Gaussian distribution.jl

### A Pluto.jl notebook ###
# v0.20.4

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)
    #! format: off
    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
    #! format: on
end

# ╔═╡ db1a7392-825a-454c-a777-954c05a6a310
using RxInfer

# ╔═╡ f4e55044-ff65-4c21-aeb3-2d8be2b97cf4
using Random

# ╔═╡ 452300fb-2cf2-4fab-8923-b847ed4b0c97
using Distributions, Plots, LaTeXStrings, PlutoUI

# ╔═╡ 636389e3-f169-4255-9b4a-8d3426d042b0
using Integrals

# ╔═╡ 5062b49b-34d0-4364-a9b3-7d29119ed599
using PlutoTeachingTools

# ╔═╡ aa4eb90e-2166-4407-97aa-666f309d5e34
md"""
``a + \mu + μ`` and ``\boldsymbol{a} + \boldsymbol{\mu} + \boldsymbol{μ}``
"""

# ╔═╡ 470a0c46-3eae-490b-b13b-5a74eae16a3d
md"""
``\def\S{\boldsymbol{\Sigma}}\def\m{\boldsymbol{\mu}} \S + \m_a`` and ``\foo + \bar``
"""

# ╔═╡ 6108deb0-2c3f-41c2-819d-106f48d210e7
md"""
# This lecture:


## Gaussian distribution is useful

### Occurance in nature, occurance in math, Entropy

### Preserved in common math (addition, ~mult~, fourier transform)

See the parameters of the new distribution





## Gaussian distribution in code

pdf

cdf

integral

RxInfer





## Gaussian distribution in math



pdf * pdf = pdf



## Bayesian inference

Model:

```math
x \sim \mathcal{N}(\mu,\sigma^2)
```

with ``\sigma^2 \in \mathbb{R}_{\geq 0}`` and ``\mu \in \mathbb{R}``.

Given samples from ``x``, we want to infer the value of ``\mu``, when the value of ``\sigma^2`` is known.

As prior, we set:

```math
\mu \sim \mathcal{N}(0, \sigma_{\mu}^2)
```

with a chosen ``\sigma_{\mu}^2 \in \mathbb{R}_{\geq 0}``.

---

ACTUALLY i want to immediately do this for multiple observations, that makes more sense with data

---

So the model is

```math
x \sim \mathcal{N}(\mu,\sigma^2)
```

with 

```math
\mu \sim \mathcal{N}(0, \sigma_{\mu}^2)
```

and given ``\sigma^2, \sigma_{\mu}^2 \in \mathbb{R}_{\geq 0}``


---


The result is:





Derivation:

Easy because you can sum precisions in the posterior.




OKE i will check Bishop for how to properly write this

Would be nice to have a general scheme:

- model
- priors
- data
- knowns
- (bayes)
- posterior


### Inferring sigma (or both)

Optional section

Inferring both might be hard closed form?, also did not manage in RxInfer.



## 
"""

# ╔═╡ a6e21f35-5563-43b4-a6cf-8526be3b5f25


# ╔═╡ 1b26aadc-62a1-4ac1-8cf4-ecdbac7ef63f


# ╔═╡ 2033d8df-7e4c-462f-9c82-f2452460e80a


# ╔═╡ 82d49f7f-cd7f-4406-a883-641069025699


# ╔═╡ 806c47e9-6415-424a-9ee3-67699dbca385


# ╔═╡ 0dd544c8-b7c6-11ef-106b-99e6f84894c3
md"""
# Continuous Data and the Gaussian Distribution

"""

# ╔═╡ a9ab8f29-d72b-472a-b038-c5a884808e2f


# ╔═╡ b738c85f-09f1-4c99-801e-daa90f476fce
convert(NormalWeightedMeanPrecision, Normal(3,0.2))

# ╔═╡ a0427b32-7ce2-46a9-b962-7fab500c9b2f


# ╔═╡ a72dadff-e813-4b18-8fa9-86c9b1515fe2
md"""
The **Gaussian Distribution** is very common in science! What makes this distribution so special?
- It occurs naturally in many processes (see the [Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem)).
- Gaussian distrbutions are easy to work with mathematically.
"""

# ╔═╡ ce5cf62a-bd84-4abe-8cef-07fde003624d
md"""
## Sampling from the Gaussian distribution

In Julia, we can sample from a Normal distribution using the `rand` function and the Distributions.jl package:
"""

# ╔═╡ 41bc3e62-e454-4f14-9d3c-c725a5a93aa8
sample_gaussian_data = rand(Distributions.Normal(5.0, 0.1), 100)

# ╔═╡ 24125436-1b30-4d05-8502-3c8ede5cc00f
md"""
This gives us a Vector of **`100` numbers**, sampled from the distribution **`Normal(5.0, 0.1)`**.

