Dependence in PI is More Optimal Than Independence and Applications to Brownian Motion and Poisson Processes

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/07/04 

Date: Sat, 7 Aug 2004 18:04:07 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE

Copyright by Owner Osher Doctorow Ph.D.
Dependence in PI is More Optimal Than Independence and Applications
to Brownian Motion and Poisson Processes
First published 2004

 From the inequality:

1) 1 + P(A)P(B) - P(A) < = 1 + P(AB) - P(A) for positive quadrant
                           dependence

it follows that dependence in PI is more optimal than statistical
independence in PI. Since statistical independence in PI is more
optimal than in BCP ((Bayesian) Conditional Probability-Statistics)
and IPS (Independent Probability-Statistics) from previous postings
of mine, the optimality extends "across the board".

Example. Brownian Motion/Wiener Processes and Poisson processes
are respectively defined among other conditions by:

2) W(t) - W(s) ~ N(0, o^2(t-s)) for s < = t and {W(ti) - W(ti-1)}
are independent, ti nondecreasing in time t, i = 1 to n

3) X(t) - X(s) ~ Poisson(lambda(t-s)) 0 < = s < = t and {X(ti) -
X(ti-1)} are independent analogously to (2) above

where ~ means "is distributed as" and o is "sigma" ("standard
deviation").

The presence of differences of random variables in both (2) and (3),
or "increments" such as W(ti) - W(ti-1), suggests that PI is
involved. If we "standardize" these increments and truncate them
into [0, 1], normality is not much affected and independence is
retained, but then let's change their independence to positive
quadrant dependence, that is to say P(AB) > = P(A)P(B) or
F(x,y) > = FX(x)FY(y) or f(x,y) > = fX(x)fY(y). We would expect
optimality.

Osher Doctorow
                           

Relevant Pages

  • Re: Statistical Dependence in PI vs Conditional Probability/Bayesian
    ... >>From Osher Doctorow mdoctorow@xxxxxxxxxxx ... "Independence" in probability-statistics is ... not measured on a continuous scale but on an "all or nothing" scale, ... conditional/Bayesian probability-statistics. ...
    (sci.physics)
  • Re: Small Markov models problem
    ... Ross Clement (Email address invalid - do not use) wrote: ... is calculated with an independence assumption between two variables ... degrees of dependence between the two variables, ... probabilities leaves the extra step of having to fix-up the marginal ...
    (sci.stat.math)
  • Re: FFT test with few kbits
    ... >> Italian mathematician to hear his thought. ... (The standard english mathematical term is "necessary" condition) ... > in other word, increasing n, the independence is stronger. ... dependence between two Fourier components becomes ...
    (sci.crypt)
  • Re: Independence Further Examined in PI
    ... especially the case for "independence" in statistics, ... "dependence" appears with qualifiers in statistics are almost ... negative quadrant dependence is also of considerable interest. ...
    (sci.stat.math)
  • Re: Independent random variables versus non correlated variables
    ... > So this means that in terms of independence of events, ... For each combination of two values x_j and y_k, you can assess the ... any statement about the dependence and independence of two variables ... you could employ a Multinomial model. ...
    (sci.stat.math)