Appendix A. Tree growth inference and prediction.

Gibbs sampling is a Markov chain Monte Carlo (MCMC)
technique that involves alternate sampling from each of the full conditional
distributions of all unknowns in the model, including parameters and latent
variables (Gelfand and Smith 1990, Clark 2005). Here we provide conditional
distributions, together with algorithms used to sample from them. In this
appendix, *p*^{(g)} designates the value of parameter *p* in the *g ^{th}* step of the Gibbs sampler, and

The conditional distribution of growth increments for
individual *i* in stand *j* is

where

for years in which growth increment observations are available, and

for years in which increment
observations are unavailable. Likewise, the density for _{} is included only for
years in with diameter measurements are available. The solutions for these
conditional distributions are provided in Clark (2007). A Metropolis sample
from this distribution begins with a proposal for the first diameter value from
a lognormal density that is truncated at two cm in width centered on the first
observation. Then all diameter increments are proposed from a lognormal (to
insure that they are positive) for the years in which individuals were alive
and included in the study. Proposals are accepted on a tree-by-tree basis. In
other words, blocking is by tree, but there are no loops over trees or years.

Parameters involved in Eqs. 1 and 2 are all sampled directly from conditional Gaussian densities. For the overall mean growth rate, the conditional posterior, integrated over random effects, is

where *n _{j}* is the number of trees in stand

Year effects are sampled from

where *I*(*A*) is the indicator variable equal
to one, when event *A* is true, and zero, otherwise. We impose a
sum-to-zero constraint on the year effects by subtracting the mean at each
Gibbs step.

Variance parameters are all directly sampled from inverse gamma distributions. These are

Observation errors on growth are sampled from

where _{} is the set of growth
observations on tree *ij*. Observation errors on diameter are sampled
from

where _{} is the set of diameter
observations on tree* ij*.

Because increment cores can shrink with moisture loss, a bias parameter can be included in the data model. A multiplicative bias on diameter increment (additive on log diameter increment is

Then the conditional density has parameters

The conditional density for the bias parameters is

and the error conditional is

We do not prove convergence, but extensive simulation indicates that the algorithm does converge to appropriate estimates. The latent diameter states were initialized by interpolation, subject to the constraint that there can be no decreasing diameters. The latent growth rates are initialized as the differenced initial diameter series.

The Gibbs sampler was run 500,000 iterations to insure
convergence and then run an additional 500,000 iterations to obtain posterior
estimates. Because of the large number of estimates (11,444 tree-years), we
did not retain all 500,000 times 19,840 estimates each for diameter and growth. Rather, we retained
the sums and the sums of the squared diameter and log growth estimates from
every 20^{th} Gibbs step. Because both variables were approximately
Gaussian, 95% credible intervals were calculated as __+__ 1.96 standard
deviations, which came, in turn, from first and second moments. These are the
intervals shown in Figs. 4 and 5. For all other parameters credible
intervals are obtained as percentiles of the Gibbs output.

Predictive loss *D _{m}* is the sum of a
goodness of fit term and a penalty term,

_{}

This can be done for both data sets. For diameter measurements, goodness of fit is taken to be the error sum of squares about the predictive mean diameter,

The model is penalized for imprecise predictions, given by the predictive variance,

The quantities _{} and _{} are the predictive
means and variances that are obtained from repeated simulations of *D _{ij}*

LITERATURE CITED

Clark, J. S. 2005. Why environmental scientists are becoming Bayesians. Ecology Letters 8:2–14.

Clark, J. S. 2007. Models for Ecological Data. Princeton University Press. Princeton, New Jersey, USA.

Gelfand, A. E., and A. F. M. Smith. 1990. Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85:398–409.