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Derive Yourself a Kalman Filter

ngr.yt

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Re: Derive Yourself a Kalman Filter

#51

Post by kgwgk »

bonoboTP wrote: post #49
kgwgk wrote: post #48

Earlier quoted context omitted.

If you mean that the space for this single experiment composed of two rolls (random variables X and Y) is the cartesian product of {x=1,x=2,x=3,x=4,x=5,x=6} and {y=1,y=2,y=3,y=4,y=5,y=6}, then I agree. But the fact that each variable alone is defined on the "same" sample space {1,2,3,4,5,6} is irrelevant. The situation is no different from the joint probability for random variables X and Z corresponding to a single e…

Let's consider something with less independence, because it makes things harder to notice. Temperature indoors T1, temperature outdoors T2, IsOvercast O. Let's say T2|O=1 is a "conditional random variable". Let's consider the average temperature indoors and outdoors. What would ((T1|O=1) + T2)/2 even mean? How could you use the two "variables" in the same expression? What is even their joint distribution? They are de…

> How could you use the two "variables" in the same expression?

Do you expect to be able to use every random variable which can be conceived in the same expression?

If you object to the name “conditional random variable” [+] that’s debatable, but if you say that the resulting thing is not a random variable I think you are wrong.

Another thing that is a random variable, even though I suspect you may not like it, is the probability distribution of a random variable.

[+] which I don’t think was actually used by the OP, by the way.

Re: Derive Yourself a Kalman Filter

#52

Post by srean »

bonoboTP wrote: post #50
srean wrote: post #45

Earlier quoted context omitted.

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 th…

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…

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.

Re: Derive Yourself a Kalman Filter

#53

Post by bonoboTP »

srean wrote: post #52
bonoboTP wrote: post #50

Earlier quoted context omitted.

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…

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 pl…

> 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?

Re: Derive Yourself a Kalman Filter

#54

Post by srean »

bonoboTP wrote: post #53
srean wrote: post #52

Earlier quoted context omitted.

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 pl…

> 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?

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.

Re: Derive Yourself a Kalman Filter

#55

Post by kgwgk »

srean wrote: post #54
bonoboTP wrote: post #53

Earlier quoted context omitted.

> 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?

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.

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).

Re: Derive Yourself a Kalman Filter

#56

Post by srean »

kgwgk wrote: post #55
srean wrote: post #54

Earlier quoted context omitted.

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.

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 subse…

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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