distribution according to random measures
$begingroup$
Is a random probability measure a probability measure, i.e. if $X_i$ is random variable on some probability space $(Omega,mathcal F,mathbb P)$ could its induced distribution be a random probability measure $mu$, say $X_isim mu$?
Then, as $mu$ is itself random, it has an own probability distribution, $musim Q$.
For example let's take the empirical measure $P_n(A)=frac{1}{n}sum_{i=1}^n delta_{X_i}(A)$. Then the empirical distribution function $F_n(x):=frac{1}{n}sum_{i=1}^n 1{X_ile x}$ is a random probability measure as it is an empirical measure indexed by the class $mathcal C={(-infty,x]: xin mathbb R}$? Can one say $X_isim P_n$?
probability probability-theory measure-theory random-variables random
asked Jan 3 at 17:17
lemontreelemontree
1608
$endgroup$
add a comment |
$begingroup$
Is a random probability measure a probability measure, i.e. if $X_i$ is random variable on some probability space $(Omega,mathcal F,mathbb P)$ could its induced distribution be a random probability measure $mu$, say $X_isim mu$?
Then, as $mu$ is itself random, it has an own probability distribution, $musim Q$.
For example let's take the empirical measure $P_n(A)=frac{1}{n}sum_{i=1}^n delta_{X_i}(A)$. Then the empirical distribution function $F_n(x):=frac{1}{n}sum_{i=1}^n 1{X_ile x}$ is a random probability measure as it is an empirical measure indexed by the class $mathcal C={(-infty,x]: xin mathbb R}$? Can one say $X_isim P_n$?
probability probability-theory measure-theory random-variables random
asked Jan 3 at 17:17
lemontreelemontree
1608
$endgroup$
add a comment |
$begingroup$
Is a random probability measure a probability measure, i.e. if $X_i$ is random variable on some probability space $(Omega,mathcal F,mathbb P)$ could its induced distribution be a random probability measure $mu$, say $X_isim mu$?
Then, as $mu$ is itself random, it has an own probability distribution, $musim Q$.
For example let's take the empirical measure $P_n(A)=frac{1}{n}sum_{i=1}^n delta_{X_i}(A)$. Then the empirical distribution function $F_n(x):=frac{1}{n}sum_{i=1}^n 1{X_ile x}$ is a random probability measure as it is an empirical measure indexed by the class $mathcal C={(-infty,x]: xin mathbb R}$? Can one say $X_isim P_n$?
probability probability-theory measure-theory random-variables random
asked Jan 3 at 17:17
lemontreelemontree
1608
$endgroup$
Is a random probability measure a probability measure, i.e. if $X_i$ is random variable on some probability space $(Omega,mathcal F,mathbb P)$ could its induced distribution be a random probability measure $mu$, say $X_isim mu$?
Then, as $mu$ is itself random, it has an own probability distribution, $musim Q$.
For example let's take the empirical measure $P_n(A)=frac{1}{n}sum_{i=1}^n delta_{X_i}(A)$. Then the empirical distribution function $F_n(x):=frac{1}{n}sum_{i=1}^n 1{X_ile x}$ is a random probability measure as it is an empirical measure indexed by the class $mathcal C={(-infty,x]: xin mathbb R}$? Can one say $X_isim P_n$?
probability probability-theory measure-theory random-variables random
probability probability-theory measure-theory random-variables random
asked Jan 3 at 17:17
lemontreelemontree
1608
asked Jan 3 at 17:17
lemontreelemontree
1608
asked Jan 3 at 17:17
lemontreelemontree
1608
asked Jan 3 at 17:17
lemontreelemontree
1608
asked Jan 3 at 17:17
lemontreelemontree
1608
1608
add a comment |
add a comment |
1 Answer
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$begingroup$
For a random measure, it is just the case that $Omega = mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition.
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $pi: mathcal{P}(mathbb{R}) mapsto mathcal{S}$, where $mathcal{S}$ is the collection of step functions in $mathbb{R}$.
$Xsim mu$ is for a fixed measure $mu$. In your example, you cannot say $X_i sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
answered Jan 3 at 18:05
AragonAragon
363
$endgroup$
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
add a comment |
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1 Answer
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1 Answer
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$begingroup$
For a random measure, it is just the case that $Omega = mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition.
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $pi: mathcal{P}(mathbb{R}) mapsto mathcal{S}$, where $mathcal{S}$ is the collection of step functions in $mathbb{R}$.
$Xsim mu$ is for a fixed measure $mu$. In your example, you cannot say $X_i sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
answered Jan 3 at 18:05
AragonAragon
363
$endgroup$
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
add a comment |
$begingroup$
For a random measure, it is just the case that $Omega = mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition.
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $pi: mathcal{P}(mathbb{R}) mapsto mathcal{S}$, where $mathcal{S}$ is the collection of step functions in $mathbb{R}$.
$Xsim mu$ is for a fixed measure $mu$. In your example, you cannot say $X_i sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
answered Jan 3 at 18:05
AragonAragon
363
$endgroup$
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
add a comment |
$begingroup$
For a random measure, it is just the case that $Omega = mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition.
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $pi: mathcal{P}(mathbb{R}) mapsto mathcal{S}$, where $mathcal{S}$ is the collection of step functions in $mathbb{R}$.
$Xsim mu$ is for a fixed measure $mu$. In your example, you cannot say $X_i sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
answered Jan 3 at 18:05
AragonAragon
363
$endgroup$
For a random measure, it is just the case that $Omega = mathcal{P}$ the set of all probability measures on space. For example $P_n$ in your definition.
The empirical measure is a random variable on the space of functions. This measure is obtained via taking the push forward measure of the mapping $pi: mathcal{P}(mathbb{R}) mapsto mathcal{S}$, where $mathcal{S}$ is the collection of step functions in $mathbb{R}$.
$Xsim mu$ is for a fixed measure $mu$. In your example, you cannot say $X_i sim P_n$ because the distribution $X_i$'s is something from which you have drawn the random variables $X_i$'s to construct the measure in the first place ($P_n$ is just a function of $X_i$'s). But, if you wanted mean that $X$ is a "sample" from the empirical measure, then you have to keep in mind that there are two steps of randomness. The first step to construct the measure, and the second is taking a sample from $P_n$ conditionally on the $X_i$'s in the first step. So one needs to write the distribution of $X$ using a conditional distribution.
answered Jan 3 at 18:05
AragonAragon
363
answered Jan 3 at 18:05
AragonAragon
363
answered Jan 3 at 18:05
AragonAragon
363
answered Jan 3 at 18:05
AragonAragon
363
363
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
add a comment |
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
Consider you have a sample $X=(X_1,dots, X_n).$ Now assume $X_imid mu sim mu$ and $musim Q$, where $mu$ is a random probability measure and $Q$ is the Distribution of the random probability measure. I think that is an example for the "two steps of randomness" you talked about. Can you explain what $X_imid mu sim mu$ means and why we can't use/ what's the difference to $X_isim mu$? Many thanks!
$endgroup$
– lemontree
Jan 3 at 18:52
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
$begingroup$
The conditional distribution will be a discrete measure. Suppose you toss a coin 6 times and you get TTHTTH. So your empirical measure $P_n$ will give mass 1/3 on head and 2/3 on tail. But $X_1$ is Bernoulli$(1/2)$. What you can do is take a new coin with respect to the observed empirical measure. The outcome $Y$ will be Bernoulli $(1/3)$ conditional on the event that there are exactly $2$ heads among $(X_1,dots,X_6)$.
$endgroup$
– Aragon
Jan 3 at 19:36
add a comment |
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