If I know $P(x)$ and $P(y|x)$, how can I know $P(y)$?

Question

If I need to compute $P(y)$ and I only know $P(y|x)$ and $P(x)$.

I know that:

$$P(y|x)=\frac{P(y,x)}{P(x)}$$

This is the contingency table for $P(y|x)$:

 P(y|x) |   +x   |  ¬x  |
-------------------------
   +y   |  0.91  | 0.03 |
-------------------------
   ¬y   |  0.09  | 0.97 |
-------------------------

And the probability of $x$:

        |   +x   |  ¬x  |
-------------------------
  P(x)  |  0.05  | 0.95 |
-------------------------

How can I compute $P(y)$?

There is insufficient information - for example if P(y|x) were 1, that could mean that P(y) =1 or it could also mean P(y) = P(x) if totally dependant. — Cato, Jun 03 '19 at 15:27
@Cato, I have updated my question. If you need more info, please tell me. Thanks. — VansFannel, Jun 03 '19 at 15:34
You have an equation $a=b/c$ and you know $a$ and $c$. So you can get $b$ just by multiplying both sides of the equation by $c$. — Michael, Jun 03 '19 at 15:39

NicNic8 · Accepted Answer · 2019-06-03T20:47:00.227

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$P(y,x) = P(y|x) P(x)$

$P(y) = \sum_x P(y,x) = \sum_x P(y|x) P(x) $

edited Jun 03 '19 at 20:47

answered Jun 03 '19 at 15:37

NicNic8

6,951

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Your last formula is wrong. – VansFannel Jun 03 '19 at 15:56
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@VansFannel Why do you think that? – callculus42 Jun 03 '19 at 16:37
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I think is $P(y) = \sum_x P(y,x) = \sum_x P(y|x) P(x) $ (Law of total probability). I need to compute $P(y)$. – VansFannel Jun 03 '19 at 16:40
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@VansFannel You are right with your formula. And you are right, that the formula of NicNic8 is wrong. – callculus42 Jun 03 '19 at 16:44
@VansFannel My former formula was correct for P(x). I have now fixed it for P(y). – NicNic8 Jun 03 '19 at 20:47
@NicNic8 No, your formula was wrong: $P(x) = \sum_y P(y,x) = \sum_y P(y|x) P(x) $. Or $P(x) = \sum_y P(y|x) P(x) $. You are trying to compute $P(x)$, using $P(x)$. – VansFannel Jun 06 '19 at 07:57
I see. It’s correct now. – NicNic8 Jun 07 '19 at 00:59

If I know $P(x)$ and $P(y|x)$, how can I know $P(y)$?

1 Answers1