What Is the Integral Formulation of the Chapman-Kolmogorov Formula?

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The discussion focuses on the Chapman-Kolmogorov formula in the context of continuous probability and its integral formulation. The formula expresses the probability of transitioning from point A to point B through an intermediate point C, highlighting the relationship between these probabilities. The integral formulation is presented as an equation involving conditional probabilities and integration over the state space. Participants seek clarification on how to derive this integral or differential form, especially when all probability distributions are known. Understanding this formulation is crucial for applications in fields like quantum mechanics and Bayesian tracking.
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Hello..where i could find information about the Chapmann-Kolmogorov formula for continuous probability..i have hear something when taking a course of QM...something about this...if you want to go from A point to B point with a certain probability crossing a point C then:

P(A,B)=P(A,C)P(B,C)

My question is what is the Integral or differential formulation of this law?..considering we know all the probability distributions..thanks.
 
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the Chapman-Kolmogorov equation
p(\mathbf{x}_{k}|\mathbf{z}_{1:k-1})= \int<br /> p(\mathbf{x}_{k}|\mathbf{x}_{k-1})p(\mathbf{x}_{k-1}|\mathbf{z}_{1:k-1})d\mathbf{x}_{k-1}as an example, from "A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking (2001)"
 
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