451 research outputs found
Quasilinear eigenvalues
In this work, we review and extend some well known results for the
eigenvalues of the Dirichlet Laplace operator to a more general class of
monotone quasilinear elliptic operators. As an application we obtain some
homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea
Information and Entropy
What is information? Is it physical? We argue that in a Bayesian theory the
notion of information must be defined in terms of its effects on the beliefs of
rational agents. Information is whatever constrains rational beliefs and
therefore it is the force that induces us to change our minds. This problem of
updating from a prior to a posterior probability distribution is tackled
through an eliminative induction process that singles out the logarithmic
relative entropy as the unique tool for inference. The resulting method of
Maximum relative Entropy (ME), which is designed for updating from arbitrary
priors given information in the form of arbitrary constraints, includes as
special cases both MaxEnt (which allows arbitrary constraints) and Bayes' rule
(which allows arbitrary priors). Thus, ME unifies the two themes of these
workshops -- the Maximum Entropy and the Bayesian methods -- into a single
general inference scheme that allows us to handle problems that lie beyond the
reach of either of the two methods separately. I conclude with a couple of
simple illustrative examples.Comment: Presented at MaxEnt 2007, the 27th International Workshop on Bayesian
Inference and Maximum Entropy Methods (July 8-13, 2007, Saratoga Springs, New
York, USA
Neural networks can detect model-free static arbitrage strategies
In this paper we demonstrate both theoretically as well as numerically that
neural networks can detect model-free static arbitrage opportunities whenever
the market admits some. Due to the use of neural networks, our method can be
applied to financial markets with a high number of traded securities and
ensures almost immediate execution of the corresponding trading strategies. To
demonstrate its tractability, effectiveness, and robustness we provide examples
using real financial data. From a technical point of view, we prove that a
single neural network can approximately solve a class of convex semi-infinite
programs, which is the key result in order to derive our theoretical results
that neural networks can detect model-free static arbitrage strategies whenever
the financial market admits such opportunities
From Information Geometry to Newtonian Dynamics
Newtonian dynamics is derived from prior information codified into an
appropriate statistical model. The basic assumption is that there is an
irreducible uncertainty in the location of particles so that the state of a
particle is defined by a probability distribution. The corresponding
configuration space is a statistical manifold the geometry of which is defined
by the information metric. The trajectory follows from a principle of
inference, the method of Maximum Entropy. No additional "physical" postulates
such as an equation of motion, or an action principle, nor the concepts of
momentum and of phase space, not even the notion of time, need to be
postulated. The resulting entropic dynamics reproduces the Newtonian dynamics
of any number of particles interacting among themselves and with external
fields. Both the mass of the particles and their interactions are explained as
a consequence of the underlying statistical manifold.Comment: Presented at MaxEnt 2007, the 27th International Workshop on Bayesian
Inference and Maximum Entropy Methods (July 8-13, 2007, Saratoga Springs, New
York, USA
Updating Probabilities with Data and Moments
We use the method of Maximum (relative) Entropy to process information in the
form of observed data and moment constraints. The generic "canonical" form of
the posterior distribution for the problem of simultaneous updating with data
and moments is obtained. We discuss the general problem of non-commuting
constraints, when they should be processed sequentially and when
simultaneously. As an illustration, the multinomial example of die tosses is
solved in detail for two superficially similar but actually very different
problems.Comment: Presented at the 27th International Workshop on Bayesian Inference
and Maximum Entropy Methods in Science and Engineering, Saratoga Springs, NY,
July 8-13, 2007. 10 pages, 1 figure V2 has a small typo in the end of the
appendix that was fixed. aj=mj+1 is now aj=m(k-j)+
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