Conditioning

2021 ◽  
pp. 190-212
Author(s):  
James Davidson
Keyword(s):  
Conditional Expectation ◽  
Conditional Independence ◽  
Theoretic Approach ◽  
Conditional Distributions

This chapter deals in depth with the concept of conditional expectation. This is defined first in the traditional “naïve” manner, and then using the measure theoretic approach. A comprehensive set of properties of the conditional expectation are proved, generalizing several results of Ch. 9, and then multiple sub‐σ‎‎‐fields and nesting are considered, concluding with a treatment of conditional distributions and conditional independence.

scholarly journals On the conditional independence implication problem: A lattice-theoretic approach

2013 ◽  
Vol 202 ◽  
pp. 29-51 ◽  
Author(s):  
Mathias Niepert ◽  
Marc Gyssens ◽  
Bassem Sayrafi ◽  
Dirk Van Gucht
Keyword(s):  
Conditional Independence ◽  
Theoretic Approach ◽  
Implication Problem

scholarly journals A relation between positive dependence of signal and the variability of conditional expectation given signal

2006 ◽  
Vol 43 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Toshihide Mizuno
Keyword(s):  
Conditional Expectation ◽  
Marginal Distribution ◽  
Stochastic Order ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Distributions ◽  
Convex Order ◽  
Positive Dependence ◽  
Dispersive Order ◽  
Conditional Expectations

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

scholarly journals On the conditional distributions of spatial point processes

2011 ◽  
Vol 43 (2) ◽  
pp. 301-307 ◽  
Author(s):  
François Caron ◽  
Pierre Del Moral ◽  
Arnaud Doucet ◽  
Michele Pace
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Theoretic Approach ◽  
Conditional Distributions ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Observation Process ◽  
Original Analysis ◽  
Latent Process

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

scholarly journals On the conditional distributions of spatial point processes

2011 ◽  
Vol 43 (02) ◽  
pp. 301-307 ◽  
Author(s):  
François Caron ◽  
Pierre Del Moral ◽  
Arnaud Doucet ◽  
Michele Pace
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Theoretic Approach ◽  
Conditional Distributions ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Observation Process ◽  
Original Analysis ◽  
Latent Process

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

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