### A Pluto.jl notebook ###
# v0.17.7

using Markdown
using InteractiveUtils

# ╔═╡ e1b18f33-19ba-4fb3-a476-47cfeb6345af
begin
	using CSV, HTTP, DataFrames, Random
	using Turing, Distributions, MCMCChains
	using MLDataUtils: rescale!
	using Plots, StatsPlots
	using StatisticalRethinking
end

# ╔═╡ eef1e512-8579-11ec-1837-0f04dc618fcb
md"""
## Bayesian inference

#### Bayes rule(theorem)

$$P(\theta|D, m) = \frac{P(D|\theta, m) P(\theta|m)}{P(D|m)}$$

-  $P(\theta|D, m)$ ` `` ` **Posterior** of $\theta$ given data D
-  $P(D|\theta, m)$ ` `` ` **Likelihood** of parameters $\theta$ in model m
-  $P(\theta|m)$ ` ` ` ` **Prior** probability of $\theta$
-  $P(D|m)$ ` `` `**Marginal likelihood** equal to $\int P(D|\theta, m)P(\theta|m)\,\mathrm{d}\theta$


###### Prediction : $P(x|D, m) = \int P(x|\theta, D, m)\,P(\theta|D, m)\, \mathrm{d}\theta$
#### 
##### Two simple rule
####
**Sum rule**: $$P(x) = \displaystyle\sum_{y} P(x, y)$$

**Produt rule**: $$P(x, y) = P(y|x)P(x)$$
#
Bayesian approach is very natural way of logical thinking which is flexible and permissive. Furthermore it can allow us to reflect the **uncertainty**.
Bayes rule tell us how to do inference about hypotheses(belief) from data.

Let me tell you a story

`One night you are suddenly awakened by a bright light at your window. You jumped up from bed and look out to see a large object in the sky that can only be described as saucer shaped. You are generally a skeptic and have never believed in alien, but, completely perplexed by the scene outside, you find yourself thinking, Could this be a UFO?!`

![ufo](https://c4.wallpaperflare.com/wallpaper/217/545/167/tv-show-rick-and-morty-ufo-wallpaper-preview.jpg)

`As you look toward the bright light outside, you notice more lights in the area. You also see that the large saucer-shaped object is held up by wires, and notice a camera crew. You hear a loud clap and someone call out "Cut!"`

![film](https://res.cloudinary.com/twenty20/private_images/t_standard-fit/photosp/0b6f5e3b-56be-41cd-84c4-a19a5d23134a/stock-photo-outdoor-behind-the-scenes-video-production-film-production-shooting-video-movie-film-b-roll-film-studio-set-film-industrial-0b6f5e3b-56be-41cd-84c4-a19a5d23134a.jpg)

You have, very likely, instantly changed your mind(**update your belief**) about what you think is happening in this second paragraph. Bayesian reasoning tends to be happen so quickly that you don't have any time to analyze your own thinking.

###### Learning model structure
You may wonder what is m in that bayes rule. you can consider m as so called *hyperparameter* of models. For example, when you meet clustering problems with k-means or mixture model, 'how many clusters in the data?' or 'what is the intrinsic dimensionality of the data?' in PCA. Futhermore, model selection is related to well-known *overfitting*.

###### Priors and Marginal likelihood
Priors can be subjective and there is no right prior. It can be very generous and non-informative prior like Uniform distribution or it can reflect our domain knowledge and belief into the model. For instance, there is no such thing like negative variance so that we set our prior to be lognormal or beta distribution. Prior is not the key factor of bayesian modeling, but informative prior can be useful if we set the reasonable one.

Marginal likelihood is interpreted in information field as $\log_2(\frac{1}{P(D|m)})$ , which is the number of bits of surprise at observing data D under the model m. The problem is this can be computationally intractable and often go very high dimension. That's why we use approximation methods like MCMC and Variational bayesian(inference).
These approximation methods are certainly the paramount subject to bayesian community.

