Applied Bayesian Modelling - Free

DISCRETE OUTCOMES 195One might also envisage (in terms of its compatibility with the BUGS computingenvironment) having a single mean m t for y t , but composed of the survival term r 1 8y t1 ,and an additional series v t , that follows a positive density (e.g. a gamma). For astationary INAR(1) model, this leads toy t Poi(m t )m t ˆ r8Y t1 v tv t G(bu(1 r), b)(5:23)5.3.3 Continuity parameter modelsHarvey (1989) and Ord et al. (1993) also propose a model for count series combiningtwo sources of randomness. One concerns changes in the underlying level (as does thesurvival term of an INAR model) and the other refers to the distribution of observationsaround that level. Thus, with Poisson sampling, the mean is itself gamma distributedwith time evolving parameters (a t , b t ),y t Poi(m t )(5:24)m t G(a t , b t )The gamma parameters are related to previous parameters (a t1 , b t1 ) via a commoncontinuity parameter f. This takes values between 0 and 1 that applies to both scale andindex of the gamma. Thusa t ˆ fa t1(5:25)b t ˆ fb t1The initial values a 0 , b 0 are assigned a prior ensuring positive values (e.g. log-normal orgamma). To avoid improper priors for later time periods, one may modify Equation(5.25) by adding a small constant ± indicating the minimum prior scale and index in theprior for the m t . This might be taken as an extra parameter or preset. Thusa t ˆ fa t1 cb t ˆ fb t1 cIt is also possible to drop the constraint f < 1 and assess the probability that f is in factconsistent with information loss (i.e. with accumulated discounting of past observationsas f < 1 implies).5.3.4 Multiple discrete outcomesThe approach of Equations (5.24)±(5.25) may be extended to multivariate count series,y kt , k ˆ 1, : : K, t ˆ 1, : : Tby modelling the total count at time tY t ˆ Xin the same way as a univariate count with parameter f 1 . The disaggregation to theindividual series is modelled via a multinomial-Dirichlet model, with the evolution ofthe Dirichlet parameters governed by a second parameter f 2 .ky kt