Recall the globe tossing model from the chapter. Compute and plot the grid approximate posterior distribution for each of the following set of observations. In each case, assume a uniform prior for p.
We want to estimate the probability of Water $p$ as parameter of the model. The parameter is at least 0 and at most 1.
grid_size = 10
step_size = 1.0 / grid_size.to_f
grid = 0.step(by: step_size, to: 1).to_a
prior = grid.each_with_object({}) do |x, hash|
hash[x] = 1.to_f / grid_size.to_f
end
# In contrast to the original notebook,
# we're using Highcharts here:
chart = LazyHighCharts::HighChart.new('2m-1') do |f|
f.title(text: "Prior distribution")
f.yAxis(min: 0, max: 1)
f.series(name: "Prior probability of parameter value", yAxis: 0, data: prior.values)
f.chart({defaultSeriesType: "line"})
end
= high_chart("2m1-1", chart)
factorial = ->(n) do
return 1 if n < 1
n.to_i.downto(1).inject(:*)
end
likelihood = ->(w, l, p) do
(factorial[w+l].to_f / (factorial[w] * factorial[l])).to_f * (p**w) * ((1-p)**l)
end
Now, let's compute the grid aprroximation of the posterior for each of the cases. The difference is only the data input we give in terms of "count of Water" versus "count of Land" of our tossing result given in the exercise.
# For case (1)
w = 3
l = 0
posterior = ->(w, l) do
unstandardized_posterior = grid.each_with_object({}) do |x, hash|
hash[x] = prior[x] * likelihood[w, l, x]
end
unstandardized_posterior.map do |x, y|
standardized = (y.to_f / unstandardized_posterior.values.sum.to_f).round(6)
[x, standardized]
end.to_h
end
chart = LazyHighCharts::HighChart.new('graph') do |f|
f.title(text: "Posterior distribution")
f.yAxis(min: 0, max: 1)
f.series(name: "Posterior probability of parameter value", yAxis: 0, data: posterior[w, l].values)
f.chart({defaultSeriesType: "line"})
end
= high_chart("2m1-2", chart)
# For case (2)
w = 3
l = 1
# For case (3)
w = 5
l = 2