compute_DIC¶

compute_DIC(model: Model, print_summary: bool = True) → dict[str, float]

It computes and prints the Deviance Information Criterion (DIC) for the fitted model.

Parameters¶

modelModel: The model with data, regressors, response variable and priors to be solved through Monte Carlo sampling.
print_summarybool, optional: If True prints the deviance summary report. Default is True.

Returns¶

dict

Dictionary with deviance summary. It contains:

key 'Deviance at posterior means',
key 'Posterior mean deviance',
key 'Effective number of parameters',
key 'Deviance Information Criterion'.

Raises

TypeError

If model is not a Model,
if print_summary is not a bool.

ValueError

If a model.posteriors is None because the sampling has not been done yet,
if a posterior key is not a column of model.data,
if model.data is an empty pandas.DataFrame,
if model.response_variable is not a column of model.data.

Notes

The DIC measures posterior predictive error by penalizing the fit of a model (deviance) by its complexity, determined by the effective number of parameters. Comparing some alternative models, the smaller the DIC of a model, the better the model. Consider a linear regression of the response variable \(y\) with respect to regressors \(X\), according to the following model:

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

\[\mu = \beta_0 + B X = \beta_0 + \sum_{j = 1}^m \beta_j x_j\]

then the likelyhood is:

\[p \left( y \left\vert B,\sigma^2 \right. \right) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{- \frac{\left(y - \mu \right)^2}{2 \sigma^2}} .\]

The deviance [1] [2] is defined as:

\[D \left( y, B, \sigma^2 \right) = -2\log p \left( y \left\vert B,\sigma^2 \right. \right) .\]

The deviance at posterior mean of \(B\) and \(\sigma^2\), denoted by \(\overline{B}\) and \(\overline{\sigma^2}\) is:

\[D_{{\overline{\beta}}, \overline{\sigma^2}} (y) = D \left( y, \overline{B}, \overline{\sigma^2} \right)\]

while the posterior mean deviance is:

\[\overline{D} \left( y, B, \sigma^2 \right) = E \left( D(y, B, \sigma^2) \left. \right\vert y \right) .\]

and the effective number of parameter is defined as:

\[pD = \overline{D} \left( y, B, \sigma^2 \right) - D_{{\overline{\beta}}, \overline{\sigma^2}} (y) .\]

The Deviance Information Criterion [1] is:

\[DIC = 2 \overline{D} \left( y, B, \sigma^2 \right) - D_{{\overline{\beta}}, \overline{\sigma^2}} (y) = \overline{D} \left( y, B, \sigma^2 \right) + pD = D_{{\overline{B}}, \overline{\sigma^2}} (y) + 2pD .\]

References