As noticed in the abstract, an interactive domain is each research domain in
which the observed system is intrinsically influenced by the observation
itself. Important examples of these domains are quantum mechanics3,
the study of psychological decision processes 4 and the analysis of
ultra-weak signals 5, for instance in tomography and remote-sensing.
As shown in 4 ,these research domains cannot be investigated by
means of classical statistical procedures because these methods do not model an
interaction between the observer and the system under investigation. Thus, the
aim of this project is the construction of a statistical method that can handle
this kind of situation and a corresponding analysis to compare the practical
behaviour of interactive systems. As a working hypothesis, that can be
immediately put in a mathematical procedure, we consider the construction of a
more general statistics by supposing that we have an incomplete knowledge of
the way we collect the information, or, otherwise, of the interaction between
the observer and the observed. This is in sharp contrast with the usual
statistical hypothesis of an incomplete knowledge of the system we consider.
The above hypothesis has been studied in the two-dimensional case (that is,
questions with only two possible answers) with very interesting
results4. The only knowledge we assume to posses is the magnitude of
the influence of the observation on the observed system. We have called this
magnitude of the interaction epsilon and the accompanying model the
epsilon-model. Epsilon can vary between 0 and 1: in the case of epsilon equals
one we have a maximal interaction; for epsilon equal to 0 there is no
interaction. It seems now that the case epsilon=1 is isomorphic to a quantum
mechanical experiment on a two-dimensional Hilbert space and that the case
epsilon=0 agrees with classical (or kolmogorovian) statistics. For the other
values of epsilon we found a new kind of statistics not isomorphic with either
of the two former cases. In other words, this is the first statistics mentioned
in literature that describes the continuous transition of a microscopical
(quantum mechanics)
to a macroscopical (classical mechanics) situation. However, it follows from
our hypothesis that this statistics in the case of epsilon=1 is not only
appropriate for the microscopical world, but in the region of social sciences
too. To check this idea one has to extend the exising model to a more
dimensional model. Mathematically, this means that the computations for the
two-dimensional case should be extended to the more dimensional model. The idea
is sufficiently precise to be implemented in a relative short time (9 to 15
months) on a personal computer. Because explicit expressions are easier to
manipulate, we want to investigate which results can be analytically solved.
The results we obtain will be tested in three ways:
1) Concrete experimental situations.
2) The two limit cases (of maximal and minimal interaction) should agree with a
quantum mechanical description and a classical description respectively.
3) Results should be consistent with the results for the two-dimensional
model.
The above hypothesis allows us to reinterprete the origin of probabilities
(ontological or epistemological) in quantum mechanics. Although many
researchers have the feeling that they have an epistemological origin, it is
generally accepted they are of an ontological nature. The reason for this
discord was that it seemed impossible to construct a statistical theory
consistent with quantum mechanics from the hypothesis of a lack of knowledge on
the observed entities. A group of six researchers (Dirk Aerts, Thomas Durt, Bob
Coecke, Frank Valckenborgh, Sven Aerts, Bart D'Hooghe) investigates the
possibility to find back the quantum mechanical probabilities by replacing the
hypothesis just above by the aforementioned working hypothesis. If the results
of this research continues as to-day, the philosophical background of quantum
theory should be reconsidered.
In psychological research one regularly works with questionnaires where the
participants have to choose between several options. If the choice of the test
person is not contained within the different options (maybe he thinks the
question to be badly posed) he'll have to make a choice at the very moment
itself. It is clear that his choice will have a contextual component: who asks
this question, why, in what circumstances, ... The fluctuations that enter the
measurement in this way are due to the measurement process (and not of the
people that are questioned) and as such they have their moment of action during
the measurement. In other words: one and the same person can can give a
different answer to same question merely because the context is different. It
is exactly this kind of fluctuations we wish to model because it are these
fluctuations that give rise to a statistical structure that is much closer to
quantum statistics than to classical statistics. In a further phase of the
project we want to apply this approach to complex problems that involve
correlations between different measurements, because it is known from quantum
theory that it is in these situations that one encounters the most profound
differences between the two types of statistics. More specifically, we wish to
consider the applicability of this approach in the item-response theory and
multi-dimensional scaling models[6] . It will be clear that if the introduction of
quantum statistics in the field of psychology is succesful then this would
constitute a promising new methodology.
In a review article by Malley and Hornstein [7] it is shown that the use of quantum statistical
methods in the area of weak-signal analysis leads to a risc-curve that is lower
everywhere than the usual Bayes-inference risc-curve.
The method they use (named quantum Statistical Inference) was developed by
engineers who wanted to use the advantages of LASER in communicationsystems.
The fluctuations in the workings of the LASER (whoes basic principles are
dictated by quantum rules) demanded of a new statistical analysing method,
capable of depicting transmission efficiency. The same methods are now applied
to as diverse fields as tomography, optical-imaging, remote-sensing,... The
use of these new methods has lead to an improvement of the signal-noise ratio
by as much as 20dB. Because these are all areas where one formerly used to
apply classical statistics, our conjecture is that these areas do not posses a
maximal fluctuations structure for the measurement, that is, we believe that
the more appropriate statistical model for these situations is neither a
quantum mechanical nor a classical one but an intermediate one. If this is the
case then it must be possible to improve the signal-noise ratio even further by
using the probabilities of the epsilon-model.
This project also has the following two connections with the international
project Principia Cybernetica ( VUB representants are F. Heylighen and J.
Bollen).
1) From cybernetics it is well-known that a subsystem never has complete
knowledge of the system it belongs to. This is because there is a feedback
(interaction) between the two systems.
2) The evolution of a knowledge system is not only modelled in Principia
Cybernetica: an adaptive semantical network has been implemented on the
Internet. The evolution of this semantical network is being examined. This
should be accompanied by an analysis. An important question that arises is the
following: to what extent is the self-organising process that is now going on
on the Internet influenced by the initial choices we gave the users of the
Internet? This question can only be answered in a framework that takes into
account the interaction.
As such, this project is a test for the development of the semantical network
as much as it is an experimental test for the statistical techniques here
proposed.
In the first year of the project the two-dimensional model will be elaborated
to the three-dimensional case. This has the advantage that one doesn't need yet
to overcome the full complexity of the n-dimensional case while it can be shown
that the three-dimensional case already displays all the particularities of the
n-dimensional one. The second project year will be dedicated to the analysis of
the data from experimental set-ups where interactivity plays an important
role.
Computer analyses will be made of this data taking into account the working
hypothesis. A start for the n-dimensional case will be made. Parameters that
are evidently important in the experiments will play a role in the development
of the n-dimensional case. In the third year of the project we will consider
how the statistical techniques can be adapted to the more dimensional
epsilon-model and to what extent this constitutes an improvement to the
techniques that are being used now. The last year serves as an evaluation year
where a case-study shall be made.