Bayesian Approach and Poppers Falsificationism

In science and engineering an improvement of knowledge is derived from the comparison between theories and experiments. While theories give the general framework for human understanding, experiments represent absolutely necessary tests for establishing the required correspondence between theories and real world. In most cases, the properties of the world are tentatively described by one or more alternative theories that generally contain one or more parameters not exactly known, so that they may be considered to be adjustable. Thus, the expected results are the identification of the best available theory and the evaluation of the best estimates of the relevant adjustable parameters. 

When the significance and the reliability of the correspondence between theories and experiments are considered, two main alternatives are available. The Standard View, based on the ideas of logical empiricism, assumes that the experiments can confirm a scientific theory, i.e., that they can increase its probability (here intended as logical confidence in its truth, i.e., in its correspondence with the real world). On the contrary, Falsificationism, first proposed by Karl Popper [17], claims that experiments cannot demonstrate the truth of a theory but can only falsify the theory, i.e., demonstrate that a theory is unfit to describe an experimental result. 

In its simpler form, the idea of falsification can be derived from the laws of Logic in fact, if Th stands for theory and Ex for experimental result, the modus tollens gives 

When only two alternatives are possible (Th or —Th), it is possible to demonstrate a theory Th from the falsity of a consequence of —Th, as in the ex absurdo demonstrations used in mathematics. Unfortunately, things are not so simple in the physical world, where ex absurdo demonstrations are not acceptable since the alternatives to Th may be infinite (and actually not all known) therefore, we can never rely on a theory to be a definitive description of reality. 

Moreover, in the physical world, the observed experimental results are biased by the presence of experimental errors, so that the concept of truth must be intended in a less strong sense. Finally, a theory Th is not a simple statement but rather a complex set of hypotheses (in some case not all explicitly stated, but situated in the background of our knowledge). If the theory disagrees with experiments, which of the hypotheses is falsified by the experiment  

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Chapter 3 provides an introduction to the identification of mathematical models for reactive systems and an extensive review of the methods for estimating the relevant adjustable parameters. The chapter is initiated with a comparison between Bayesian approach and Poppers falsificationism. The aim is to establish a few fundamental ideas on the reliability of scientific knowledge, which is based on the comparison between alternative models and the experimental results, and is limited by the nonexhaustive nature of the available theories and by the unavoidable experimental errors. 

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