Maximum entropy, fluctuations and priors

The method of maximum entropy (ME) is extended to address the following problem: Once one accepts that the ME distribution is to be preferred over all others, the question is to what extent are distributions with lower entropy supposed to be ruled out. Two applications are given. The first is to the theory of thermodynamic fluctuations. The formulation is exact, covariant under changes of coordinates, and allows fluctuations of both the extensive and the conjugate intensive variables. The second application is to the construction of an objective prior for Bayesian inference. The prior obtained by following the ME method to its inevitable conclusion turns out to be a special case of what are currently known under the name of entropic priors.Comment: presented at MaxEnt 2000, the 20th International Workshop on Bayesian Inference and Maximum Entropy Methods (July 8-13, Gif-sur-Yvette, France)

Geometry from Information Geometry

We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry -- the geometry up to local changes of scale. Remarkably, this is precisely what is needed to model "physical" space in general relativity.Comment: Presented at MaxEnt 2015, the 35th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (July 19-24, 2015, Potsdam NY, USA

Change, time and information geometry

Dynamics, the study of change, is normally the subject of mechanics. Whether the chosen mechanics is ``fundamental'' and deterministic or ``phenomenological'' and stochastic, all changes are described relative to an external time. Here we show that once we define what we are talking about, namely, the system, its states and a criterion to distinguish among them, there is a single, unique, and natural dynamical law for irreversible processes that is compatible with the principle of maximum entropy. In this alternative dynamics changes are described relative to an internal, ``intrinsic'' time which is a derived, statistical concept defined and measured by change itself. Time is quantified change.Comment: Presented at MaxEnt 2000, the 20th International Workshop on Bayesian Inference and Maximum Entropy Methods (July 8-13, 2000, Gif-sur-Yvette, France

Entropic Dynamics

Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In this paper we review three examples of entropic dynamics. First we tackle the simpler case of a standard diffusion process which allows us to address the central issue of the nature of time. Then we show that imposing the additional constraint that the dynamics be non-dissipative leads to Hamiltonian dynamics. Finally, considerations from information geometry naturally lead to the type of Hamiltonian that describes quantum theory.Comment: Invited contribution to the Entropy special volume on Dynamical Equations and Causal Structures from Observation

Towards a Statistical Geometrodynamics

Can the spatial distance between two identical particles be explained in terms of the extent that one can be distinguished from the other? Is the geometry of space a macroscopic manifestation of an underlying microscopic statistical structure? Is geometrodynamics derivable from general principles of inductive inference? Tentative answers are suggested by a model of geometrodynamics based on the statistical concepts of entropy, information geometry, and entropic dynamics.Comment: Invited talk at the Decoherence, Information, Entropy, and Complexity Workshop, DICE02, September 2000, Piombino, Ital

Entropic Dynamics

I explore the possibility that the laws of physics might be laws of inference rather than laws of nature. What sort of dynamics can one derive from well-established rules of inference? Specifically, I ask: Given relevant information codified in the initial and the final states, what trajectory is the system expected to follow? The answer follows from a principle of inference, the principle of maximum entropy, and not from a principle of physics. The entropic dynamics derived this way exhibits some remarkable formal similarities with other generally covariant theories such as general relativity.Comment: Presented at MaxEnt 2001, the 21th International Workshop on Bayesian Inference and Maximum Entropy Methods (August 4-9, 2001, Baltimore, MD, USA

From Objective Amplitudes to Bayesian Probabilities

We review the Consistent Amplitude approach to Quantum Theory and argue that quantum probabilities are explicitly Bayesian. In this approach amplitudes are tools for inference. They codify objective information about how complicated experimental setups are put together from simpler ones. Thus, probabilities may be partially subjective but the amplitudes are not.Comment: 10 pages, 2 figures. Invited paper presented at the International Conference on Foundations of Probability and Physics - 4 (Vaxjo University, Sweden, 2006). The various versions reflect my attempts to include the figures in the main body of the pape

The Entropic Dynamics of Relativistic Quantum Fields

The formulation of quantum mechanics within the framework of entropic dynamics is extended to the domain of relativistic quantum fields. The result is a non-dissipative relativistic diffusion in the infinite dimensional space of field configurations. On extending the notion of entropic time to the relativistic regime we find that the field fluctuations provide the clock that sets the scale of duration. We also find that the usual divergences that affect all quantum field theories do not refer to the real values of physical quantities but rather to epistemic quantities invariably associated to unphysical probability distributions such as variances and other measures of uncertainty.Comment: 10 pages. Presented at MaxEnt 2012, The 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, (July 15-20, 2012, Garching, Germany