100,679 research outputs found
Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum Probability
We develop a systematic approach to quantum probability as a theory of
rational betting in quantum gambles. In these games of chance the agent is
betting in advance on the outcomes of several (finitely many) incompatible
measurements. One of the measurements is subsequently chosen and performed and
the money placed on the other measurements is returned to the agent. We show
how the rules of rational betting imply all the interesting features of quantum
probability, even in such finite gambles. These include the uncertainty
principle and the violation of Bell's inequality among others. Quantum gambles
are closely related to quantum logic and provide a new semantics to it. We
conclude with a philosophical discussion on the interpretation of quantum
mechanics.Comment: 21 pages, 2 figure
Conjugate Projective Limits
We characterize conjugate nonparametric Bayesian models as projective limits
of conjugate, finite-dimensional Bayesian models. In particular, we identify a
large class of nonparametric models representable as infinite-dimensional
analogues of exponential family distributions and their canonical conjugate
priors. This class contains most models studied in the literature, including
Dirichlet processes and Gaussian process regression models. To derive these
results, we introduce a representation of infinite-dimensional Bayesian models
by projective limits of regular conditional probabilities. We show under which
conditions the nonparametric model itself, its sufficient statistics, and -- if
they exist -- conjugate updates of the posterior are projective limits of their
respective finite-dimensional counterparts. We illustrate our results both by
application to existing nonparametric models and by construction of a model on
infinite permutations.Comment: 49 pages; improved version: revised proof of theorem 3 (results
unchanged), discussion added, exposition revise
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