Parsimony rule

Applications of the Theory.—As an illustration of the application of the foregoing prindples some predictions may be made regarding the structure of cyanite, andalusite and sillimanite, the three forms of Al2Si05. From the rule of parsimony we expect all aluminum octahedra to be similar and all silicon tetrahedra to be similar. Let the number of octahedra one comer of which is formed by the ith oxygen ion be a,- then the stoichiometries oxygen-aluminum ratio, 5 2, requires that... 

A one standard error rule is described in Hastie et al. (Hastie et al. 2001). It is assumed that several values for the measure of the prediction error at each considered model complexity are available (this can be achieved, e.g., by CV or by bootstrap, Sections 4.2.5 and 4.2.6). Mean and standard error (standard deviation of the means, s) for each model complexity are computed, and the most parsimonious model whose mean prediction error is no more than one standard error above the minimum mean prediction error is chosen. Figure 4.4 (right) illustrates this procedure. The points are the mean prediction errors and the arrows indicate mean plus/minus one standard error. 

The rule of parsimony. The number of essentially different kinds of constituents in a crystal tends to be small. 

These principles are phrased in the language of the ionic model, but they provide a simpler and more explicit description of stable structures than that given by the ionic model s energy minimization principle. Among the important ideas captured by Pauling s rules are those of local charge neutrality, the definition of electrostatic bond strength, and the rule of parsimony which is closely... 

Perhaps the more important questions raised by Rule 5, as Burdett and McLarnan (1984) point out, concern the extent to which it really is borne out by observation. For example, Baur et al. (1983) have developed a numerical index for the degree of parsimony in a crystal structure and have shown that, using this measure, many crystal structures are not parsimonious but lavish in their use of different local environments. Also, the dominance of short-range forces is by no means obvious when ordered structures with extremely large unit cells are observed (e.g., a c dimension of 1500 A in some SiC polytypes Shaffer, 1969). The explanation of such structures poses problems for electrostatic as well as covalent models. 

The choice of the right model to use to describe experimental results is one of the trickiest, and most interesting, tasks in scientific work, and this is a subject that can only be touched on here. As discussed above, we are guided by the Principle of Parsimony, that in science one should seek the simplest explanation for phenomena. In the present context, that means that we should define models with as few parameters as possible, consistent with obtaining a satisfactory description of the data. This is a sensible approach, because if a simple model fits the data adequately, then so necessarily must more complicated versions of that model. It follows that experimental observations can only serve to rule out models, often, but not always, because they are oversimplified the data can never prove that a model is correct The question naturally arises at this stage about how one can establish whether or not a model is successful in accounting for the data. There are several criteria for assessing the quality of a model. 

The metabolic pattern of extinct ancient organisms can be predicted from the rule of parsimony (Appendix 10). In addition the existence of unique structures with very wide systematic occurrence (Appendix 11) can be explained by the concept of similar receptor structures in the different receptor families sharing affinity to certain chemical entities. 

For model selection, the prediction rule with the smallest CV error averaged over the V evaluation sets is labeled as the best model. Nonetheless, in practice a 1-SErule is used, which suggests selection of the most parsimonious model whose error is no more than one standard error above that of the best model s. 

In defining a feature system properly, one must formally define the attributes and the type of values they can take. So far we have described a somewhat traditional set of features (maimer, place etc) each of which can take a multiple values (e.g. dental, alveolar, or palatal) and have made use of descriptive devices, such as assimilation to describe how sounds change in particular environments. In phonology researchers attempt to improve on this, by defining features and rule mechanisms in an attempt to explain these sorts of effects in a more elegant and parsimonious manner. 

Rule 5. There are a limited number of different kinds of coordination environments for an ion in a crystal. This rule, also known as the rule of parsimony, suggests that there is beauty in simplicity. It would be highly unlikely, for instance, to observe a crystalline structure where some AP+ ions are tetrahedral and others are octahedral, even though the energy difference between CN =4 and CN = 6 is relatively small. 

In addition to its reliance on the principle of parsimony Boyle s approach assumes that the factitious character of such manuhictured substances would rule out the possibility that they had substantial forms on the assunqmon that only natural things can have substantial forms. This assumption was rendered problematic however, by the elasticity of the ardhcial-natuial dichotomy. For a discussion of this weakness in Boyle s argument, see Newman, Promethean Ambitions pp. 271-283. 

FIGURE 6.4 Fifty percent majority rule consensus tree of 2348 most parsimonious trees (L = 808, Cl = 54, RI = 79) of analysis of 86 terminals. Jackknife (in italics) and Bremer support values are shown below branches those clades that appear also in the strict consensus topology are shown with 100 above the branch. 

---