3,817 research outputs found
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
Categorical Aspects of Parameter Learning
Parameter learning is the technique for obtaining the probabilistic
parameters in conditional probability tables in Bayesian networks from tables
with (observed) data --- where it is assumed that the underlying graphical
structure is known. There are basically two ways of doing so, referred to as
maximal likelihood estimation (MLE) and as Bayesian learning. This paper
provides a categorical analysis of these two techniques and describes them in
terms of basic properties of the multiset monad M, the distribution monad D and
the Giry monad G. In essence, learning is about the reltionships between
multisets (used for counting) on the one hand and probability distributions on
the other. These relationsips will be described as suitable natural
transformations
Facts, Values and Quanta
Quantum mechanics is a fundamentally probabilistic theory (at least so far as
the empirical predictions are concerned). It follows that, if one wants to
properly understand quantum mechanics, it is essential to clearly understand
the meaning of probability statements. The interpretation of probability has
excited nearly as much philosophical controversy as the interpretation of
quantum mechanics. 20th century physicists have mostly adopted a frequentist
conception. In this paper it is argued that we ought, instead, to adopt a
logical or Bayesian conception. The paper includes a comparison of the orthodox
and Bayesian theories of statistical inference. It concludes with a few remarks
concerning the implications for the concept of physical reality.Comment: 30 pages, AMS Late
Concerning Dice and Divinity
Einstein initially objected to the probabilistic aspect of quantum mechanics
- the idea that God is playing at dice. Later he changed his ground, and
focussed instead on the point that the Copenhagen Interpretation leads to what
Einstein saw as the abandonment of physical realism. We argue here that
Einstein's initial intuition was perfectly sound, and that it is precisely the
fact that quantum mechanics is a fundamentally probabilistic theory which is at
the root of all the controversies regarding its interpretation. Probability is
an intrinsically logical concept. This means that the quantum state has an
essentially logical significance. It is extremely difficult to reconcile that
fact with Einstein's belief, that it is the task of physics to give us a vision
of the world apprehended sub specie aeternitatis. Quantum mechanics thus
presents us with a simple choice: either to follow Einstein in looking for a
theory which is not probabilistic at the fundamental level, or else to accept
that physics does not in fact put us in the position of God looking down on
things from above. There is a widespread fear that the latter alternative must
inevitably lead to a greatly impoverished, positivistic view of physical
theory. It appears to us, however, that the truth is just the opposite. The
Einsteinian vision is much less attractive than it seems at first sight. In
particular, it is closely connected with philosophical reductionism.Comment: Contribution to proceedings of Foundations of Probability and
Physics, Vaxjo, 200
Disintegration and Bayesian Inversion via String Diagrams
The notions of disintegration and Bayesian inversion are fundamental in
conditional probability theory. They produce channels, as conditional
probabilities, from a joint state, or from an already given channel (in
opposite direction). These notions exist in the literature, in concrete
situations, but are presented here in abstract graphical formulations. The
resulting abstract descriptions are used for proving basic results in
conditional probability theory. The existence of disintegration and Bayesian
inversion is discussed for discrete probability, and also for measure-theoretic
probability --- via standard Borel spaces and via likelihoods. Finally, the
usefulness of disintegration and Bayesian inversion is illustrated in several
examples.Comment: Accepted for publication in Mathematical Structures in Computer
Scienc
- …