Derive Yourself a Kalman Filter

Re:Derive Yourself a Kalman Filter by srean on Monday May 13, 2019 @02:34PM (#19902337)

Billingsley is pretty darn good. It might have left the connection as a dotted line given that the notion is no different from conditional expectation. The only connection you have to make is conditional expectation is a function and a random variable. You must have seen expectation taken of a conditional expectation. That should should convince you that condititional expectation is indeed a random variable. Since that r.v. was obtained by conditioning its not a stretvh to call it a conditioned r.v.

Any book that explains conditioning over a sigma algebra should suffice. You could try Loeve, Dudely or Neveu but dont remember if its mentioned explicitly.

BTW conditional expectation is really more fundamental than conditional probability. Its the former that yields the latter in measure theoretic probability. If you want to drink from the source that would be Kolmogorov.

Finally if you are reading Billingsley you are adequately qualified to call yourself a mathematician.

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Re:Derive Yourself a Kalman Filter by bonoboTP on Monday May 13, 2019 @09:54PM (#19906102)

It's getting a little tedious. Please show me a concrete citation of a serious textbook (not a tutorial/handout by a grad student or a paper by a random researcher) that puts the three words "conditional random variable" next to each other (consistently, not simply as a one-off potential mistake). Google doesn't show serious sources for it.

While I agree with isolated points of your comment I think it doesn't add up to a useful/coherent concept of conditional random variable.

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Re:Derive Yourself a Kalman Filter by srean on Tuesday May 14, 2019 @12:17PM (#19910878)

Thats a little too much to ask, perhaps if they were grep'able I could have obliged, unfortunately I dont have a photographic memory.

More concretely its just another name for conditional expectation. I am assuming you are aware that conditional expectation is a random variable obtained via conditioning (equivalently as a piecewise approximation in L_2). If you arent familiar with that view point that would be the place to start. Kolmogorov, Neveu, Dudely, Billingsley will all cover that view point.

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Re:Derive Yourself a Kalman Filter by bonoboTP on Wednesday May 15, 2019 @12:45PM (#19921217)

> I am assuming you are aware that conditional expectation is a random variable

That's not what we're considering here, but things of the form X|Y=y for a concrete y. Even as E[X|Y=y], that's not a function, y is specified. Do you agree we shouldn't call X|Y=y a conditional random variable?

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Re:Derive Yourself a Kalman Filter by srean on Wednesday May 15, 2019 @02:50PM (#19922497)

Oh absolutely for a specific y its not function (or a random variable) one usually thinks of Y as a variable and not a constant.

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Re:Derive Yourself a Kalman Filter by kgwgk on Wednesday May 15, 2019 @04:22PM (#19923346)

The expectation E[X|Y=y] is a fixed value. (Edit: it’s the expectation of the random variable “X|Y=y”, while E[X|Y] is a random variable because it’s a function of the random variable Y: for each element in the sample space there is a corresponding value of “y” and in turn there is a value of the expectation E[X|Y=y].)

X|Y=y (as used in the blog post being discussed) is a random variable: it’s a function from a subset of the original sample space (corresponding to the elements for which the value of the random variable Y is y) to real values (or whatever the image of the X random variable is).

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Re:Derive Yourself a Kalman Filter by srean on Thursday May 16, 2019 @01:53PM (#19931183)

Yes you are right. I had messed up in the comment above. It continues to be a function on the restriction Y=y

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