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Copy pathBayesianMandMs.jl
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BayesianMandMs.jl
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### A Pluto.jl notebook ###
# v0.19.14
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 8e03213b-7fa1-4254-8a20-09135ce8caea
begin
using Distributions
using PlutoUI
using Plots
using StatsPlots
using LaTeXStrings
PlutoUI.TableOfContents(title = "Contents")
end
# ╔═╡ 3a40d036-6565-499a-9d35-b5d176773bcc
html"""
<style>
pluto-helpbox {
display: none;
}
</style>
"""
# ╔═╡ 9ef33e78-7b11-46e2-8500-57fc89e591e1
html"<button onclick=present()> present</button>"
# ╔═╡ 068c302e-c364-11ed-0542-af0d94a7d1bf
md"""
# Bayesian exercise with m&m's
Before we get started, let's review some of the notation:
$\theta \rightarrow$ percentage of blue m&m's. This is the parameter we are trying to infer.
$n \rightarrow$ total number of m&m's in the bag
$y \rightarrow$number of blue m&m's in the bag
"""
# ╔═╡ ac0b423c-4037-4002-9377-d1afe864f17f
md"""
## 1. Defining the prior distribution
As discussed, you are going to use the conjugate prior to the binomial distribution --- the beta distribution --- to quantify your prior information on the percentage $\theta$ of blue m&m's. Recall that the beta distribution has two parameters that define its shape, $\alpha$ and $\beta$, and that it has the form:
$p(\theta) \propto \theta^{\alpha -1}(1 - \theta)^{\beta -1}$
To get a feel for how the values of the parameters $\alpha$ and $\beta$ "shape" the distribution, look at the example below.
"""
# ╔═╡ 89ae5cdb-d360-465a-84a5-ffd3427524b2
md"""
Create a few sets of values for ( $\alpha$, $\beta$):
"""
# ╔═╡ f88a54a0-c023-4400-bf87-e9304a5d8f72
begin
# alpha and beta both equal to one
a1 = 1;
b1 = 1;
# alpha and beta are equal to each other but not equal to one
a2 = 5;
b2 = 5;
# alpha < beta
a3 = 1;
b3 = 5;
# alpha > beta
a4 = 5;
b4 = 1;
beta_dist = [Beta(a1,b1),Beta(a2,b2),Beta(a3,b3),Beta(a4,b4)]
end;
# ╔═╡ 2e7ca9bf-6703-4cc1-8356-537968a9cd76
@bind beta_sel Radio(["1" => "𝛼 = $a1, 𝛽 = $b1", "2" => "𝛼 = $a2, 𝛽 = $b2", "3" => "𝛼 = $a3, 𝛽 = $b3", "4" => "𝛼 = $a4, 𝛽 = $b4", "all" => "all together"], default="1")
# ╔═╡ 9d5fb101-a814-45a5-9f62-85b6c78075a4
md"Now plot the different beta distributions that these parameter values create:"
# ╔═╡ bf102ad2-7bd5-4684-89b3-79e5fc24f70e
if beta_sel != "all"
sel = parse(Int, beta_sel)
plot(beta_dist[sel], 0, 1, ylims=(0,5),lw=2, lc=:black, ls=:dashdot, title="Beta Distribution for different parameters",legend=:top, label=L"\alpha = %$a4, \beta = %$b4")
xlabel!(L"\theta")
ylabel!(L"p(\theta)")
else
plot(Beta(a1,b1), linewidth=2, linecolor=:blue, ls=:solid, label=L"\alpha = %$a1, \beta = %$b1")
plot!(Beta(a2,b2),lw=2, lc=:orange, ls=:dash, label=L"\alpha = %$a2, \beta = %$b2")
plot!(Beta(a3,b3), lw=2, lc=:green, label=L"\alpha = %$a3, \beta = %$b3")
plot!(Beta(a4,b4), 0, 1,lw=2, lc=:darkviolet, ls=:dashdot, title="Beta Distribution for different parameters",legend=:top, label=L"\alpha = %$a4, \beta = %$b4")
xlabel!(L"\theta")
ylabel!(L"p(\theta)")
end
# ╔═╡ 9422470f-148a-4afd-8c16-4780e5d313d9
md"""
For reference, the mean and variance of the beta distribution are given by
$\mathbb{E}[\theta] = \frac{\alpha}{\alpha+\beta}$
$Var[\theta] = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha+ \beta+1)}$
"""
# ╔═╡ b0c18fec-8855-4841-8fe8-b857ef927e70
md"
