TABLE 11.

Average realized and predicted discrepancy measures for the autoregressive multinomial and Dirichlet‐multinomial models, average posterior two‐sided p‐values for data from Lombardy

Autoregressive model
Multinomial	$\widehat{\text{Dist}}$	$\widetilde{\text{Dist}}$	p‐value
Model 1 (2nd order, without constraints)	2487.417	147.896	0.000
Model 2 (2nd order, with constraints)	3868.846	70.011	0.000
Model 3 (3rd order, without constraints)	2532.507	139.870	0.000
Model 4 (3rd order, with constraints)	3855.632	71.237	0.000
Dirichlet‐multinomial	$\widehat{\mathbf{\text{Dist}}}$	$\widetilde{\mathbf{\text{Dist}}}$	p‐value
Model 5 (2nd order, without constraints)	4487.878	5911.391	0.765
Model 6 (2nd order, with constraints)	5210.567	4960.395	0.388
Model 7 (3rd order, without constraints)	4346.473	6514.379	0.646
Model 8 (3rd order, with constraints)	4957.757	5034.154	0.427