Kevin Gross, Anthony R. Ives, and Erik V. Nordheim. 2005. Estimating fluctuating vital rates from time-series data: a case study of aphid biocontrol. Ecology 86:740–752.

Appendix D. Priors. See Appendix H for references cited.

Here, we describe the prior distributions on all the parameters except for the time-varying vital rates, which are discussed in the main text.

Initial densities

To facilitate MCMC mixing, the initial state vector X₁ is re-parameterized into the following six components: total aphid density, fraction of aphids that are juveniles, fraction of juveniles parasitized, fraction of adults parasitized, total mummy density, and fraction of mummies hyperparasitized. Priors for each of the fractional components are Uniform(0,1). The prior for the total aphid density is Uniform(0,50) and for the total mummy density is Uniform(0,10).

Process model parameters that do not depend on cutting cycle

Ives et al. (1999) showed that parasitoids preferentially attack juvenile aphids. We assume that the ratio of attacks on adults to attacks on juveniles is a constant
, and use the data in Ives et al. to build an informative Beta prior for ,  with prior mode 0.064 and prior standard deviation 0.025. For the reduction in aphid fecundity due to parasitism, , we use a vague prior, Uniform on (0,1).

Parameters in the observation model

There are 8 parameters in , and the priors for all are mutually independent.  For the overdispersion parameters in the negative binomial distributions, k_A and k_M, stem count data from 1996 were used to calculate the priors, with prior modes set equal to their MLEs from 1996, and prior variances inflated substantially compared to the sampling variances of the MLEs. Both priors were Gamma distributions, for k_A with mean 7.63 and standard deviation 7.07, and for k_M with mean 29.60 and standard deviation 22.36.

For the conversion parameters, _s and _tc, priors were modified triangular distributions, with uniform prior density on a range covering all reasonable values, and tapering prior density at the boundaries of the prior support (Fig. F1, Appendix F). The prior support for _s is informed by 1996 data, and the prior support for
_tc is based on empirical observation. 

Priors for the overdispersion parameters _W and _H are  triangular priors on (0,1), with prior modes at 0 (Fig. F1). Priors for the emergence parameters _W and _H are Uniform(0,1).