Let's look at these numbers in a histogram:
"""

# ╔═╡ 29f9c097-8bca-4339-a927-28d101cb3a22
histogram(
	sample_gaussian_data;
	bins=3.975:.05:6,
	size=(650,90), legend=nothing, color="black"
)

# ╔═╡ 7fad63f9-c986-438f-99ec-bc8b8bf70ce5
md"""

## Inference



In the code above, we were **given** the numbers ``(\mu, \sigma)``, and we **generated** 100 numbers.


The **inference** task is the opposite: **given** 100 numbers sampled from a distribution, we try to **infer** the parameters ``(\mu, \sigma)`` were used to generate that data.

Let's look at two methods of inference:
- **ML estimator** gives the most likely **value** of ``(\mu, \sigma)``.
- *(later)* **Bayesian inference** gives **posterior distributions** for ``\mu`` and ``\sigma``.


### ML estimator

We can compute the statistical mean and standard deviation of our data:
"""

# ╔═╡ cc17e180-89f1-48b7-a9dd-fa1e7814ceb7
mean(sample_gaussian_data)

# ╔═╡ b9aa17ac-ff44-4899-b15a-e2136eaadbed
std(sample_gaussian_data)

# ╔═╡ fa5e50f5-56b8-48c0-83a0-031fade560d6
md"""

In this simple Gaussian case, the statistical mean and standard deviation are TODO WHAT IS ML


These values give our **best guess** for the parameters that were used to generate the data. But **how sure are we about these estimates?** Can we give a margin of error?


"""

# ╔═╡ 108ba4f6-9292-45a5-a4f3-bb560b43d31b


# ╔═╡ 52f308bd-3740-4e2c-98d8-794d02697770
md"""
## Sampling a Multivariate Normal distribution

Similar to the 1-dimensional case, we can use the `rand` function to sample from a multivariate normal distribution. Instead of a number and a number, we use a `Vector` for the mean, and a `Matrix` for the covariance:
"""

# ╔═╡ 1225c975-bbd5-4f17-82ff-a072b4e5bad9
sample_mv_gaussian_data = rand(
	Distributions.MvNormal([5.0, 7.0], [
		2.0  0.8
		0.8  0.35
	]),
	1000
)

# ╔═╡ c07f2474-601d-4bce-98d1-5970b580094b
md"""
The result is a **`2×100` matrix of numbers**, each column corresponding to one sample.

Let's make a scatter plot. Each point corresponds to one sample.
"""

# ╔═╡ daa65444-810d-49af-8e2d-a0af285cd8d4
scatter(
	sample_mv_gaussian_data[1,:], sample_mv_gaussian_data[2,:];
	color="black", markersize=3, opacity=.5,
	legend=nothing,
)

# ╔═╡ 3a99575c-2773-4b86-814c-465a74d297cf
md"""
## Short inference example

In the example below, we take a subset from our dataset. We can then use Bayesian Inference to infer the parameters (mean, variance) from the dataset.
"""

# ╔═╡ 89c630f6-e7ef-4284-86cd-5eaa3383735e
@bindname data_size Slider(
	[1,10,20,50,100,200,500,1000,2000,5000,10_000,20_000]; 
	show_value=true,
	default=10,
)

# ╔═╡ 3271e5a2-d749-4a7c-801f-f1cb692c74ba


# ╔═╡ 21f842fe-e37a-4154-8561-ef814ccd3e54
md"""
With Bayesian machine learning, we can use this data to **infer** the generating distribution.


As more data is provided, our inference gives a more precise result!
"""

# ╔═╡ 57df5ab6-a9ae-4c8f-8147-78b7e16988d4


# ╔═╡ 9e0d5434-6953-4e7c-884b-9ef607441e1b


# ╔═╡ 7384f4ee-a889-44db-842b-df30ed1e6a49
full_data = rand(
	# 🤫 these parameters are secret! we will try to infer them from the data
	Normal(10, 7),
	20_000
)

# ╔═╡ adac696f-1b5e-4af4-bd45-4a13e8606332
data = full_data[1:data_size]

# ╔═╡ b3f91134-c64c-49cc-b9a6-4da1fe2a37ad
histogram(data; bins=-30.5:30, xlim=(-30,30), size=(650,90), legend=nothing, color="black")

# ╔═╡ bc13ef80-d16f-48c2-b699-3c1043927c8e
md"""
### Three variances

In each of the three charts, take a look at the **variance**. What does the variance mean in this chart?

#### Chart 1 & 2: **data variance**
In the first chart, you can see the variance of our observed variable, which is σ² = 7². 

In the second chart, you see our inference results! We tried to **infer** the standard deviation from our data. 


