Comparing posterior and likelihood estimates for proportions (off topic)

Date: Tue, 05 Dec 2006 17:08:27 -0500
From: "Doran, Harold" <HDoran at air.org>
Sender: r-help-bounces at stat.math.ethz.ch
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Thread-topic: Comparing posterior and likelihood estimates for proportions	(off
 topic)
Thread-index: AccYuQPSLwIrLya5T4ivem8lVU99aQ==

This question is slightly off topic, but I'll use R to try and make it
as relevant as possible. I'm working on a problem where I want to
compare estimates from a posterior distribution with a uniform prior
with those obtained from a frequentist approach. Under these conditions
the estimates should agree.

Specifically, I am asking the question, "What is the probability that
the true proportion of students passing a test is 50% when the observed
proportion for that school is only 38%?"

For my example, there are 100 students in the school and 38 of them
passed an exam. For conjugacy, if we choose a beta prior, then posterior
in this case is also a beta distribution. Now, I believe the a and b
parameters for a beta with a uniform prior is a=1 and b=1, or 1/(1+1)

Here is my R code for the posterior with a flat prior

n <- 100 # Total number of individuals
y <- 38  # Number of successes
a <- 1   # Parameter 1 for Beta prior
b <- 1   # Parameter 2 for Beta prior
theta <- .38 # Proportion passing

pbeta(.50, a + y, b+n-y, lower.tail=FALSE)
[1] 0.008253

Now, the binomial distribution gives

dbinom(50, 100, .38)

[1] 0.0040984

Obviously, the results don't agree. So, I'm wondering if I have 

A) made a computational error
B) have an error in my assumption that the results should agree in this
case

Thanks for any reactions
Harold

Windows XP

version

               _                           
platform       i386-pc-mingw32             
arch           i386                        
os             mingw32                     
system         i386, mingw32               
status                                     
major          2                           
minor          4.0                         
year           2006                        
month          10                          
day            03                          
svn rev        39566                       
language       R                           
version.string R version 2.4.0 (2006-10-03)

 





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