Appendix 2. Bayesian forecasting.

In these models, agents use Bayesian models to update their information about the ecosystem or the management system, and to forecast the future state of the ecosystem or future management decisions. In general, a polynomial model

 Xt = Xt-1 + at-1 + R(Xt-1) (A.2.1) R(Xt-1) = B0 + B1Xt-1 + B2Xt-12 + . . . . (A.2.2)

is used to forecast the system state, X. Depending on the context, X can be the pollutant level in the ecosystem, the payoff from a market, or the regulations set by an agency. At each time step t, the new Xt is observed and used to calculate the updated parameters Bt and their covariances. The procedure, known as Bayesian updating, is based on a statistical model (Pole et al. 1994).

 Xt = F't Bt + vt (A.2.3) Bt = Bt-1 + wt (A.2.4)

where v and w are independent Student's t-distributed errors, F' = [1 Xt Xt2 . . .] and B' = [B0 B1 B2 . . .]. Eq. A.2.3 is simply Eq. A.2.1 written as a linear regression with errors v. According to Eq. A.2.4, values of the parameters change through time as a random walk. The inverse variance of v is assumed to follow a gamma distribution with parameters n and d. C is the covariance matrix of w.

Eqs. A.2.3 and A.2.4 account for the dynamics of the system state and parameter estimates, but do not account for temporal change in uncertainty. The development of uncertainty is modeled using a discount (or memory) parameter D, 0 < D £ 1 (Pole et al. 1994). In practice, we usually set 0.8 < D £ 1. Low values of D cause estimates of B to be quite sensitive to short-term fluctuations in X. Values of D near 1 cause estimates of B to change gradually, with little response to short-term variation in X. For a given data set, D can be estimated by maximum likelihood (West and Harrison 1989). In contrast, we use D as a parameter that controls the sensitivity of the learning system to accumulated information over many time steps, rather than the most recent observations.

Before making a new observation, we know Bt-1, nt-1, dt-1, and Ct-1. Given a new observation Xt, we calculate the forecast error vt and update Ft. We update B and the variances as follows.

 St-1 = dt-1/nt-1 (A.2.5) nt = D nt-1 + 1 (A.2.6) Rt = D-1 Ct-1 (A.2.7) Qt = F'tRt Ft + St-1 (A.2.8) dt = D dt-1 + [(St-1 vt2) / Qt] (A.2.9) At = Rt Ft / Qt (A.2.10) Ct = (St / St-1) [Rt - (At A't Qt)] (A.2.11) Bt = Bt-1 + At vt. (A.2.12)

The new information available for decision making is contained in Bt and Ct. These estimates can be used to predict the Student's t distribution of Xt+1 as a function of input targets, a, with mean Xt+1 and scale factor Qt+1, calculated as

 Xt+1 = F't+1 Bt (A.2.13) Qt+1 = F't+1 (D-1 Ct ) Ft+1 + dt/nt. (A.2.14)

Degrees of freedom are nt.

.

In the Market Manager and Governing Board models, agents update parameters Bi for a model of pollutant dynamics in the water that includes a term for recycling,

 Xt = Xt-1 + at-1 - B1,t-1 Xt-1 + B2,t-1 g(Xt-1) (A.2.15) g(X) = Xq / (1 + Xq). (A.2.16)

These equations are similar to the nondimensional model of Appendix 1. The model is an approximation that might be reasonable in a situation in which the dynamics of the slow variable and mechanism of recycling are unknown. Because updating depends on assimilation of new observations of X, the learning process will always lag behind the changes in the ecosystem.

At each time step t, the new Xt is observed and used to calculate the updated parameters Bt and their covariances. As in Eqs. A2.3 and A2.4, we have

 Xt = F't Bt + vt (A.2.17) Bt = Bt-1 + wt (A.2.18)

where v and w are independent Student's t-distributed errors, F' = [a X g(X)], and B' = [1 B1 B2]. The updating of parameters B and their covariance matrix C is accomplished via Eqs. A.2.5 to A.2.12. Predictions also follow the procedure of Eqs. A.2.13 and A.2.14.

In practice, estimates of phosphorus recycling in lakes are done by intensive experimentation over a limited period of time (e.g., Soranno et al. 1997), or by predictions using regressions fit to literature data (e.g., Nürnberg 1984). Time series of phosphorus recycling rate are measured rarely. The instantaneous recycling rate is proportional to P in the water, and time series of P in the water are often known. In fitting Eq. A.2.15 to observed time series, estimates of B1 and B2 will be highly correlated. In this situation, researchers may choose to fix B2 at a value known from the literature, and update B1 based on regular observations of Xt. We modeled this approximation by setting B2 = 1 so that Eq. A.2.15 becomes

 Xt = Xt -1 + at-1 - B1,t-1 Xt-1 + g(Xt-1). (A.2.19)

In Eqs. A.2.3 and A.2.4, we now have B' = [1 B1 1]. The updating and prediction procedures are unchanged.

We refer to B1 of Eq. 2.19 as the reversibility parameter for the following reason. The lake can have one or three steady states, depending on the value of B1 (Carpenter et al. 1999). For sufficiently high values of B1, the lake has a single stable steady state and the P level is smoothly reversible by changing a. For intermediate levels of B1, there are two stable steady states and the transition between them is hysteretic. For sufficiently low levels of B1, there are two stable states, but the high-P state is irreversible. Once in the high-P state, there is no feasible value of a that can be used to reach the low-P state in only a few years. Instead, it is necessary to hold a at low levels for decades until the level of phosphorus in the sediment has declined. Thus, the value of B1 tells us whether the P level is smoothly reversible by changing a, hysteretic (delayed response to reduction in a), or irreversible (extended period of very low a is required to reduce levels of pollutant in the sediments).