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Is there a difference between the two? Can a "kernel" be seen as the distribution of a random "object" given a realization of another variable?

ToniAz
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    Maybe my answer here helps? http://math.stackexchange.com/questions/33430/whats-the-intuition-behind-and-some-illustrative-applications-of-probability-ke/33472#33472 –  Mar 01 '16 at 02:24
  • @ByronSchmuland Thank you. I guess I can think of a transition kernel as some kind of conditional probability to make things simpler for me, because they both describe transitioning how a random object transitions. – ToniAz Mar 01 '16 at 15:54

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Like much of statistics, this is a parameter versus estimator issue. Consider the probability distribution of a random variable $X$ as a parameter itself (which, under non-parametric statistics is reasonable). Thus, this parameter is the function $P_X(x).$ THe kernel of this probability is the estimator of the distribution, $\hat{P_X}(x)$. Kernel density estimation is the non-parametric procedure used to find a realization of this estimator.

Michael
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