Talk:Chapman–Kolmogorov equation

I'm guessing that the Chapman in the title might be Sydney Chapman? [1] -- The Anome 19:52, Jan 26, 2005 (UTC) Amended, from http://members.aol.com/jeff570/c.html --Adoniscik 17:17, 3 November 2005 (UTC)

Merge with master equation

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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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 mention and/or a link.

Undefined variables

In the Application to Markov chains section, the variable $s$ is not defined anywhere. —Preceding unsigned comment added by 46.115.120.113 (talk) 16:18, 6 June 2010 (UTC)

Proof

What exactly is the form for the proof of the Chapman-Kolmogorov equation?

Isn't it just marginalization?

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?

All the proofs I can find assume some Markovian process.

Does someone know a reference to Chapman's or Kolmogorov's work w.r.t. this?

An example for a non-Markovian process where this equation is not merely the law of total probability would also be clarifying.

Anne van Rossum (talk) 12:56, 21 December 2014 (UTC)

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:

$p(x_{1},t_{1})=\int dx_{2}\ p(x_{1},t_{1};x_{2},t_{2})$

which corresponds to the equation in the current article given by:

$p_{i_{1},\ldots ,i_{n-1}}(f_{1},\ldots ,f_{n-1})=\int _{-\infty }^{\infty }p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})\,df_{n}$

However, Gardiner does not call that equation "the Chapman-Kolmogorov Equation".

Marginalization also proves the equation

$p(x_{1},t_{1}\ |\ x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1};x_{2},t_{2}|x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1}|x_{2},t_{2};x_{3},t_{3})\ p(x_{2},t_{2}|x_{3},t_{3})$

Gardiner says "This equation is also always valid. We now introduce the Markov assumption. If $t_{1}\geq t_{2}\geq t_{3}$ we can drop the $t_{3}$ dependence in the double conditioned probability and write

$p(x_{1},t_{1}|x_{3},t_{3})=\int dx_{2}\ p(x_{1},t_{1}|x_{2},t_{2})p(x_{2},t_{2}|x_{3},t_{3})$ which is the Chapman-Kolmogorov equation."

So Gardiner's definition of the Chapman-Kolmogorov equation is more restrictive than the definition given in the current article.

Tashiro~enwiki (talk) 09:31, 29 November 2015 (UTC)

Easier definition

I've added an item to Further reading (Introduction to Probability Models) 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. — Preceding unsigned comment added by 2A00:1028:83D4:42DE:225:22FF:FEF6:293 (talk) 14:57, 20 March 2016 (UTC)

References for this general formulation

Indeed, as pointed out by @Tashiro~enwiki and @Anne van Rossum 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.

I am going to check original works of Chapman and Kolmogorov.

Moreover, references should be provided for this equation which has been introduce by @Miguel~enwiki saying "Form of CK equation needed for Kolmogorov construction is more general than CK for Markov processes" ReHoss (talk) 13:12, 1 October 2024 (UTC)

So after investigation, here is my feedback:

I checked in the two seminal papers of Chapman and Kolmogorov:

- S. Chapman - On the Brownian Displacements and Thermal Diffusion of Grains Suspended in a Non-Uniform Fluid (1928)

- A. Kolmogorov - "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On the Analytical Methods in the Theory of Probability) (1931)

In those two papers, the "marginalised version" is not given. Instead, the standard Markovian integral form involving two continuous time kernels is given.

Moreover, I also checked in the thesis directed by M. Frechet:

- Étude de l’équation fonctionnelle de Chapman-Kolmogoroff dans le cas d’un domaine d’intégration illimité à une dimension - Thèses de l’entre-deux-guerres, 1936

that the CK equation is also considered as the standard integral form involving two continuous time Markov kernels. This thesis may reveal what was considered as a CK equation back then.

Consequently, I think the equation only involving $i_{3}$ and $i_{1}$ in the Application to time-dilated Markov chains should be given as definition!

Best,2A01:CB06:B802:DC0F:AF75:E89E:A3A:2A66 (talk) 13:20, 4 October 2024 (UTC)