This property lies at the root of the theory of Markov processes, which are characterized by the condition that past and future are conditionally independent, given the present. The notions of past, present, and future are somewhat elastic. One may have an extended present, a finite or infinite past or future. The mathematical patterns of past, present, and future provide important results on the "behavior" of Markov processes.
Definition . The pair
is
conditionally independent, givenZ , denoted
iff
An arbitrary class
of random vectors is conditionally independent, give
Z ,
iff such a product rule holds for each finite subclass or two or more members of the class.
Remark . The expression “for all Borel sets
,” here and elsewhere, implies the sets are
on the appropriate codomains. Also, the expressions below “for all Borel functions
g ,” etc.,
imply that the functions are real-valued, such that the indicated expectations are finite.
The following are equivalent . Each is necessary and sufficient that
.
****
For any Borel function
g , there exists a Borel function
e
g such that
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Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
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A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,