5.2 Measures of Model Specificity
To see what is involved here, we exhibit in figure 1 a model representing a target variable T as the dependent variable and the decision variables C1 and C2. The cylinder represents a mapping from control variables C1 and C2 into the target variable T. The domain of the relation is the ellipse in the plane C1OC2. The height of the cylinder is the precision of the model. That is, given any pair of values of C1 and C2, say
and
,the value of T will be determined by some point within the cylinder on a line perpendicular to the C1OC2 plane at (
,
). The relation does not indicate which point on that line will yield the realized value of T. Since this is true for any point within the cylinder, the size of the collection of such points is the volume of the cylinder. Since the cylinder is the mapping between the independent and the dependent variables, its volume is naturally identified with the volume of the relation.
We shall say that the agent has a universal model which comprises the whole of the C1,C2,T)-space except in the domain of the cylinder. It will be in keeping with natural use of language if we call the cylinder a special model for the domain represented by the ellipse in the C1OC2-plane. The default values continue to apply everywhere else in (C1,C2,T)-space. The volume of the universal model is now the volume of the feasible space minus the volume of that space which is in the domain of the cylindrical model plus the volume of that model (the volume of the cylinder). In other words, the volume of the agent's universal model is reduced by the volume of the space above and below the cylindrical special model. Clearly, the volume of the universal model can be reduced further either by increasing either the domain of the cylindrical model or its precision. Moreover, special models with different domains from that of the cylindrical model will also reduce the volume of the universal model. So, too, will models of lesser dimension than the whole of the (C1,C2,T)-space. If a relationship between (say) C1 and T can be found over some domain of C1 and some range of T, then in (C1,C2,T)-space the special model is a bar, rectangular in cross-section projected perpendicularly from the rectangle in the C1OT plane implied by the domain and range of the two-dimensional model. The volume of the universal model is therefore reduced by the volumes above and below that bar in the feasible space.
The perfect model has perfect precision and, therefore, zero volume in space of any dimension -- just as a point, a line or a plane have zero volume in three dimensions. The volume of the least restrictive model is the volume of the feasible part of a space with dimensionality equal to the number of all of the variables which define the state of the environment. Reducing the dimensionality of the model, increasing the domain of the special models and/or increasing the precision of special models all reduce the volume of the agent's universal model. We take reduction in model volume to be a criterion of improvement and, therefore, a guide to model adaptation.
The default assumption is that anything is possible. Only those possibilities within the domain of the model description that are not predicted by that description are ruled out. So if the domain of applicability of the model description is narrowed (by, say, adding extra conditions) then fewer possibilities will be ruled out - the total volume increases. If the precision of the prediction of the model description is increased, then within this domain more possibilities are ruled out - the total volume decreases. Thus by these definitions, condition 3 above implies that models which give more precise predictions and those with wider conditions of application are preferable. Also models with a smaller volume are more easily falsifiable, in the sense that a random possibility is more likely to lie outside the volume of possibilities predicted by the model and thus show it to be wrong.
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