<p>From <a href="https://www.dropbox.com/s/hi9gc9ky5n16a5x/ch4.pdf?e=1&amp;dl=0" rel="nofollow noreferrer">Blackwell's theorem</a>, it is possible to draw an equivalence between one signal being a garbling of another and the statistical notion of a Mean Preserving Spread.</p>
<p>Consequently, at least in the context of Blackwell's Theorem, the following is an alternative definition of a MPS:</p>
<p>For a pair of distributions <span class="math-container">$F$</span> and <span class="math-container">$F'$</span> induced by two signals <span class="math-container">$\sigma' : \Theta \to \Delta(S')$</span> and <span class="math-container">$\sigma : \Theta \to \Delta(S')$</span>, <span class="math-container">$F$</span> is Mean Preserving Spread of <span class="math-container">$F'$</span> if there exists a Markov Kernel <span class="math-container">$\gamma : S \to \Delta(S')$</span> such that:</p>
<p><span class="math-container">$$\sigma'(s' | \theta) = \int_{s \in S} \gamma(s' |s) \sigma(s | \theta) ds$$</span></p>
<p>I know this is true due to the theorem, but I'm trying to prove the result directly, using the following (standard) definition of a Mean Preserving Spread:
For a pair of random variables from the pair of distributions <span class="math-container">$Z \sim F$</span> and <span class="math-container">$\tilde{Z} \sim \tilde{F}$</span>, <span class="math-container">$F$</span> is a MPS of <span class="math-container">$\tilde{F}$</span> if:
<span class="math-container">$$\mathbb{E}(Z | \tilde{Z}) = \tilde{Z}$$</span></p>
<p>I have made the following partial attempt.</p>
<p>First, I note that in general:
<span class="math-container">$$\mathbb{E}(X|Y) = \int_A \dfrac{f_{X,Y}(x,y)}{f_Y(y)}$$</span></p>
<p>Where <span class="math-container">$A$</span> is the <span class="math-container">$\sigma$</span>-algebra generated by <span class="math-container">$Y$</span>.</p>
<p>In this case, I think (though am far from sure) that:
<span class="math-container">$$f_{X,Y}(x,y) = \gamma(s' |s) \sigma(s|\theta)$$</span></p>
<p><span class="math-container">$$f_Y(y) = \gamma(s|s')$$</span></p>
<p>Consequently:</p>
<p><span class="math-container">$$\begin{align}
\mathbb{E}(Z|\tilde{Z}) &amp;= \int_{s' \in S'} \dfrac{\sigma'(s'|\theta)}{\gamma(s|s')} .ds'\\
&amp;= \int_{s' \in S'} \dfrac{\sigma(s|\theta) \gamma(s|s')}{\gamma(s|s')} .ds' \\
&amp;= \sigma(s|\theta)
\end{align}$$</span></p>
<p>Would someone be able to confirm that I've specified the densities correctly here, and if I have not, suggest what they should be?</p>

CC BY-SA 4.0
2024-05-21T06:38:13.727Z
The notation $\int_A$ for $A$ a $\sigma$-algebra is rarher unusual. More importantly, where do the densities come from?
Michael Greinecker
CC BY-SA 4.0
2024-07-23T09:36:59.317Z
@matthewoulton Are the notes in your link from some book? Are these some personal notes? They seem to be nice written and if it was some book I would like to read it more thouroughly. Recently, I became much interested in Blackwell theory about informational experiments and I see this is used so mych in Bayesian persuasion that is a field I find it challenging to understand. Could you please give me some information on the topic about your link? 

Thank you in advance!
Oliver Queen
CC BY-SA 4.0
2024-08-11T15:35:51.747Z
They're my own notes, but based on Annie Liang's, which you can find here:
https://www.anniehliang.com/lecture-notes
I think he notes are really well written and would highly recommend.
matthewoulton
<mathematical-economics>
Mean Preserving Spreads as a Markov Kernel
matthewoulton
# Mean Preserving Spreads as a Markov Kernel

(Asked by: username_0 on 2024-05-21)

From [Blackwell's theorem](https://www.dropbox.com/s/hi9gc9ky5n16a5x/ch4.pdf?e=1&dl=0), it is possible to draw an equivalence between one signal being a garbling of another and the statistical notion of a Mean Preserving Spread.

Consequently, at least in the context of Blackwell's Theorem, the following is an alternative definition of a MPS:

For a pair of distributions $F$ and $F'$ induced by two signals $\sigma' : \Theta \to \Delta(S')$ and $\sigma : \Theta \to \Delta(S')$, $F$ is Mean Preserving Spread of $F'$ if there exists a Markov Kernel $\gamma : S \to \Delta(S')$ such that:

$$\sigma'(s' | \theta) = \int\_{s \in S} \gamma(s' |s) \sigma(s | \theta) ds$$

I know this is true due to the theorem, but I'm trying to prove the result directly, using the following (standard) definition of a Mean Preserving Spread:
For a pair of random variables from the pair of distributions $Z \sim F$ and $\tilde{Z} \sim \tilde{F}$, $F$ is a MPS of $\tilde{F}$ if:
$$\mathbb{E}(Z | \tilde{Z}) = \tilde{Z}$$

I have made the following partial attempt.

First, I note that in general:
$$\mathbb{E}(X|Y) = \int\_A \dfrac{f\_{X,Y}(x,y)}{f\_Y(y)}$$

Where $A$ is the $\sigma$-algebra generated by $Y$.

In this case, I think (though am far from sure) that:
$$f\_{X,Y}(x,y) = \gamma(s' |s) \sigma(s|\theta)$$

$$f\_Y(y) = \gamma(s|s')$$

Consequently:

$$\begin{align}
\mathbb{E}(Z|\tilde{Z}) &= \int\_{s' \in S'} \dfrac{\sigma'(s'|\theta)}{\gamma(s|s')} .ds'\\
&= \int\_{s' \in S'} \dfrac{\sigma(s|\theta) \gamma(s|s')}{\gamma(s|s')} .ds' \\
&= \sigma(s|\theta)
\end{align}$$

Would someone be able to confirm that I've specified the densities correctly here, and if I have not, suggest what they should be?

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: The notation $\int_A$ for $A$ a $\sigma$-algebra is rarher unusual. More importantly, where do the densities come from?  
> **username_2**: @username_0 Are the notes in your link from some book? Are these some personal notes? They seem to be nice written and if it was some book I would like to read it more thouroughly. Recently, I became much interested in Blackwell theory about informational experiments and I see this is used so mych in Bayesian persuasion that is a field I find it challenging to understand. Could you please give me some information on the topic about your link?   Thank you in advance!  
> **username_0**: They're my own notes, but based on Annie Liang's, which you can find here: https://www.anniehliang.com/lecture-notes I think he notes are really well written and would highly recommend.  

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## 0 answers

