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September 8th, 2019, 08:35 AM   #1
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Bayesian inference problem

I thought this concept was simple so I haven't studied it in depth, but apparently I was wrong. This is from "Thinking Fast and Slow".

A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city.

- 85% of the cabs in the city are Green (15% are Blue)
- A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is P(cab involved in accident was Blue)?
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September 8th, 2019, 11:57 AM   #2
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I've never studied Bayesian inference, but a quick glance at the Wikipedia page tells me this might be consistent with what it was saying. Please do let me know if this is the answer he gave.

P(cab was green, identified as green) = 0.85*0.8 = 0.68
P(cab was green, identified as blue) = 0.85*0.2 = 0.17
P(cab was blue, identified as blue) = 0.15*0.8 = 0.12
P(cab was blue, identified as green) = 0.15*0.2 = 0.03
P(total) = 0.68+0.17+0.12+0.03 = 1 (phew!)

P(cab was blue, given constraint identified as blue) = 0.12/(0.12+0.17) = 0.414.
Basically, combining the high reliability of the witness with the low probability of it being a blue cab in the first place.
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September 8th, 2019, 04:13 PM   #3
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Yeah that looks right. I can usually figure out stuff like this intuitively, but I was stuck on this. I like to be able to "feel" the answer rather than just plugging in the digits. I'm still having a hard time "feeling" .12/(.12+.17), but I'm working on it thanks
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September 8th, 2019, 06:53 PM   #4
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When you have eliminated the impossible, whatever remains, however improbable, must be the truth.

Either he saw a green cab and was wrong about the colour, or he saw a blue cab and was right. Out of the two normally unlikely occurrences, the latter is a little less likely.


...Now, what if, on top of all that, someone determined there was a 5% chance the witness was lying?
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