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What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?
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What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
- what a probability space $(Omega, mathcalF, mathbbP)$ refers to
- probability without measure theory really well (i.e., discrete and continuous random variables)
- what the measure-theoretic definition of a random variable is
- that a Lebesgue integral has something to do with linear combinations of step functions, and intuitively, it involves partitioning the $y$-axis (as opposed to the Riemann integral, which partitions the $x$-axis).
I haven't had time to learn measure-theoretic probability lately due to graduate school creeping in as well as other commitments, and conditional expectation is often covered as one of the last topics in every measure-theoretic probability text I've seen.
I have seen notations such as $mathbbE[X mid mathcalF]$, where I assume $mathcalF$ is some sort of $sigma$-algebra - but of course, this looks very different from, say, $mathbbE[X mid Y]$ from what I saw in my non-measure-theoretic treatment of probability, where $X$ and $Y$ are random variables.
I was also surprised to see that one book I have (Essentials of Probability Theory for Statisticians by Proschan and Shaw (2016)), if I recall correctly, explicity states that conditional expectation is defined as a conditional expectation, rather than the conditional expectation, which implies to me that there's more than one possible conditional expectation when given two pairs of random variables. (Unfortunately, I don't have the book on me right now, but I can update this post later).
The Wikipedia article is quite dense, and I see words such as "Radon-Nikodym" which I haven't learned yet, but I would at least like to get an idea of what the intuition of conditional expectation is in a measure-theoretic sense.
probability-theory measure-theory intuition conditional-expectation
asked Aug 2 at 11:59
Clarinetist
10.3k32767
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up vote
4
down vote
favorite
What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
- what a probability space $(Omega, mathcalF, mathbbP)$ refers to
- probability without measure theory really well (i.e., discrete and continuous random variables)
- what the measure-theoretic definition of a random variable is
- that a Lebesgue integral has something to do with linear combinations of step functions, and intuitively, it involves partitioning the $y$-axis (as opposed to the Riemann integral, which partitions the $x$-axis).
I haven't had time to learn measure-theoretic probability lately due to graduate school creeping in as well as other commitments, and conditional expectation is often covered as one of the last topics in every measure-theoretic probability text I've seen.
I have seen notations such as $mathbbE[X mid mathcalF]$, where I assume $mathcalF$ is some sort of $sigma$-algebra - but of course, this looks very different from, say, $mathbbE[X mid Y]$ from what I saw in my non-measure-theoretic treatment of probability, where $X$ and $Y$ are random variables.
I was also surprised to see that one book I have (Essentials of Probability Theory for Statisticians by Proschan and Shaw (2016)), if I recall correctly, explicity states that conditional expectation is defined as a conditional expectation, rather than the conditional expectation, which implies to me that there's more than one possible conditional expectation when given two pairs of random variables. (Unfortunately, I don't have the book on me right now, but I can update this post later).
The Wikipedia article is quite dense, and I see words such as "Radon-Nikodym" which I haven't learned yet, but I would at least like to get an idea of what the intuition of conditional expectation is in a measure-theoretic sense.
probability-theory measure-theory intuition conditional-expectation
asked Aug 2 at 11:59
Clarinetist
10.3k32767
2
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
1
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
- what a probability space $(Omega, mathcalF, mathbbP)$ refers to
- probability without measure theory really well (i.e., discrete and continuous random variables)
- what the measure-theoretic definition of a random variable is
- that a Lebesgue integral has something to do with linear combinations of step functions, and intuitively, it involves partitioning the $y$-axis (as opposed to the Riemann integral, which partitions the $x$-axis).
I haven't had time to learn measure-theoretic probability lately due to graduate school creeping in as well as other commitments, and conditional expectation is often covered as one of the last topics in every measure-theoretic probability text I've seen.
I have seen notations such as $mathbbE[X mid mathcalF]$, where I assume $mathcalF$ is some sort of $sigma$-algebra - but of course, this looks very different from, say, $mathbbE[X mid Y]$ from what I saw in my non-measure-theoretic treatment of probability, where $X$ and $Y$ are random variables.
