Recall that if x is the number of successes in n trials of a binomial setting with probability of success p on each trial, the likelihood function, p( x p), can be expressed as
p(x|π) = πx (1 – π)n-x
Consider four possible outcomes of a ganzfeld experiment: x = 9 hits in n = 25 sessions, x = 18 hits in n = 50 sessions, x = 36 hits in n = 100 sessions, and x = 72 hits in n = 200 sessions.
a. For each of the four possible experimental outcomes, what is the observed hit rate?
b. Construct a graph of the likelihood function corresponding to each of the four experimental outcomes. What do you observe about the shape of the likelihood function as the sample size increases? Does it become more flat or more peaked? What do you observe about the likely values of p as the sample size increases? Does the most likely value of p appear to change for the different likelihood functions?
c. Now let’s consider parameter settings for a beta prior distribution for π for the skeptic: α = 27 and β = 81. Graph the posterior distribution of π for the skeptic. Now examine how the posterior estimate of π is related to the “peakedness” of the likelihood function. If the likelihood function has a sharp peak, does the prior for π have much of an effect on the posterior estimate p*? What does this tell you about the sensitivity of the posterior to the choice of prior if the likelihood function is very peaked? Very spread out?