# Setup
source('https://pbreheny.org/7110/f26/notes/fun.R')
source('https://pbreheny.org/7110/f26/notes/radford-neal.R')

# An inconsistent MLE
set.seed(1)
x <- gen(2, 10)
x[7] <- 0.29
theta <- sort(c(seq(-1, 3, len = 99), x))
l <- sapply(theta, function(a) log.lik(x, a))
plotL(theta, exp(l))
abline(v = 2, col = 'gray', lty = 2, lwd = 2)
plotl(theta, l)
abline(v = 2, col = 'gray', lty = 2, lwd = 2)

# An inconsistent MLE: Increasing_the_sample_size
x <- gen(2, 40)
theta <- sort(c(seq(-1, 3, len = 99), x))
l <- sapply(theta, function(a) log.lik(x, a))
plotL(theta, exp(l - max(l)))
abline(v = 2, col = 'gray', lty = 2, lwd = 2)
plotl(theta, l)
abline(v = 2, col = 'gray', lty = 2, lwd = 2)

# Entropy
x <- seq(0, 1, len = 101)
H <- function(p) {
  out <- -p * log(p) - (1 - p) * log(1 - p)
  out[p == 0] <- 0
  out[p == 1] <- 0
  out
}
plot(
  x,
  H(x),
  type = 'l',
  ylim = c(0, log(2)),
  bty = 'n',
  las = 1,
  xlab = expression(pi),
  ylab = expression(H(pi))
)

# Multiple dimensions: Circle figure
theta <- seq(0, 2 * pi, len = 99)
plot(
sin(theta),
cos(theta),
bty = 'l',
las = 1,
type = 'l',
axes = FALSE,
ann = FALSE,
col = 'slateblue',
lwd = 3
)
points(1, 0, col = 'slateblue', pch = 16, cex = 2)
points(0, 1, col = 'slateblue', pch = 16, cex = 2)
points(-1, 0, col = 'slateblue', pch = 16, cex = 2)
points(0, -1, col = 'slateblue', pch = 16, cex = 2)
points(0, 0, col = 'slateblue', pch = 16, cex = 2)
text(0.15, 0.15, expression(theta^'*'))
mtext(expression(theta[1]), side = 1, line = 1)
mtext(expression(theta[2]), side = 2, las = 1, line = 1)