"""

# ╔═╡ 3b7e0e99-5e48-4141-ac36-d855ba7aa0bf
md"""
### Let's see with a linear regression example
"""

# ╔═╡ edec49f8-576f-470a-bcae-2040a688e987
md"""
cons

    consumption of ice cream per head (in pints);
income

    average family income per week (in US Dollars);
price

    price of ice cream (per pint);
temp

    average temperature (in Fahrenheit);
"""

# ╔═╡ 9d556aef-e773-4f8d-b220-4f0de53d5c41
begin
	# Import the "Default" dataset.
	url = "https://raw.githubusercontent.com/delphinH/imagetoy/main/dataset/Icecream.csv"
	data = CSV.read(HTTP.get(url).body, DataFrame)
	
	# Show the first six rows of the dataset.
	first(data, 6)
	
end

# ╔═╡ d36eab23-6ae5-4e54-be78-8901839bbc49
describe(data)

# ╔═╡ 4f7b0614-729f-4745-9f1a-e3eb255ea7cc
size(data)

# ╔═╡ 7c307991-87d0-4a85-a3fc-26ed21ae319d
#data in matrix form
begin
	Xs = data[:, :Temp]
	ys = data[:, :Cons]
end

# ╔═╡ f57e4624-d679-4866-b95d-f0b2bc4436e0
gr()

# ╔═╡ d9e79634-8d9f-460a-bd1e-e3471dc82bcd
p=scatter(Xs, ys, color = :teal, legend=false, fmt = "png") # scatter plot of data

# ╔═╡ 9ad28f14-1545-4141-a192-6d3adc1b4f48
#define linear model with priors
@model function lin_reg(x, y)
	#priors
	α ~ Normal(mean(y), 10)
	β ~ Normal(0, 1)
	σ ~ Exponential(1)

	μ = α .+ β * (x .- mean(x))
	
	y ~ MvNormal(μ, σ)
end
	

# ╔═╡ 19954116-525f-4caa-b66d-748dd581bfaa
model = lin_reg(Xs, ys)

# ╔═╡ 22e0b5a2-663a-49eb-a4f6-fa36353186fc
chain = sample(model, NUTS(), MCMCThreads(), 1000, 2) #approximation sampling

# ╔═╡ 11c077f4-5294-4d26-98fa-a35e3e8116f1
plot(chain)

# ╔═╡ 115341f3-7d46-456f-9d7a-3ed41fbdafcd
summarystats(chain)

# ╔═╡ c6e67570-ab0b-4213-8bbf-f9b0361ba802
mean(chain[:α])

# ╔═╡ 4e8a4bef-c071-4934-91b8-aee0416e342a
begin
	Random.seed!(12)
	alpha = 0.36
	beta = 0.00347
	nid = 50
	L = Uniform(20, 80)
	L_data = rand(L, nid)
end

# ╔═╡ 23d7011f-514a-4eb5-a509-5db22e852b0e
mu = alpha .+ beta * (L_data .- mean(L_data))

# ╔═╡ a0862af8-39c9-426b-a979-3ea1af8ed579
begin
	sd = Normal(0, 0.048)
	sde = rand(sd, nid)
end

# ╔═╡ e0084ff1-0443-418d-a36a-fd67d0c61659
W_data = mu .+ sde #generate data given posterior

# ╔═╡ 16285a48-97b4-4bcf-b69a-40d4d8e3d575
begin
	#compare with observed and generated data
	scatter(Xs, ys, color = :teal, legend=false, fmt = "png")
	scatter!(L_data, W_data)
end

# ╔═╡ 5cea4149-3003-4716-8db8-fb684f8db7c4
md"""
###### Reference
Zoubin Ghahramani: MLSS 2015 Tubingen

Will Kurt: Bayesian statistics the fun way
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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[[LazyStack]]
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[[LeftChildRightSiblingTrees]]
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[[LibCURL_jll]]
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[[LibGit2]]
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[[LibSSH2_jll]]
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[[Libgcrypt_jll]]
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[[Libglvnd_jll]]
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[[Libmount_jll]]
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[[Libtiff_jll]]
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[[LineSearches]]
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[[MCMCChains]]
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[[MKL_jll]]
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[[MbedTLS]]
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[[MbedTLS_jll]]
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[[Measures]]
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[[MicroCollections]]
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[[Missings]]
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[[MonteCarloMeasurements]]
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[[NamedArrays]]
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[[NetworkOptions]]
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[[Ogg_jll]]
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[[OpenBLAS_jll]]
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[[OpenLibm_jll]]
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[[OpenSSL_jll]]
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[[OpenSpecFun_jll]]
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[[Optim]]
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[[Opus_jll]]
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[[PCRE_jll]]
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[[PDMats]]
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[[Parameters]]
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[[ParetoSmooth]]
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[[ParetoSmoothedImportanceSampling]]
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[[Parsers]]
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[[Pixman_jll]]
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[[PlotThemes]]
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[[Plots]]
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[[PooledArrays]]
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[[PositiveFactorizations]]
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[[ProgressLogging]]
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[[Qt5Base_jll]]
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[[QuadGK]]
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version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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