Deriving posterior probability from frequencies

#1
I'm working through an introductory Bayesian stats text: Doing Bayesian Analysis. Having been away from stats for a while and having never working from a Bayesian framework it's a bit challenging.

Right now I'm stumbling on a problem about deriving a posterior probability from a small amount of neuroscience frequency data about a part of the brain being active during certain activities. The data and target answer can be found in Poldrack(2006)[PDF]

The exercise asks:

Given the table showing brain activity during certain activities; suppose a new study finds that the ROI (region of interest) is activated. If the prior probability that the task involves language processing is .5, what is the posterior probability, given that the ROI is activated? A study done by Poldrack(2006) reports that the posterior probability is 0.69.

The frequency data:
Code:
	                Language Study	Not Language Study	Total
Activated	        166         	199	                365
Not Activated	    703	            2154	            2857
Total	            869	            2353	            3222
And I calculated the conjoint probabilities:
Code:
	        LS+	    LS-	    Marginal Activated
Activated+	0.052	0.062	0.113
Activated-	0.218	0.669	0.887
Marginal LS	0.270	0.730	1
But when I apply Bayes' rule to calculate p(LS+|A+) I get .227

from this:
p(A+|LS+)p(LS+=.5) / p(A+)

(.052*.5) / .113

Would someone be so kind as to point out what I'm misunderstanding?
 
Last edited:

vinux

Dark Knight
#2
I guess there is some problem with your calculation.

p(LS+ | A+)=p(A+|LS+)p(LS+) / p(A+)
=p(A+|LS+)p(LS+) / [ p(A+|LS+)p(LS+) +p(A+|LS-)p(LS-) ]
=\( \dfrac{0.052/0.270 * 0.5}{0.052/0.270 * 0.5 + 0.062/0.730 * 0.5}\)
=0.6939671
 

hlsmith

Less is more. Stay pure. Stay poor.
#6
In medicine, there are positive or negative likelihoods and also likelihood ratios. Looking these terms up can help lay the foundation for problems like these. If the rest of you post was going to say "all!"