I was also surprised to see that one book I have (Essentials of Probability Theory for Statisticians by Proschan and Shaw (2016)), if I recall correctly, explicity states that conditional expectation is defined as a conditional expectation, rather than the conditional expectation, which implies to me that there's more than one possible conditional expectation when given two pairs of random variables. (Unfortunately, I don't have the book on me right now, but I can update this post later).
The Wikipedia article is quite dense, and I see words such as "Radon-Nikodym" which I haven't learned yet, but I would at least like to get an idea of what the intuition of conditional expectation is in a measure-theoretic sense.
probability-theory measure-theory intuition conditional-expectation
asked Aug 2 at 11:59
Clarinetist
10.3k32767
What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
- what a probability space $(Omega, mathcalF, mathbbP)$ refers to
- probability without measure theory really well (i.e., discrete and continuous random variables)
- what the measure-theoretic definition of a random variable is
- that a Lebesgue integral has something to do with linear combinations of step functions, and intuitively, it involves partitioning the $y$-axis (as opposed to the Riemann integral, which partitions the $x$-axis).
I haven't had time to learn measure-theoretic probability lately due to graduate school creeping in as well as other commitments, and conditional expectation is often covered as one of the last topics in every measure-theoretic probability text I've seen.
I have seen notations such as $mathbbE[X mid mathcalF]$, where I assume $mathcalF$ is some sort of $sigma$-algebra - but of course, this looks very different from, say, $mathbbE[X mid Y]$ from what I saw in my non-measure-theoretic treatment of probability, where $X$ and $Y$ are random variables.
I was also surprised to see that one book I have (Essentials of Probability Theory for Statisticians by Proschan and Shaw (2016)), if I recall correctly, explicity states that conditional expectation is defined as a conditional expectation, rather than the conditional expectation, which implies to me that there's more than one possible conditional expectation when given two pairs of random variables. (Unfortunately, I don't have the book on me right now, but I can update this post later).
The Wikipedia article is quite dense, and I see words such as "Radon-Nikodym" which I haven't learned yet, but I would at least like to get an idea of what the intuition of conditional expectation is in a measure-theoretic sense.
probability-theory measure-theory intuition conditional-expectation
asked Aug 2 at 11:59
Clarinetist
10.3k32767
asked Aug 2 at 11:59
Clarinetist
10.3k32767
asked Aug 2 at 11:59
Clarinetist
10.3k32767
asked Aug 2 at 11:59
Clarinetist
10.3k32767
10.3k32767
2
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
1
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41
add a comment |Â
2
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
1
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41
2
2
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
1
1
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41
add a comment |Â
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2
In a Hilbert space you can think of it as a projection similar to what you might have seen in least-squares context like briefly discussed for example in this text people.ku.edu/~t926s829/math865-2015spring/conditionalexpL2.pdf
– WalterJ
Aug 2 at 12:11
On cond exp or $sigma$-algebras: stats.stackexchange.com/questions/192179 math.stackexchange.com/questions/2048219 quant.stackexchange.com/questions/37497 math.stackexchange.com/questions/1273287 math.stackexchange.com/questions/2711361 math.stackexchange.com/questions/375994 math.stackexchange.com/questions/26733 math.stackexchange.com/questions/77757
– BCLC
Aug 2 at 12:33
The expression $mathbb E[Xmid Y]$ is shorthand for $mathbb E[Xmid sigma(Y)]$, where $$ sigma(Y) = Y^-1(B) : Binmathcal B(mathbb R) $$ is the $sigma$-algebra generated by $Y$.
– Math1000
Aug 2 at 12:37
1
Also, in a "measure-theoretic sense" a conditional expectation IS a Radon-Nikodym derivative. These notes may lend some intuition however: ma.utexas.edu/users/gordanz/notes/conditional_expectation.pdf
– Math1000
Aug 2 at 12:41