Mathematics: Compute the mutual information between x and y where x and y are binary and with joint distribution specified as p(0, 0) = p(0, 1) = p(1, 0) = 1/3.

Compute the mutual information between x and y

Ответ:

fbillinton

13.03.2022

Mathematics

The mutual information

$MI=\displaystyle\sum_{x,y}p(x,y)\ln\dfrac{p(x,y)}{p(x)p(y)}$

First compute the marginal distributions for

and

.

$p(x)=\begin{cases}\frac23&\text{for }x=0\\\frac13&\text{for }x=1\end{cases}$

has the same marginal distribution (replace

with

above).

The support for the joint PMF are the points (0,0), (1,0), and (0,1), so this is what you sum over. We get

$MI=p(0,0)\ln\dfrac{p(0,0)}{p_X(0)p_Y(0)}+p(1,0)\ln\dfrac{p(1,0)}{p_X(1)p_Y(0)}+p(0,1)\ln\dfrac{p(0,1)}{p_X(0)p_Y(1)}$
$MI=\dfrac13\ln\dfrac{\frac13}{\frac23\cdot\frac23}+\dfrac13\ln\dfrac{\frac13}{\frac13\cdot\frac23}+\dfrac13\ln\dfrac{\frac13}{\frac23\cdot\frac13}$
$MI\approx0.174$

Be sure to check how mutual information is defined in your text/notes. I used the natural logarithm above.

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Ответ:

shira35

25.02.2023

Mathematics

Hey there!

CarMart's rate of sales is 11 cars per day.

Winston's rate of sales is 10 cars per day.

Compared with CarMart, Winston is selling fewer cars per day.

Compared with CarMart, Winston started with 2 fewer cars.

Let me know if this helps :)

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Compute the mutual information between x and y where x and y are binary and with joint distribution specified as p(0, 0) = p(0, 1) = p(1, 0) = 1/3.

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