Meaning of $X(B_1)$ where X is a random variable
$begingroup$
I'm studying the formal definition a Dirichlet process:
$${if } X sim operatorname{DP}(H,alpha)$$
$$text{then }(X(B_1),dots,X(B_n)) sim operatorname{Dir}(alpha H(B_1),dots, alpha H(B_n))$$
where $$B_1,...,B_n text{ are the partitions of a measurable set S} $$
What does $X(B_n)$ mean? Could you provide me an example?
I can't really see what does notation represents.
Hitherto I have always seen random variables such $X$ "on their own" and the notation $X(B_n)$ really confuses me.
random-variables set-partition
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
$endgroup$
add a comment |
$begingroup$
I'm studying the formal definition a Dirichlet process:
$${if } X sim operatorname{DP}(H,alpha)$$
$$text{then }(X(B_1),dots,X(B_n)) sim operatorname{Dir}(alpha H(B_1),dots, alpha H(B_n))$$
where $$B_1,...,B_n text{ are the partitions of a measurable set S} $$
What does $X(B_n)$ mean? Could you provide me an example?
I can't really see what does notation represents.
Hitherto I have always seen random variables such $X$ "on their own" and the notation $X(B_n)$ really confuses me.
random-variables set-partition
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
$endgroup$
add a comment |
$begingroup$
I'm studying the formal definition a Dirichlet process:
$${if } X sim operatorname{DP}(H,alpha)$$
$$text{then }(X(B_1),dots,X(B_n)) sim operatorname{Dir}(alpha H(B_1),dots, alpha H(B_n))$$
where $$B_1,...,B_n text{ are the partitions of a measurable set S} $$
What does $X(B_n)$ mean? Could you provide me an example?
I can't really see what does notation represents.
Hitherto I have always seen random variables such $X$ "on their own" and the notation $X(B_n)$ really confuses me.
random-variables set-partition
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
$endgroup$
I'm studying the formal definition a Dirichlet process:
$${if } X sim operatorname{DP}(H,alpha)$$
$$text{then }(X(B_1),dots,X(B_n)) sim operatorname{Dir}(alpha H(B_1),dots, alpha H(B_n))$$
where $$B_1,...,B_n text{ are the partitions of a measurable set S} $$
What does $X(B_n)$ mean? Could you provide me an example?
I can't really see what does notation represents.
Hitherto I have always seen random variables such $X$ "on their own" and the notation $X(B_n)$ really confuses me.
random-variables set-partition
random-variables set-partition
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
asked Jan 8 at 22:56
Tommaso BendinelliTommaso Bendinelli
14110
14110
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In the article you have quoted it says : a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. For each sample point $omega$, $X(omega)$ is not a number but a probability measure. $X(B)$ is defined by $X(B)(omega)=X(omega) (B)$, the measure of $B$ under $X(omega)$. For example, you can have something like $X(omega) (B)=int_B phi(omega,t) dt$. If you fix $B$ you have an ordinary random variable and if you fix $omega$ you get a measure $mu$ defined by $mu (B)=int_B phi(omega,t) dt$.
An example: let ${W_t}$ be standard Brownian motion. Define $X(omega) ([a,b))=W_b(omega)-W_a (omega)$. You can extend this to $X(omega) (A)$ when $A$ is a finite union of half closed intervals. Using some standard techniques you can extend this to $X(omega) (A)$ for any Borel set $A$. It is customary to write $X(A)(omega)$ for $X(omega) (A)$.
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
$endgroup$
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
add a comment |
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1 Answer
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1 Answer
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$begingroup$
In the article you have quoted it says : a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. For each sample point $omega$, $X(omega)$ is not a number but a probability measure. $X(B)$ is defined by $X(B)(omega)=X(omega) (B)$, the measure of $B$ under $X(omega)$. For example, you can have something like $X(omega) (B)=int_B phi(omega,t) dt$. If you fix $B$ you have an ordinary random variable and if you fix $omega$ you get a measure $mu$ defined by $mu (B)=int_B phi(omega,t) dt$.
An example: let ${W_t}$ be standard Brownian motion. Define $X(omega) ([a,b))=W_b(omega)-W_a (omega)$. You can extend this to $X(omega) (A)$ when $A$ is a finite union of half closed intervals. Using some standard techniques you can extend this to $X(omega) (A)$ for any Borel set $A$. It is customary to write $X(A)(omega)$ for $X(omega) (A)$.
