# 21.2: Estimating Posterior Distributions (Section 20.4)


# create a table with results
nResponders <- 64
nTested <- 100

drugDf <- tibble(
outcome = c("improved", "not improved"),
number = c(nResponders, nTested - nResponders)
)

Computing likelihood

likeDf <-
tibble(resp = seq(1,99,1)) %>%
mutate(
presp=resp/100,
likelihood5 = dbinom(resp,100,.5),
likelihood7 = dbinom(resp,100,.7),
likelihood3 = dbinom(resp,100,.3)
)

ggplot(likeDf,aes(resp,likelihood5)) +
geom_line() +
xlab('number of responders') + ylab('likelihood') +
geom_vline(xintercept = drugDf$number[1],color='blue') + geom_line(aes(resp,likelihood7),linetype='dotted') + geom_line(aes(resp,likelihood3),linetype='dashed') Computing marginal likelihood # compute marginal likelihood likeDf <- likeDf %>% mutate(uniform_prior = array(1 / n())) # multiply each likelihood by prior and add them up marginal_likelihood <- sum( dbinom( x = nResponders, # the number who responded to the drug size = 100, # the number tested likeDf$presp # the likelihood of each response
) * likeDf$uniform_prior ) Comuting posterior bayesDf <- tibble( steps = seq(from = 0.01, to = 0.99, by = 0.01) ) %>% mutate( likelihoods = dbinom( x = nResponders, size = 100, prob = steps ), priors = dunif(steps) / length(steps), posteriors = (likelihoods * priors) / marginal_likelihood ) # compute MAP estimate MAP_estimate <- bayesDf %>% arrange(desc(posteriors)) %>% slice(1) %>% pull(steps) ggplot(bayesDf,aes(steps,posteriors)) + geom_line() + geom_line(aes(steps,priors), color='black', linetype='dotted') + xlab('p(respond)') + ylab('posterior probability of the observed data') + annotate( "point", x = MAP_estimate, y = max(bayesDf$posteriors),
shape=9,
size = 3
)

21.2: Estimating Posterior Distributions (Section 20.4) is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.