Table 3. Summary of USA/Canada influenza data inference.

Posterior distribution quantile summaries for BEAST implementation of BNPR-PS models: with no covariates (model: {γ(t)}), with an ordinary covariate (model: {γ(t), −t}), and with seasonal indicator covariates (model: {γ(t), I_winter, I_autumn, I_summer}). Using a fixed tree, we also include marginal likelihoods (MLs) for all of the above models, plus a model with N_e(t) and λ_s(t) estimated independently and nonparametrically (Unrelated N_e(t) and λ_s(t)) and a conditional model combined with a constant sampling intensity model for sampling times (Constant λ_s(t)).

Model	ML	Coef	Q0.025	Median	Q0.975
Unrelated Ne(t) and λs(t)	-732	—	—	—	—
Constant λs(t)	-975	—	—	—	—
{γ(t)}	-626	γ(t)	1.11	1.45	2.01
{γ(t), −t}	-632	γ(t)	1.21	1.52	2.00
		−t	-0.10	-0.02	0.07
{γ(t), Iwinter, Iautumn, Isummer}	-604	γ(t)	0.72	0.92	1.21
		Iwinter	1.91	2.79	3.83
		Iautumn	1.88	2.85	3.85
		Isummer	0.44	1.52	2.58