Bayesian Probabilities and the Histories Algebra

Abstract

We attempt a justification of a generalisation of the consistent histories programme using a notion of probability that is valid for all complete sets of history propositions. This consists of introducing Cox's axioms of probability theory and showing that our candidate notion of probability obeys them. We also give a generalisation of Bayes' theorem and comment upon how Bayesianism should be useful for the quantum gravity/cosmology programmes.Comment: 10 pages, accepted by Int. J. Theo. Phys. Feb 200

Bayesian analog of Gleason’s theorem

“…where αtm are Schrödinger picture operators. Our novel peg assignments, those that we suggested in [1], are…”

“…One tentative partial order might look something like Fig. (1). This partial order 2 has the added advantage as we are unable to relate p(⊳α|I) and p(α|I) using it (the partial ordering is symmetric in the real axis and complex conjugation represents time reversal).…”

“…Once we introduce further axioms that these pegs ought to obey-other than (3)-then we will be able to speculate what this partial order on the peg space might be. We proposed these pegs for the history algebra P(V) because they are additive for disjoint homogeneous history propositions [1]-thus these complex pegs seem to behave something like we expect probabilities should. Now we aim to show that we can derive these pegs from axioms analogous to Cox's axioms of probability theory applied to the HPO algebra using an analogue of Gleason's theorem.…”

“…In [1] we postulated a novel notion of probability by generalising Cox's axioms of probability [2,3] in a manner appropriate to quantum theory. In this paper we wish to go one step further; we will present a uniqueness proof analogous to Gleason's theorem for our postulated generalised probabilities.…”

“…We will be helped considerably by another analogue of Gleason's theorem in the literature [4] which is applied to the decoherence functional in the History Projection Operator (HPO) formulation of the consistent histories programme [5]. First we will review results previously discussed [1] and then we will outline the relevant Gleason-like theorem, its interpretation and, for completeness, its proof. We will then propose a generalised entropy.…”