Structural hypotheses, mathematical model

Mathematical models of biological processes are often used for hypothesis testing and process optimization. Using physical interpretation of results to obtain greater insight into process behavior is only possible when structured models that consider several parts of the system separately are employed. A number of dynamic mathematical models for cell growth and metabolite pro-... 

Usually, a mathematical model simulates a process behavior, in what can be termed a forward problem. The inverse problem is, given the experimental measurements of behavior, what is the structure A difficult problem, but an important one for the sciences. The inverse problem may be partitioned into the following stages hypothesis formulation, i.e., model specification, definition of the experiments, identifiability, parameter estimation, experiment, and analysis and model checking. Typically, from measured data, nonparametric indices are evaluated in order to reveal the basic features and mechanisms of the underlying processes. Then, based on this information, several structures are assayed for candidate parametric models. Nevertheless, in this book we look only into various aspects of the forward problem given the structure and the parameter values, how does the system behave ... 

Calculating Possibilities. Consider the sequence-structure relation in Figure 2. A uniform probability of is assigned to each sequence. Note that the uniform distribution is not a requirement of the mathematical model. It was assumed that all sequences could adopt the same conformation. The possibility for each structural hypothesis to contain the conformation was computed using Zadeh s theory of possibility. 

In the history of mathematics, uncertainty was approached in the XVlP century by Pascal and Fermat who introduced the notion of probability. However, probabilities do not allow one to process subjective beliefs nor imprecise or vague knowledge, such as in computer modeling of three-dimensional structure. Subjectivity and imprecision were only considered from 1965, when Zadeh, known for his work in systems theory, introduced the notion of fuzzy set. The concept of fuzziness introduces partial membership to classes, admitting intermediary situations between no and full membership. Zadeh s theory of possibility, introduced in 1977, constitutes a framework allowing for the representation of such uncertain concepts of non-probabilistic nature (9). The concept of fuzzy set allows one to consider imprecision and uncertainty in a single formalism and to quantitatively measure the preference of one hypothesis versus another. Note, however, that Bayesian probabilities could have been used instead. 

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