A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMC

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.

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PROBABILITY DISTRIBUTIONS RELATED TO THE LAW OF THE ITERATED LOGARITHM

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.62.1.11 ◽

1969 ◽

Vol 62 (1) ◽

pp. 11-13 ◽

Cited By ~ 3

Author(s):

H. Robbins ◽

D. Siegmund

Keyword(s):

Probability Distributions ◽

Iterated Logarithm ◽

The Law

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

International Astronomical Union Colloquium ◽

10.1017/s0252921100014718 ◽

1997 ◽

Vol 161 ◽

pp. 197-201 ◽

Cited By ~ 1

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.

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Ethical Challenges: Less About Moral Wrongdoing and More About Communication Breakdown

Perspectives on Gerontology ◽

10.1044/gero20.3.72 ◽

2015 ◽

Vol 20 (3) ◽

pp. 72-84 ◽

Cited By ~ 2

Author(s):

Paula Leslie ◽

Mary Casper

Keyword(s):

Ethical Challenges ◽

Speech Language Pathology ◽

Thickened Liquids ◽

Communication Breakdown ◽

Professional Duty ◽

Language Pathology ◽

The Law ◽

Personal Case

“My patient refuses thickened liquids, should I discharge them from my caseload?” A version of this question appears at least weekly on the American Speech-Language-Hearing Association's Community pages. People talk of respecting the patient's right to be non-compliant with speech-language pathology recommendations. We challenge use of the word “respect” and calling a patient “non-compliant” in the same sentence: does use of the latter term preclude the former? In this article we will share our reflections on why we are interested in these so called “ethical challenges” from a personal case level to what our professional duty requires of us. Our proposal is that the problems that we encounter are less to do with ethical or moral puzzles and usually due to inadequate communication. We will outline resources that clinicians may use to support their work from what seems to be a straightforward case to those that are mired in complexity. And we will tackle fears and facts regarding litigation and the law.

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