Is P(A|B) a function of A or B?
my doubt is P(A) is function of event A outcome. What about P(A|B)? is it funtion of A outcome? i think no right? P(A|B) is what fraction of P(A,B) in P(B)?? Please elaborate.
I am a beginer. If i am making any mistake in terminology or thinking, please correct.
$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
Another doubt i have is why we wrte the above as
$P(A|B) \propto P(B|A)P(A)$
Kindly explain
On
$P(A)$ is the probability of event A alone (eg the probability of rain on any day)
$P(B)$ is the probability of event B alone (eg the probability of rainbow on any day)
$P(A | B)$ is the probability of event A given event B has occurred. (eg the probability rain on a day with rainbow)
$P(B | A)$ is the probability of event B given event A has occurred. (eg the probability of a rainbow on a day with rain)
All these would be just fractions or numbers.
Regarding $P(A|B)∝P(B|A)P(A)$ when the denominator increases the value ultimately decreases thus directly propotional to $P(B|A)P(A)$
In our context, $P(A|B)\propto{}P(B|A)P(A)$ means that, regarded as a function of the event $A$, $A\mapsto{}P(A|B)$ is proportional to $A\mapsto{}P(B|A)P(A)$.
In practice, in the Bayesian setting, this relationship comes to: the posterior distribution is proportional to the product of the likelihood (regarded as a function of the paramaters to infer) and of the prior distribution.