It would be better in my opinion if you had a bullet list with the different options and you could click on the chosen answer.
You should add a way for people to send you comments about the specific examples
For instance, I think a proper explanation of the base fallacy rate should include the proper bayesian analysis. In fact, let’s try to do that with the example that you give:
Numbers taken from a quick google search. We are going to suppose that this takes place in US
Farmer population in the states ~2.5 * 10^6
Number of symphony orchestra 1224
Number of trumpets in a symphony orchestra 4
Total number of trumpets in a symphony orchestra ~5000
Odds farmer/trumpet player = 500:1
This gives us the prior odds. We have to multiply the prior odds by the likelihood ratio. This is tricky, but let’s put some numbers anyway just for the sake of explanation. For instance, we could assume that 80% of trumpet players are keen on Opera, 50% enjoy visiting museums and 20% grew up playing chess. We will assume also that for farmers, the numbers are 10% opera, 20% museums, 5% chess (this is assuming independence among the different factors, which is probably a bit of a stretch)
For farmers: 10% enjoy opera, 20% enjoy visiting museums, 5% grew up playing chess = 0.001
Good point about the independence, I added a note. Do you think it would be possible to come up with a better estimate somehow of the likelihood ratio?
Fanastic idea! I have just signed up.
Two comments:
It would be better in my opinion if you had a bullet list with the different options and you could click on the chosen answer.
You should add a way for people to send you comments about the specific examples
For instance, I think a proper explanation of the base fallacy rate should include the proper bayesian analysis. In fact, let’s try to do that with the example that you give:
Numbers taken from a quick google search. We are going to suppose that this takes place in US
Farmer population in the states ~2.5 * 10^6
Number of symphony orchestra 1224
Number of trumpets in a symphony orchestra 4
Total number of trumpets in a symphony orchestra ~5000
Odds farmer/trumpet player = 500:1
This gives us the prior odds. We have to multiply the prior odds by the likelihood ratio. This is tricky, but let’s put some numbers anyway just for the sake of explanation. For instance, we could assume that 80% of trumpet players are keen on Opera, 50% enjoy visiting museums and 20% grew up playing chess. We will assume also that for farmers, the numbers are 10% opera, 20% museums, 5% chess (this is assuming independence among the different factors, which is probably a bit of a stretch)
For farmers: 10% enjoy opera, 20% enjoy visiting museums, 5% grew up playing chess = 0.001
Trumpet players = 80% opers, 50% enjoy visiting museums, 20% chess = 0.08
L_r farmer/trumpet = 0.001/0.08 = 0.0125 (this is the likelihood ratio)
Posterior = 500:1 * 0.001:0.08 = 6.25
In this case, we can see that it is around 6 times more likely that the person is a farmer than a trumpeter.
However, if in our model we make the number of farmers who like opera just 1%, in this case, the posterior would favour the trumpeters.
I doubt these are independent.
I realise you have the math right further down, but this should be ~5000. (I assume typo)
yes typo. Thanks! Corrected
Good point about the independence, I added a note. Do you think it would be possible to come up with a better estimate somehow of the likelihood ratio?
I would do additional conditioning. So P(opera | farmer), P(museum | opera, farmer), P(chess | museum, opera, farmer), etc.
My guess would it would look something like:
P(opera | farmer) = 5% (does anyone actually like opera?)
P(museums | opera, farmer) = 95%
P(chess | m, o, f) = 40%
So 5% * 95% * 40% = 1.9% of farmers...
P(o | t) = 80%
P(m | o, t) = 50%
P(c | m, o, t) = 20%
So 80% * 50% * 20% = 8% of trumpet players...
Which is a likelihood ratio ~.25 so I end up with something like 125 to 1 that we’re talking to a farmer.
Thank you for the feedback!
In terms of convenience (doing the exercises on the site), I’m looking into it.
Your comment about allowing people to send comments for specific examples is a great idea.
Would you mind if I included your explanation of the example on the site?
please do! let me know if I can lend a hand somehow ;)