Suppose 3 boolean RVs, there should be $2^N = 2^3 = 8$ params, could someone please give a concrete example to enumerate what are they?

Question

This CMU Machine Learning Course gives a formula to compute the number of params in an estimation.

$2^n$ for the distribution $P(Y|X_1, X_2, ... X_n)$

per that formula, the number of params in the estimation for this distribution

$P(Y|X_1, X_2, X_3)$

is equal to $2^n = 2^3 = 8$.

to get more intuitive understanding, I am trying to enumerate all the params for n=3.

Could someone please give a concrete example to enumerate what those 8 params are?

Answer 1

It says every $X$ and $Y$ are booleans. Which means each of them has exactly two possible values.

So, you have $X_{i} = \{x_{i}, \bar{x}_{i}\}$, $i = 1,2,3$. Your 8 possibilites are:

$x_{1},x_{2},x_{3}$

$x_{1},x_{2},\bar{x}_{3}$

$x_{1},\bar{x}_{2},x_{3}$

$x_{1},\bar{x}_{2},\bar{x}_{3}$

$\bar{x}_{1},x_{2},x_{3}$

$\bar{x}_{1},x_{2},\bar{x}_{3}$

$\bar{x}_{1},\bar{x}_{2},x_{3}$

$\bar{x}_{1},\bar{x}_{2},\bar{x}_{3}$

That's easy straight forward. I hope that helps.

Answer 2

This part gives an example of 3 boolean-valued variables, gender, hours_worked, wealth.

8 possible values are:

   gender hours_worked wealth      ratio
0  Female           v0   poor   0.253122
1  Female           v0   rich  0.0245895
2  Female           v1   poor  0.0421768
3  Female           v1   rich  0.0116293
4    Male           v0   poor   0.331313
5    Male           v0   rich  0.0971295
6    Male           v1   poor   0.134106
7    Male           v1   rich   0.105933