I'm a bit confused with arranging the Bayes equation to update probability. Say, I have the following data:
$P(\text{blue birds in the whole study area}) = 0.16$; $P(\text{all except blue colored birds in the whole study area}) = 0.84$; $P(\text{birds in NW of the study area is blue}) = 0.22$; and $P(\text{blue birds outside the NW part of the study area}) = 0.11$;
Is that correct if I write:
$P(\text{blue|NW}) = \frac{P(\text{blue}) \cdot P(\text{NW|blue})}{P(\text{blue}) \cdot P(\text{NW|blue}) + P(\neg \text{blue}) \cdot P(\text{blue|}\neg\text{NW})} = \frac{0.16 \cdot 0.22}{0.16 \cdot 0.22 + 0.84 \cdot 0.1} = 0.28$?
Therefore, the probability of finding a blue bird in the NW of the study site has increased from the prior estimate of $16%$ to $28%$.
The confusing part is that we also know: $P(\text{birds in NW that are not blue}) = 0.78$ and if I use this information in the equation as $P(\neg\text{blue|NW})$, then the calculation stands as: $0.16*0.22/(0.16*0.22 + 0.84*0.78) = 0.054$ or $5.4\%$ only (though the probability of finding a blue bird in the NW is supposed to increase)?!?
In sum, which one is correct to use: $P(\text{blue|}\neg\text{NW})$ or $P(\neg\text{blue|NW})$ in this particular case or the whole idea is wrong??
Thanks.
Well you are defining P(blue|NW) in a wrong way! P(blue|NW) can be interpreted as probability of seeing blue bird given you are looking in NW area.
I would say that you are considering events in quite wrong fashion thus getting paradoxical answers! To make understanding better consider following events and then try to get answer: blue -> event that you see blue bird (in entire area) NW -> event that you are looking in NW area Hence P(blue) = 0.16 and P(~blue) = 0.84 Then P(NW) = ?? (We don't know or you didn't state) Then P(blue|NW) = 0.22 P(blue|~NW) = 0.11 and like this define other probabilities and then try to use Bayes inference!
If you still don't get answer please feel free to ask, as I can give you more detailed explanation.