#### Now find $\alpha$ and $\beta$ values that make a beta distribution that looks like the one you sketched.
"
# ╔═╡ f5d29d99-4f50-4a95-90ce-957f273d757e
md"""
mya =
$(@bind mya Slider(1:20, default = 2, show_value=true))
myb =
$(@bind myb Slider(1:20, default = 9, show_value=true))
"""
# ╔═╡ 3c729348-e431-405e-a42b-0ac8b3127c49
plot(Beta(mya, myb), xlim=(0,1), ylim=(0,5), title="Beta Distribution", xlabel = L"\theta", ylabel=L"p(\theta)", linewidth = 2, linecolor=:black, legend=:none)
# ╔═╡ 771acf9d-3e0f-46b5-8a99-7568c9751ac8
md"""
The mean and variance of your prior distribution:
"""
# ╔═╡ 482fdb02-7f11-448e-87af-24f3263ce8b0
begin
priormean = mean(Beta(mya,myb))
L"""$\mathbb{E}[\theta] = %$priormean """
end
# ╔═╡ d7017fa4-8108-43a2-8833-f3df303297d6
begin
priorvar = var(Beta(mya,myb))
L"""$Var[\theta] = %$(priorvar) """
end
# ╔═╡ 3d46052a-2bd6-4dc6-98c4-cc6c21a29722
md"""
## 2. Product of likelihood and prior
Simplify the product of the likelihood (binomial distribution) and the prior (beta distribution), and identify the kernel of the posterior distribution,
$p(\theta|y) \propto \theta^{y+\alpha -1}(1 - \theta)^{n-y+\beta -1}$
The above equation still has the form of a beta distribution, except now the parameters defining the distribution are $(y+\alpha)$ and $(n - y -\beta)$. Recall that the values of $n$ and $y$ come from the data.
"""
# ╔═╡ 2803e3f1-add1-485c-b3e6-e4500ded1b6a
md"""
## 3. Open the m&m's bag and record the data
Collect the data below:
"""
# ╔═╡ 3d7c6468-1612-491a-be64-af8306ca376d
md"""
Total number of m&ms:
ntotal = $(@bind ntotal NumberField(0:100; default=21) )
Number for each colour:
🔵 = $(@bind nblue NumberField(0:100; default=6) ),
🟠 = $(@bind norange NumberField(0:100; default=7) )
🟢 = $(@bind ngreen NumberField(0:100; default=3) ),
🟡 = $(@bind nyellow NumberField(0:100; default=4) )
🔴 = $(@bind nred NumberField(0:100; default=0) ),
🟤 = $(@bind nbrown NumberField(0:100; default=1) )
"""
# ╔═╡ afd34009-36f3-4bc1-81a0-69490503c3b0
if ntotal != (nblue + norange + ngreen + nyellow + nred + nbrown)
print("the total M&Ms don't add up!")
end
# ╔═╡ 007a405b-dd40-4561-82b6-650a7a422095
md"""
## 4. Plot the posterior distribution give $n$, $y$, $\alpha$, $\beta$
Next, we will plot the posterior distribution given the data and the prior, and the mean of the posterior distribution. Recall that the posterior distribution has the kernel of the beta distribution, but is parameterized slightly differently. We already wrote functions to calculate the mean of the beta distribution given
$\alpha$ and $\beta$ values.
"""
# ╔═╡ b93c6dc4-7702-42ca-9bbc-fec6b128ac63
md"""
But now we have $\alpha^\prime$ and $\beta^\prime$ are given by
$\alpha^\prime = y + \alpha$,
and
$\beta^\prime = n - y + \beta$
(If this is confusing, then look at the equation of the posterior distribution again, and compare it to the equation for the beta distribution)."""
