---
myst:
    substitutions:
        sentence1: ""
        sentence2: "as noted at the end of this section"
---
```{include} /../core/calibration/linear_bayesian.md
```

In the specific case of a linear model, one can then write $f_\theta(\mathbf{x})=h^T(\mathbf{x})\theta$ 
where $h(\mathbf{x})$ is the regressor vector. This way of writing the model can include an "hidden 
virtual" $\theta_0 = 1$ whose purpose is to integrate a constant term into the regression (to 
describe a pedestal). Using the statistical approach introduced in [](#calibration_reminder), one 
can also define the covariance matrix of the residuals which will be written hereafter as 
$\Sigma={\rm diag}(\sigma_{\varepsilon_1},\ldots,\sigma_{\varepsilon_n})$

From there, one can construct the *design matrix* $ H = [h(\mathbf{x}_1), \ldots, h(
\mathbf{x}_n)]^T \in M_{n,p}(\mathbb{R})$ whose columns define the subspace onto which the 
model is projected. With a normal prior, which follows the form $\theta \sim \mathcal{N}(m_\theta,
\Sigma_\theta)$ the posterior is also expected to be normal, so it can be written
$\pi(\theta|\mathbf{y}) \sim \mathcal{N}(m^{post}_\theta,\Sigma^{post}_\theta)$ where its parameters 
are expressed as 

```{math}
:label: eq_linPostMeanCalib

m^{post}_{\theta} = \Big( \Sigma^{-1}_{\theta} +H^{T} \Sigma^{-1} H \Big)^{-1} \Big( m^{T}_{\theta}
\Sigma^{-1}_{\theta} + \mathbf{y}^{T} \Sigma^{-1} H \Big)^{T}
```
and 

```{math}
:label: eq_linPostStdCalib

\Sigma^{post}_{\theta} = \Big( \Sigma^{-1}_{\theta} + H^{T} \Sigma^{-1} H \Big)^{-1}
```

It is also possible, as introduced in [](#calibration_reminder_discussing_theoretical_bayesian_approach), 
to use a non-informative **prior** such as 
Jeffrey's prior: it is an improper flat prior ($\pi(\mathbf{y})\propto 1$) 
{cite}`bioche2015approximation`, whose posterior distribution (in the linear case) is also Gaussian. 
For this prior, the posterior parameters are equivalent to those obtained with a Gaussian prior, given 
in {eq}`eq_linPostMeanCalib` and {eq}`eq_linPostStdCalib` with all references to $\Sigma_\theta$ removed: 

```{math}
:label: eq_linPostMeanStdCalibJef

m^{post}_{\theta} = \Big(H^{T} \Sigma^{-1} H \Big)^{-1} H^{T} \Sigma^{-1} \mathbf{y} \;\; {\rm 
and} \; \; \Sigma^{post}_{\theta} = \Big(H^{T} \Sigma^{-1} H \Big)^{-1}
```

This final form corresponds to the expected results obtained when only considering linear regression within the weighted least squares approach {cite}`Fry2010`.

```{toctree}
linear_bayesian/prediction
```