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??



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.

  • $\begingroup$ Thanks for your response, Akshay. I'm actually looking for seeing/finding blue bird given I'm looking in NW area i.e. P(blue|NW). The idea is that I know the prior information in a regional scale (blue = 0.16, ~blue = 0.84) and now I want to update using the prior at local scale (NW). Could you please explain using the Bayes formula in more detail with the data I provided you with? I don't get what you meant by P(NW), however. $\endgroup$
    – ToNoY
    Jan 5 '13 at 19:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.