# ╔═╡ 6037455c-d0df-4217-b9c4-b4e0431bde32
md""" So to calculate the mean and variance:"""
# ╔═╡ 2faa1c69-6ad7-4928-b172-b1ed12bc92ab
begin
posteriormean = mean(Beta(nblue + mya,ntotal - nblue + myb))
L"""$\mathbb{E}_p[\theta] =\frac{\alpha^\prime}{\alpha^\prime+\beta^\prime} = %$posteriormean """
end
# ╔═╡ 146c6ccd-a01c-4d68-aab4-506ee8f703c0
begin
posteriorvar = var(Beta(nblue + mya,ntotal - nblue + myb))
L"""$Var_p[\theta] = \frac{\alpha^\prime\beta^\prime}{(\alpha^\prime + \beta^\prime)^2(\alpha^\prime+ \beta^ \prime+1)} = %$(posteriorvar) """
end
# ╔═╡ 6a34eb68-f77f-46b1-9632-c93572a8a08b
md"""
We are ready to plot the posterior distribution, add a line showing the mean, and compare it to the prior distribution."""
# ╔═╡ 831f10b1-2712-4ca9-92d7-76d430657a42
begin
plot(Beta(mya, myb), xlim=(0,1), xlabel = L"\theta", ylabel=L"p(\theta)", linewidth = 1, linecolor=:grey, linestyle=:dash, label= L"prior, $\alpha = %$mya, \beta = %$myb$", legendfontsize=11)
plot!(Beta(nblue + mya,ntotal - nblue + myb),xlabel = L"\theta", ylabel=L"p(\theta)", linewidth = 2, linecolor=:black, linestyle=:solid, label= "posterior")
vline!([posteriormean], linecolor=:red, ls=:dashdot, label="posterior mean")
title!("Posterior distribution for the probability of drawing a blue m&m", titlefontsize=12)
xlims!(0,1)
ylims!(0,10)
end
# ╔═╡ ae293a8d-6152-4b01-b3e7-5cc8e0ac4b7b
md"""
## 5. Plot the posterior distribution for different colours.
As an extra step let's plot for a different colour to see their posterior.
"""
# ╔═╡ 4cdb09de-05d7-4816-b5c7-42001792e38f
md"m&m colour:"
# ╔═╡ 314ebd13-e0c8-4483-b22f-047a0dba1c19
@bind colour_sel Radio(["orange" => "🟠", "green" => "🟢", "yellow" => "🟡",
"red" => "🔴", "brown" => "🟤"], default= "orange")
# ╔═╡ c8c3bee7-58d7-434e-80c8-8f4aeebc313d
colours = Dict("orange" => norange, "green" =>ngreen , "yellow" => nyellow,
"red"=> nred, "brown" => nbrown, "blue" => nblue);
# ╔═╡ d34302aa-e9f8-4cc3-af4f-209510899a65
all_colours = ["blue", "orange", "green", "yellow", "red", "brown"];
# ╔═╡ 763837cf-9735-4ae5-b6f6-5f2bbbb3cb82
begin
altermean = Dict();
altervar = Dict();
for i in all_colours
altermean[i] = mean(Beta(colours[i] + mya, ntotal - colours[i] + myb)) ;
altervar[i] = var(Beta(colours[i] + mya, ntotal - colours[i] + myb)) ;
end
end
# ╔═╡ 62f55225-5d37-4034-b34b-5c4a58a16052
begin
plot(Beta(mya, myb), xlim=(0,1), xlabel = L"\theta", ylabel=L"p(\theta)", linewidth = 1, linecolor=:grey, linestyle=:dash, label= L"prior, $\alpha = %$mya, \beta = %$myb$", legendfontsize=11)
plot!(Beta(colours[colour_sel] + mya, ntotal - colours[colour_sel] + myb),xlabel = L"\theta", ylabel=L"p(\theta)", linewidth = 2, linecolor=:black, linestyle=:solid, label= "posterior")
vline!([altermean[colour_sel]], linecolor=:red, ls=:dashdot, label="posterior mean")
title!("Posterior distribution for the probability of drawing: $(colour_sel) m&m", titlefontsize=11)
xlims!(0,1)
ylims!(0,15)
end
# ╔═╡ 357b4b55-3fc5-4a70-9843-c2ab1307e56b
md"""
#### How do the posterior means look like?
"""
# ╔═╡ 24d64111-d206-46db-a8e0-574357486c9d
bar([altermean[i] for i in all_colours],yticks=(1:length(all_colours), all_colours), orientation=:h, yflip=true, legend=:none, xlabel = L"\mathbb{E}_p[\theta]",
title="Posterior mean for each M&M colour" )
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
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LaTeXStrings = "~1.3.0"
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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