3. Model Fit and Selection

Laud-Ibrahim Criteria

Laud-Ibrahim Criteria is another model comparison tool.

Consider \(M\) family of models, and \(\theta^{(m)}\) are the \(m\) model’s parameters. By Bayes theorem, the posterior of \(\theta^{(m)}\) is

\[\pi(\theta^{(m)} | m, y) = \frac{f(y | \theta^{(m)},m) \pi(\theta^{(m)})}{\int f(y | \theta^{(m)},m) \pi(\theta^{(m)}) d \theta^{(m)}} \]

The posterior predictive distribution \(z|y,m\) is

\[ f(z|y,m) \sim \int f(z|\theta^{(m)},m) \pi(\theta^{(m)}|y,m)d\theta^{(m)}\]

So for a fixed \(m\), Laud-Ibrahim Criteria is

\[L_m^2 = \text{E}(Z - y)'(Z-y) = \sum_{i=1}^n (\text{E} Z_i - y_i)^2 + \text{Var}(Z_i) \]

where \(Z \sim f(z|y,m)\).

\(\text{L}_m = \sqrt{\text{L}_m^2} \) is essentially a discrepancy measure, and is in the same units as the response \(y\). The model that minimizes \(\text{L}_m\) is prefered. On the next page, we can see an example of comparing models with Laud-Ibrahim Criteria in PyMC.