distribution according to random measures












1














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







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    1














    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







    share|cite|improve this question

























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      1







      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







      share|cite|improve this question













      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






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      asked 2 days ago









      lemontree

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          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.






          share|cite|improve this answer








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          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.


















          • 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!
            – lemontree
            2 days ago










          • 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 $(X_1,dots,X_6)$.
            – Aragon
            2 days ago













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          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.






          share|cite|improve this answer








          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.


















          • 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!
            – lemontree
            2 days ago










          • 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 $(X_1,dots,X_6)$.
            – Aragon
            2 days ago


















          1














          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.






          share|cite|improve this answer








          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.


















          • 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!
            – lemontree
            2 days ago










          • 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 $(X_1,dots,X_6)$.
            – Aragon
            2 days ago
















          1












          1








          1






          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.






          share|cite|improve this answer








          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          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.







          share|cite|improve this answer








          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          share|cite|improve this answer



          share|cite|improve this answer






          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          answered 2 days ago









          Aragon

          213




          213




          New contributor




          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.





          New contributor





          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          Aragon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.












          • 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!
            – lemontree
            2 days ago










          • 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 $(X_1,dots,X_6)$.
            – Aragon
            2 days ago




















          • 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!
            – lemontree
            2 days ago










          • 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 $(X_1,dots,X_6)$.
            – Aragon
            2 days ago


















          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!
          – lemontree
          2 days ago




          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!
          – lemontree
          2 days ago












          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 $(X_1,dots,X_6)$.
          – Aragon
          2 days ago






          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 $(X_1,dots,X_6)$.
          – Aragon
          2 days ago




















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