I was reading through a paper, when I encounter p(a|s). What does this mean? It was in context with log probability but I cant find this notation anywhere.
What does this notation mean p(x|y)?
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probability
notation
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1The probability of event $a$, given that event $b$ occurs. For example, when rolling a die, $P(X\text{ is even}\mid X\text{ is prime})=\frac13$ – 2017-02-28
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0Specifically $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$. See [wiki](https://en.wikipedia.org/wiki/Conditional_probability). – 2017-02-28
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0Usually that notation is used for conditional probabilities. It is the likelihood of event $a$ happening, given that we know $s$ about it. – 2017-02-28
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0@Miemels, feel free to answer your own question, now that you know the answer! – 2017-02-28
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0I will add that $p(x\mid y)$ is too often used as shortened for $p_{X\mid Y}(x\mid y)$, especially in engineering and physics texts. $~$ In such case it usually means conditional probability *mass* function, which equals $\mathsf P(X=x\mid Y=y)$. $~$ ( Only occasionally is $p$ used to represent a probability *density* function. ) – 2017-03-01
3 Answers
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That is standard notation for conditional probability.
See here for example for further information.
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Like everyone else said, it's just notation for conditional probability. Here is the textbook that I used last semester in my probability class. Flip to page 141 and you can read all about the conditional probability, both in discrete and continuous cases.
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The notation P(x|y) means P(x) given event y has occurred, this notation is used in conditional probability.
There are two cases if x and y are dependent or if x and y are independent.
Case 1) P(x|y) = P(x&y)/P(y)
Case 2) P(x|y) = P(x)
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0"There are two cases if x and y are dependent or if x and y are independent" Actually there is only one case, when P(x|y) = P(x&y)/P(y). – 2017-02-28
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0Indeed, the "second case" is just a simplification; because independence means that $\mathsf P(x \& y)=\mathsf P(x)\cdot\mathsf P(y)$ . – 2017-02-28