sample_sigma2¶

sample_sigma2(y: Series, X: ndarray, beta: ndarray, k_1: float, theta_0: float) → float¶

Draws a sample for the variance from an inverse-gamma conditional posterior distribution at each regression iteration.

Parameters¶

ypd.Series: Vector of the response variable \(y\). It has a number of rows equal to the number of observed data.
Xnumpy.ndarray: Regressors matrix \(X\). It has a number of rows equal to the number of observed data and a number of columns equal to the number of regressors.
betanumpy.ndarray: Regressors’ parameters array of the model. It is sampled by the function sample_beta at each iteration.
k_1float: Sum of the variance prior shape and the number of observed data.
theta_0float: Variance prior scale.

The variance is drawn from an inverse-gamma distribution. The conditional posterior distribution is:

\[H \left( \sigma^2 \left\vert B, y \right. \right) \sim \text{Inv-}\Gamma \left( \frac{\kappa_1}{2}, \frac{\theta_1}{2} \right)\]

where \(B\) is the \(\beta_j\) vector, \(\kappa_1\) is:

\[\kappa_1 = \kappa_0 + n\]

where \(n\) is the number of observed data and \(\theta_1\) is:

\[\theta_1 = \theta_0 + \left( y - X B \right)^T \left( y - X B \right) .\]