linear' with covariate when you need explicit slope priors outside model['fixed'].
For bounded slopes, switch to the clinear model which enforces hard limits on $\beta$ (see CLINEAR).