Chapter 7 Exercises
Tea tasting
Solutions
In Example 7.1.1 (tea tasting), instead of assessing whether the taster has any ability at all, assess whether she has a ``practically-significant'' tasting ability. Specifically, what is the posterior probability that she correctly detects the pouring order at least twice as often as she gets it wrong, under
a) a one-parameter uniform prior,
b) a (one-parameter) sceptical prior with a 80% prior probability of p1=0.5, and a uniform distribution on p1>0.5.
model {
for (i in 1:2) {
y[i] ~ dbin(p[i], n[i])
}
p[2] <- 1 - p[1]
# a) Uniform prior
# p[1] ~ dunif(0, 1)
# b) Sceptical prior
p[1] <- theta[pick]
pick ~ dcat(q[])
q[1] <- 0.8; q[2] <- 0.2
theta[1] <- 0.5
theta[2] ~ dunif(0.5, 1)
post <- step(p[1] / p[2] - 2)
}
list(n=c(4,4), y=c(3,1))
a) Under the one-parameter uniform prior, the posterior probability is 62%.
node mean sd MC error 2.5% median 97.5% start sample
p[1] 0.6997 0.1382 0.001521 0.4045 0.7133 0.9243 501 9500
p[2] 0.3003 0.1382 0.001521 0.07567 0.2867 0.5955 501 9500
post 0.6196 0.4855 0.005326 0.0 1.0 1.0 501 9500
b) Under the sceptical prior, the posterior probability is reduced to 21%.
node mean sd MC error 2.5% median 97.5% start sample
p[1] 0.5711 0.1221 0.00118 0.5 0.5 0.8819 501 9500
p[2] 0.4289 0.1221 0.00118 0.1181 0.5 0.5 501 9500
pick 1.313 0.4639 0.004615 1.0 1.0 2.0 501 9500
post 0.2145 0.4105 0.00411 0.0 0.0 1.0 501 9500
theta[2] 0.7439 0.1356 0.001461 0.5135 0.744 0.9836 501 9500