18,582 research outputs found
Representation of Functional Data in Neural Networks
Functional Data Analysis (FDA) is an extension of traditional data analysis
to functional data, for example spectra, temporal series, spatio-temporal
images, gesture recognition data, etc. Functional data are rarely known in
practice; usually a regular or irregular sampling is known. For this reason,
some processing is needed in order to benefit from the smooth character of
functional data in the analysis methods. This paper shows how to extend the
Radial-Basis Function Networks (RBFN) and Multi-Layer Perceptron (MLP) models
to functional data inputs, in particular when the latter are known through
lists of input-output pairs. Various possibilities for functional processing
are discussed, including the projection on smooth bases, Functional Principal
Component Analysis, functional centering and reduction, and the use of
differential operators. It is shown how to incorporate these functional
processing into the RBFN and MLP models. The functional approach is illustrated
on a benchmark of spectrometric data analysis.Comment: Also available online from:
http://www.sciencedirect.com/science/journal/0925231
Lower and Upper Conditioning in Quantum Bayesian Theory
Updating a probability distribution in the light of new evidence is a very
basic operation in Bayesian probability theory. It is also known as state
revision or simply as conditioning. This paper recalls how locally updating a
joint state can equivalently be described via inference using the channel
extracted from the state (via disintegration). This paper also investigates the
quantum analogues of conditioning, and in particular the analogues of this
equivalence between updating a joint state and inference. The main finding is
that in order to obtain a similar equivalence, we have to distinguish two forms
of quantum conditioning, which we call lower and upper conditioning. They are
known from the literature, but the common framework in which we describe them
and the equivalence result are new.Comment: In Proceedings QPL 2018, arXiv:1901.0947
Interference Effects in Quantum Belief Networks
Probabilistic graphical models such as Bayesian Networks are one of the most
powerful structures known by the Computer Science community for deriving
probabilistic inferences. However, modern cognitive psychology has revealed
that human decisions could not follow the rules of classical probability
theory, because humans cannot process large amounts of data in order to make
judgements. Consequently, the inferences performed are based on limited data
coupled with several heuristics, leading to violations of the law of total
probability. This means that probabilistic graphical models based on classical
probability theory are too limited to fully simulate and explain various
aspects of human decision making.
Quantum probability theory was developed in order to accommodate the
paradoxical findings that the classical theory could not explain. Recent
findings in cognitive psychology revealed that quantum probability can fully
describe human decisions in an elegant framework. Their findings suggest that,
before taking a decision, human thoughts are seen as superposed waves that can
interfere with each other, influencing the final decision.
In this work, we propose a new Bayesian Network based on the psychological
findings of cognitive scientists. We made experiments with two very well known
Bayesian Networks from the literature. The results obtained revealed that the
quantum like Bayesian Network can affect drastically the probabilistic
inferences, specially when the levels of uncertainty of the network are very
high (no pieces of evidence observed). When the levels of uncertainty are very
low, then the proposed quantum like network collapses to its classical
counterpart
Updating the Born rule
Despite the tremendous empirical success of quantum theory there is still
widespread disagreement about what it can tell us about the nature of the
world. A central question is whether the theory is about our knowledge of
reality, or a direct statement about reality itself. Regardless of their stance
on this question, current interpretations of quantum theory regard the Born
rule as fundamental and add an independent state-update (or "collapse") rule to
describe how quantum states change upon measurement. In this paper we present
an alternative perspective and derive a probability rule that subsumes both the
Born rule and the collapse rule. We show that this more fundamental probability
rule can provide a rigorous foundation for informational, or "knowledge-based",
interpretations of quantum theory.Comment: 6+2 pages; 3 figure
- …