A Systematic Theory of Exponential Families of Probability Distributions

1966 ◽  
Vol 11 (3) ◽  
pp. 425-435 ◽  
Author(s):  
N. N. Chentsov
Keyword(s):  
Probability Distributions ◽  
Exponential Families ◽  
Systematic Theory

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

scholarly journals Divergence Functions in Dually Flat Spaces and Their Properties

2018 ◽  
Author(s):  
Tomohiro Nishiyama
Keyword(s):  
Probability Distributions ◽  
Flat Space ◽  
Exponential Families ◽  
Geometrical Approach ◽  
Bhattacharyya Distance ◽  
Dual Affine ◽  
The Law ◽  
Law Of Cosines ◽  
Dually Flat Space ◽  
Jensen Shannon Divergence

In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual affine coordinate systems and strictly convex functions called potential and a canonical divergence is naturally introduced as a function of the affine coordinates and potentials. The canonical divergence satisfies a relational expression called triangular relation. This can be regarded as a generalization of the law of cosines in Euclidean space.In this paper, we newly introduce two kinds of divergences. The first divergence is a function of affine coordinates and it is consistent with the Jeffreys divergence for exponential or mixture families. For this divergence, we show that more relational equations and theorems similar to Euclidean space hold in addition to the law of cosines. The second divergences are functions of potentials and they are consistent with the Bhattacharyya distance for exponential families and are consistent with the Jensen-Shannon divergence for mixture families respectively. We derive an inequality between the the first and the second divergences and show that the inequality is a generalization of Lin's inequality.

Coordinate systems and Taylor series in statistics

1988 ◽  
Vol 415 (1849) ◽  
pp. 445-452 ◽  
Keyword(s):  
Taylor Series ◽  
Probability Distributions ◽  
Geometric Theory ◽  
Exponential Families ◽  
Natural Coordinates ◽  
Coordinate Systems ◽  
Derivatives Of

A geometric theory of Taylor series is presented and shown to reduce to the choice of a set of coordinates at each point of a family of probability distributions, the so-called coordinate string. Any family admits a naturally defined coordinate string which generalizes the natural coordinates of exponential families. This concept gives a coordinate-free description of the Möbius derivatives of McCullagh & Cox and the connection strings of Barndorff-Nielsen.

Sample mean based index policies by O(log n) regret for the multi-armed bandit problem

1995 ◽  
Vol 27 (04) ◽  
pp. 1054-1078 ◽  
Author(s):  
Rajeev Agrawal
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Probability Distributions ◽  
Infinite Horizon ◽  
Exponential Families ◽  
Contraction Principle ◽  
Bandit Problem ◽  
Sample Mean ◽  
Seminal Work ◽  
Index Policies

We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases slowly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(log n) lower bound on the regret with a constant that depends on the Kullback–Leibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(log n) regret with a constant that is also based on the Kullback–Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a ‘contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations.

Sample mean based index policies by O(log n) regret for the multi-armed bandit problem

1995 ◽  
Vol 27 (4) ◽  
pp. 1054-1078 ◽  
Author(s):  
Rajeev Agrawal
Keyword(s):  
Lower Bound ◽  
Large Deviations ◽  
Probability Distributions ◽  
Infinite Horizon ◽  
Exponential Families ◽  
Contraction Principle ◽  
Bandit Problem ◽  
Sample Mean ◽  
Seminal Work ◽  
Index Policies

We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases slowly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(log n) lower bound on the regret with a constant that depends on the Kullback–Leibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(log n) regret with a constant that is also based on the Kullback–Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a ‘contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations.

Asteroid and Comet Impact Speeds upon Mars: Significance for Panspermia and the Supply of Organics

1997 ◽  
Vol 161 ◽  
pp. 197-201 ◽  
Author(s):  
Duncan Steel
Keyword(s):  
Probability Distributions ◽  
Regions Of Interest ◽  
Significant Fraction ◽  
Organic Chemicals ◽  
Impact Speed ◽  
The Mean ◽  
The Impact

AbstractWhilst lithopanspermia depends upon massive impacts occurring at a speed above some limit, the intact delivery of organic chemicals or other volatiles to a planet requires the impact speed to be below some other limit such that a significant fraction of that material escapes destruction. Thus the two opposite ends of the impact speed distributions are the regions of interest in the bioastronomical context, whereas much modelling work on impacts delivers, or makes use of, only the mean speed. Here the probability distributions of impact speeds upon Mars are calculated for (i) the orbital distribution of known asteroids; and (ii) the expected distribution of near-parabolic cometary orbits. It is found that cometary impacts are far more likely to eject rocks from Mars (over 99 percent of the cometary impacts are at speeds above 20 km/sec, but at most 5 percent of the asteroidal impacts); paradoxically, the objects impacting at speeds low enough to make organic/volatile survival possible (the asteroids) are those which are depleted in such species.

A Systematic Theory of Argumentation

2003 ◽  
Author(s):  
Frans H. van Eemeren ◽  
Rob Grootendorst
Keyword(s):  
Systematic Theory

An Accelerated Cue Combination Principle Accounts for Multi-cue Depth Perception

2020 ◽  
Vol 3 (1) ◽  
pp. 10501-1-10501-9
Author(s):  
Christopher W. Tyler
Keyword(s):  
Belief Propagation ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Fusion Model ◽  
Cue Combination ◽  
Depth Estimate ◽  
Appropriate Behavior ◽  
Depth Cues ◽  
Bayesian Averaging

Abstract For the visual world in which we operate, the core issue is to conceptualize how its three-dimensional structure is encoded through the neural computation of multiple depth cues and their integration to a unitary depth structure. One approach to this issue is the full Bayesian model of scene understanding, but this is shown to require selection from the implausibly large number of possible scenes. An alternative approach is to propagate the implied depth structure solution for the scene through the “belief propagation” algorithm on general probability distributions. However, a more efficient model of local slant propagation is developed as an alternative.The overall depth percept must be derived from the combination of all available depth cues, but a simple linear summation rule across, say, a dozen different depth cues, would massively overestimate the perceived depth in the scene in cases where each cue alone provides a close-to-veridical depth estimate. On the other hand, a Bayesian averaging or “modified weak fusion” model for depth cue combination does not provide for the observed enhancement of perceived depth from weak depth cues. Thus, the current models do not account for the empirical properties of perceived depth from multiple depth cues.The present analysis shows that these problems can be addressed by an asymptotic, or hyperbolic Minkowski, approach to cue combination. With appropriate parameters, this first-order rule gives strong summation for a few depth cues, but the effect of an increasing number of cues beyond that remains too weak to account for the available degree of perceived depth magnitude. Finally, an accelerated asymptotic rule is proposed to match the empirical strength of perceived depth as measured, with appropriate behavior for any number of depth cues.

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