Consistency Analysis Summary

In practice, large environmental statistical models are difficult to estimate and validate because data sets are often not large enough or complete enough to allow the use of standard statistical methods. The most common approach to model fitting with an incomplete sample is to maximize the marginal likelihood function found by integrating the full likelihood over the unobserved variables (Lehmann and Casella 1998, pp. 458-460). This estimation method will be referred to as Marginal Maximum Likelihood (MML).

Another solution to this problem is to fix some of the parameters at values based on the substantive literature and then use the sample to estimate the balance of the parameters. This approach was used by Speed (1993) to estimate a large state-space time series model of salmon population in the Pacific Northwest. Consistency analysis (Haas 1991a, 1991b, 1997) extends this idea of fixing some of the parameters to substantive theory-derived values (called hypothesis values) by producing parameter estimates such that the resultant fitted or consistent joint distribution represented by the influence diagram deviates minimally from both the joint distribution specified by the hypothesis parameter values (called the hypothesis distribution), and the empirical distribution computed from a sample taken on a subset of the influence diagram's chance nodes (called the empirical distribution).

To incorporate prior knowledge into consistency analysis then, the analyst need only supply a set of point values of all influence diagram parameters (the vector $\mbox{\boldmath$\beta$}_H$)and a priority weight (c_H) that expresses the importance to the analyst of having the consistent distribution agree with the hypothesis distribution relative to agreeing with the empirical distribution. This representation of prior knowledge through a set of point values of the parameters is in keeping with the statistical inference approach advocated by Edwards (1972) that hypotheses should be formulated as point values of a model's parameters. See the Appendix for a description of the consistency analysis parameter estimation algorithm along with a comparison of consistency analysis with bayesian, frequentist, and goodness-of-fit parameter estimation methods.