Maybe the title of this thread should be:
Arithmetic proof of predicted values or Why you shouldn't let stats lie dormant in your life for 7 years!
I'm boning up on stats and reviewing some introductory text that looked interesting. One that I'm working through is Doing Bayesian Analysis by Kruschke.
In the text ~p52 we're dealing with a coin we have some suspicion of being biased. We are most confident it is a fair coin portioning our our belief of fair and bias as follows: fair -> Θ=.5 and equal suspicion that it is biased high and low Θ=.25 or Θ=.75. If D is our data, and the likelihood function is p(y=Heads | Θ), given that in a simple coin model p(D | Θ) => Θ; and Bayes' rule for discrete variables p(y|x) = [p(x|y)p(y)] / [∑y p(x|y)p(y)] I would think that (just as the author points out) that we could back into the prior belief predicted probability of getting a Heads as (marginal probability of Heads):
p(D=Heads) = ∑,Θ p(D=Heads | Θ )p(Θ)
going something like this:
p(y=H) = p(y=H|Θ=.25)p(Θ=.25)
+ p(y=H|Θ=.50)p(Θ=.50)
+ p(y=H|Θ=.75)p(Θ=.75)
= (.25*.25)+(.5*.5)+(.75*.75)
But here is where my understanding breaks down.
1) My calculation does not deliver the .5 I was looking for, rats!
2) Turning the page I discover that I'm almost there but the author has done this:
p(y=H) = (.25*.25)+(.5*.5)+.(75*.25) <-- huh?
Granted, the product is .5 but why .25?
I can give you more info if needed. If you could help me understand this I'd be only too grateful.
Arithmetic proof of predicted values or Why you shouldn't let stats lie dormant in your life for 7 years!
I'm boning up on stats and reviewing some introductory text that looked interesting. One that I'm working through is Doing Bayesian Analysis by Kruschke.
In the text ~p52 we're dealing with a coin we have some suspicion of being biased. We are most confident it is a fair coin portioning our our belief of fair and bias as follows: fair -> Θ=.5 and equal suspicion that it is biased high and low Θ=.25 or Θ=.75. If D is our data, and the likelihood function is p(y=Heads | Θ), given that in a simple coin model p(D | Θ) => Θ; and Bayes' rule for discrete variables p(y|x) = [p(x|y)p(y)] / [∑y p(x|y)p(y)] I would think that (just as the author points out) that we could back into the prior belief predicted probability of getting a Heads as (marginal probability of Heads):
p(D=Heads) = ∑,Θ p(D=Heads | Θ )p(Θ)
going something like this:
p(y=H) = p(y=H|Θ=.25)p(Θ=.25)
+ p(y=H|Θ=.50)p(Θ=.50)
+ p(y=H|Θ=.75)p(Θ=.75)
= (.25*.25)+(.5*.5)+(.75*.75)
But here is where my understanding breaks down.
1) My calculation does not deliver the .5 I was looking for, rats!
2) Turning the page I discover that I'm almost there but the author has done this:
p(y=H) = (.25*.25)+(.5*.5)+.(75*.25) <-- huh?
Granted, the product is .5 but why .25?
I can give you more info if needed. If you could help me understand this I'd be only too grateful.