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
$endgroup$
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
add a comment |
$begingroup$
In the article you have quoted it says : a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. For each sample point $omega$, $X(omega)$ is not a number but a probability measure. $X(B)$ is defined by $X(B)(omega)=X(omega) (B)$, the measure of $B$ under $X(omega)$. For example, you can have something like $X(omega) (B)=int_B phi(omega,t) dt$. If you fix $B$ you have an ordinary random variable and if you fix $omega$ you get a measure $mu$ defined by $mu (B)=int_B phi(omega,t) dt$.
An example: let ${W_t}$ be standard Brownian motion. Define $X(omega) ([a,b))=W_b(omega)-W_a (omega)$. You can extend this to $X(omega) (A)$ when $A$ is a finite union of half closed intervals. Using some standard techniques you can extend this to $X(omega) (A)$ for any Borel set $A$. It is customary to write $X(A)(omega)$ for $X(omega) (A)$.
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
$endgroup$
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
add a comment |
$begingroup$
In the article you have quoted it says : a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. For each sample point $omega$, $X(omega)$ is not a number but a probability measure. $X(B)$ is defined by $X(B)(omega)=X(omega) (B)$, the measure of $B$ under $X(omega)$. For example, you can have something like $X(omega) (B)=int_B phi(omega,t) dt$. If you fix $B$ you have an ordinary random variable and if you fix $omega$ you get a measure $mu$ defined by $mu (B)=int_B phi(omega,t) dt$.
An example: let ${W_t}$ be standard Brownian motion. Define $X(omega) ([a,b))=W_b(omega)-W_a (omega)$. You can extend this to $X(omega) (A)$ when $A$ is a finite union of half closed intervals. Using some standard techniques you can extend this to $X(omega) (A)$ for any Borel set $A$. It is customary to write $X(A)(omega)$ for $X(omega) (A)$.
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
$endgroup$
In the article you have quoted it says : a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. For each sample point $omega$, $X(omega)$ is not a number but a probability measure. $X(B)$ is defined by $X(B)(omega)=X(omega) (B)$, the measure of $B$ under $X(omega)$. For example, you can have something like $X(omega) (B)=int_B phi(omega,t) dt$. If you fix $B$ you have an ordinary random variable and if you fix $omega$ you get a measure $mu$ defined by $mu (B)=int_B phi(omega,t) dt$.
An example: let ${W_t}$ be standard Brownian motion. Define $X(omega) ([a,b))=W_b(omega)-W_a (omega)$. You can extend this to $X(omega) (A)$ when $A$ is a finite union of half closed intervals. Using some standard techniques you can extend this to $X(omega) (A)$ for any Borel set $A$. It is customary to write $X(A)(omega)$ for $X(omega) (A)$.
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
edited Jan 9 at 8:33
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
answered Jan 8 at 23:16
Kavi Rama MurthyKavi Rama Murthy
62.3k42262
62.3k42262
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
add a comment |
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
Thank for answering, but it's still not clear. Particularly, in this contest what is a sample point $omega$? Is it a point from the measurable set S? Furthermore what is $phi(omega,t)$ ?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:02
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
There is basic probability space $(Omega,mathcal F,P)$ on which $X$ is defined. A sample point is simply an element of $Omega$. In my example $phi$ could be any measurable function on $Omega times mathbb R$ such that it is integrable w.r.t. Lebesgue measure for every $omega$.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:08
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
And how is this probability space related to the measurable set S and its partitions? What is the sample space $Omega$ of this probability space? I can't visualize it.
$endgroup$
– Tommaso Bendinelli
Jan 9 at 8:13
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
@TommasoBendinelli $X(omega)$ is a measure on some set $S$ with a sigma algebra. In my example it is the real line with Borel sigma algebra. $Omega$ and $S$ are not related in general.
$endgroup$
– Kavi Rama Murthy
Jan 9 at 8:19
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
$begingroup$
I've seen what you added and it's clearer now. B is an area, and different $omega$ lead to the different measurement of B. The sum all measurements of B has any constraints? Does it have to sum to 1 or is the probability of all $omega$ that has to sum to 1?
$endgroup$
– Tommaso Bendinelli
Jan 9 at 12:49
add a comment |
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