Editing Mathematical model (section)

===''A priori'' information===

[[File:Blackbox3D-withGraphs.svg|thumb|480px|To analyse something with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown) ''box''. The usual representation of this ''black box system'' is a [[data flow diagram]] centered in the box.]]

Mathematical modeling problems are often classified into [[black box]] or [[White box (software engineering)|white box]] models, according to how much [[a priori (philosophy)|a priori]] information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually, it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an [[exponential decay|exponentially decaying]] function, but we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.

In black-box models, one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are [[artificial neural network|neural networks]] which usually do not make assumptions about incoming data. Alternatively, the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of [[nonlinear system identification]]<ref name="SAB1">Billings S.A. (2013), ''Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains'', Wiley.</ref> can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

====Subjective information====

Sometimes it is useful to incorporate subjective information into a mathematical model.  This can be done based on [[Intuition (knowledge)|intuition]], [[experience]], or [[expert opinion]], or based on convenience of mathematical form. [[Bayesian statistics]] provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a [[prior probability distribution]] (which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads.  After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.