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Further, it is important to, at the least, ensure that both of these are "searchable" as some users may be familiar with one term but not the other, though they are related and both potentially useful to the same person, e.g.. I would suggest leaving as is and referring to the other in each via
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Whether the proof is just marginalization depends on what one calls "the
Chapman-Kolmogorov Equation". In "Handbook Of Stochastic Methods" by C.W. Gardiner, second edition, pages 43-44, marginalization is a proof for the equation:
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1419:) where the definition is clear with the proof on the same page. I'm not sure if it corresponds with what is written in this article, because I wasn't able to grasp the definition here.
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it is important to trace back the origin of this "marginalised" view of the CK equation. I personally like this general form but I am not sure it is considered as a CK equation.
1076:{\displaystyle p(x_{1},t_{1}\ |\ x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1};x_{2},t_{2}|x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1}|x_{2},t_{2};x_{3},t_{3})\ p(x_{2},t_{2}|x_{3},t_{3})}
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The two terms are not the same: In the theory of
Continuous-Time Markov Chains (CTMC) what is described under "master equation" is called the "balance equation".
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Does it start with a definition of a conditional probability? Then it requires the law of total probability and we are done? What else is needed?
711:{\displaystyle p_{i_{1},\ldots ,i_{n-1}}(f_{1},\ldots ,f_{n-1})=\int _{-\infty }^{\infty }p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})\,df_{n}}
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So
Gardiner's definition of the Chapman-Kolmogorov equation is more restrictive than the definition given in the current article.
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An example for a non-Markovian process where this equation is not merely the law of total probability would also be clarifying.
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saying "Form of CK equation needed for
Kolmogorov construction is more general than CK for Markov processes"
1379:{\displaystyle p(x_{1},t_{1}|x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1}|x_{2},t_{2})p(x_{2},t_{2}|x_{3},t_{3})}
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Gardiner says "This equation is also always valid. We now introduce the Markov assumption. If
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Moreover, references should be provided for this equation which has been introduce by @
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However, Gardiner does not call that equation "the
Chapman-Kolmogorov Equation".
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Does someone know a reference to
Chapman's or Kolmogorov's work w.r.t. this?
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What exactly is the form for the proof of the
Chapman-Kolmogorov equation?
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499:{\displaystyle p(x_{1},t_{1})=\int dx_{2}\ p(x_{1},t_{1};x_{2},t_{2})}
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which corresponds to the equation in the current article given by:
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I am going to check original works of
Chapman and Kolmogorov.
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dependence in the double conditioned probability and write
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All the proofs I can find assume some Markovian process.
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40:WikiProjects
319:—Preceding
201:Mathematics
192:mathematics
148:Mathematics
30:Start-class
1481:Categories
96:Statistics
87:statistics
59:Statistics
268:Adoniscik
260:The Anome
1423:unsigned
321:unsigned
249:guessing
293:In the
228:on the
123:on the
1465:ReHoss
36:scale.
1449:and @
339:Proof
1469:talk
1431:talk
1398:talk
366:talk
329:talk
247:I'm
258:--
220:Low
115:Low
1483::
1471:)
1433:)
1400:)
1239:∫
1116:≥
1103:≥
908:∫
801:∫
675:…
641:…
618:∞
613:∞
610:−
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266:--
255:?
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428:x
424:d
418:=
415:)
410:1
406:t
402:,
397:1
393:x
389:(
386:p
364:(
327:(
305:s
232:.
127:.
42::
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