priors

property LinearModel.priors: dict[str, dict[str, float | int]]

Priors for the regressors’ and variance parameters. Each prior is a key-value pair, where the value is a dict with:

• hyperparameter names as keys

• hyperparameter values as values.

Returns

dict

Priors for each random variable. It must contain an 'intercept' and a 'variance' keys. Each value must be a dict with hyperparameter names as key and hyperparameter values as values.

Raises

TypeError

• If priors is not a dict,

• if a priors’ value is not a dict.

ValueError

• If priors is an empty dict,

• if a priors’ value is an empty dict,

• if a 'variance' value is not positive,

• if a 'shape' value is not positive,

• if a 'scale' value is not positive.

KeyError

• If priors does not contain both 'intercept' and 'variance' keys,

• if a prior’s hyperparameters are not:

  • 'mean' and 'variance' for a regression parameter \(\beta_j\) or

  • 'shape' and 'scale' for variance \(\sigma^2\).

Notes

To each random variables is assigned a prior distribution:

• to each regressor parameter \(\beta_j\) is assigned a normal prior distribution with hyperparameters 'mean' \(\beta_j^0\) and 'variance' \(\Sigma_{\beta_j}^0\):

  \[\beta_j \sim N(\beta_j^0 , \Sigma_{\beta_j}^0)\]

• to variance \(\sigma^2\) is assigned an inverse gamma distribution with hyperparameters 'shape' \(\kappa^0\) and 'scale' \(\theta^0\):

  \[\sigma^2 \sim \text{Inv-}\Gamma(\kappa^0, \theta^0)\]

Examples

Consider a linear regression of the response_variable \(y\) with respect to regressors \(x_1\), \(x_2\) and \(x_3\), according to the following model:

\[y \sim N(\mu, \sigma^2)\]

\[\mu = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3\]

then the sampler would require priors for:

• parameter \(\beta_0\) of variable 'intercept', with 'mean' \(\beta_0^0\) and 'variance' \(\Sigma_{\beta_0}^0\)

• parameter \(\beta_1\) of variable \(x_1\), with 'mean' \(\beta_1^0\) and 'variance' \(\Sigma_{\beta_1}^0\)

• parameter \(\beta_2\) of variable \(x_2\), with 'mean' \(\beta_2^0\) and 'variance' \(\Sigma_{\beta_2}^0\)

• parameter \(\beta_3\) of variable \(x_3\), with 'mean' \(\beta_3^0\) and 'variance' \(\Sigma_{\beta_3}^0\)

• variable \(\sigma^2\), with 'shape' \(\kappa^0\) and 'scale' \(\theta^0\)

>>> model = baypy.model.LinearModel()
>>> model.priors = {
...     'intercept': {'mean': 0, 'variance': 1e6},
...     'x_1': {'mean': 0, 'variance': 1e6},
...     'x_2': {'mean': 0, 'variance': 1e6},
...     'x_3': {'mean': 0, 'variance': 1e6},
...     'variance': {'shape': 1, 'scale': 1e-6}
... }