#### Chart 3: **posterior variance**
In the third chart, you see another variance, the **variance of the posterior for μ**! A small variance of the posterior means **high precision**. Indeed, as we get more data, this variance gets very small – we are very sure about our inference 

"""

# ╔═╡ eed3b2da-8eb6-4462-a852-d39e95fdba48


# ╔═╡ 80612314-bf45-4bb8-9316-a910716ca459
@model function simple_data_model(y, a)
	μ ~ Normal(mean=0, var=a)
	σ = 7
	
    for i in eachindex(y)
        y[i] ~ Normal(mean=μ, var=σ^2)
    end
end

# ╔═╡ 25032652-bf9b-4165-9c2f-15a6298145f4
result = infer(
	model = simple_data_model(a = 1e9),
	data = (y = data,),
)

# ╔═╡ 1c432cb9-bcae-44aa-a32e-1fd142a938b1
result.posteriors

# ╔═╡ 75c3d336-f4dc-43d4-a9fe-d026d37dac66
posterior = convert(Normal, result.posteriors[:μ])

# ╔═╡ 4ffd7bad-3d46-43f5-a0f3-d8a642b98f4c
let
	p = plot(; xlim=(-30,30), ylim=(0,0.06), size=(650,90), legend=nothing, title="Some possible distributions")
	μ_samples = rand(posterior, 20)
	σ_samples = rand(Normal(7.0, 10/data_size), 20)

	# @info "hujh" σ_samples

	for (μ,σ) in zip(μ_samples, σ_samples)
		plot!(p, x -> Distributions.pdf(Normal(μ, max(2.0,σ)), x); color="black", opacity=.3)
		vline!(p, [μ]; color="red", opacity=.3)
	end

	p
end
	
	

# ╔═╡ 0501b18c-cdc9-4772-917f-195c75f05118
plot(x -> Distributions.pdf(posterior, x); xlim=(-30,30), ylim=(0,1), size=(650,90), legend=nothing, color="red", normalize=:pdf, title="Posterior of μ")

# ╔═╡ bd9e3aa4-e6a2-4b4e-bd45-ed9a80374cb4
md"""
## Multivariate Gaussian distribution


"""

# ╔═╡ 0dd817c0-b7c6-11ef-1f8b-ff0f59f7a8ce
md"""
### Preliminaries

Goal 

  * Review of information processing with Gaussian distributions in linear systems

Materials        

  * Mandatory

      * These lecture notes
  * Optional

      * Bishop pp. 85-93
      * [MacKay - 2006 - The Humble Gaussian Distribution](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Mackay-2006-The-humble-Gaussian-distribution.pdf) (highly recommended!)
      * [Ariel Caticha - 2012 - Entropic Inference and the Foundations of Physics](https://github.com/bertdv/BMLIP/blob/master/lessons/notebooks/files/Caticha-2012-Entropic-Inference-and-the-Foundations-of-Physics.pdf), pp.30-34, section 2.8, the Gaussian distribution
  * References

      * [E.T. Jaynes - 2003 - Probability Theory, The Logic of Science](http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf) (best book available on the Bayesian view on probability theory)

"""

# ╔═╡ 06cad4f9-b40c-428f-b2b9-686274749dc9
rand_deterministic(a...) = rand(MersenneTwister(1), a...)

# ╔═╡ 0dd82814-b7c6-11ef-3927-b3ec0b632c31
md"""
### Example Problem

Consider a set of observations ``D=(x_1,…,x_N)`` in ``\mathbb{R}^2`` (see Figure). All observations were generated by the same process. 

We now draw an extra observation ``x_\bullet ∈ \mathbb{R}^2`` from the same data generating process. What is the probability that ``x_\bullet`` lies within the shaded rectangle ``S \subseteq \mathbb{R}^2``?

"""

# ╔═╡ a3638872-847f-47a2-941d-94b9c7d2218b
N = 100

# ╔═╡ cd6cc0a8-eb41-4366-82e4-2b6ee92c7a37
S = ([0,1.],[2,2.])

# ╔═╡ 0aa00f04-a53e-4c2b-9b54-a9f93977c481
in_rectangle(x, rectangle=S) = all(rectangle[1] .<= x .<= rectangle[2]) 

# ╔═╡ 07b6e6be-eed0-4a89-bcdb-52f3ed79f100
@bind try_again CounterButton()

# ╔═╡ ff222c45-8909-489d-b4a2-ec7d9914ede4
generative_dist = let
	# reference 
	try_again

	# the distribution
	MvNormal(
		[0.0, 1.0], 
		[
			0.8 0.5
			0.5 1.0
		]
	)
end

# ╔═╡ 5c9b660b-d44a-402f-a1f8-738911a183b3
D = rand(generative_dist, N)

# ╔═╡ d34233a3-60fe-4ac6-9822-b6097749206b
x_dot = rand(generative_dist)

# ╔═╡ 76cb64ad-e3e7-4af1-b5b3-4bd73f68883b
let
	scatter(D[1,:], D[2,:], marker=:x, markerstrokewidth=3, label=L"D")
	scatter!([x_dot[1]], [x_dot[2]], label=L"x_\bullet")
	plot!(range(0, 2), [1., 1., 1.], fillrange=2, alpha=0.4, color=:gray,label=L"S")
	plot!(xlim=(-3,3), ylim=(-3,3))
end

# ╔═╡ 085e9093-e3d1-41b7-8b3e-ae623809f13d
md"""
### Sampling
"""

# ╔═╡ 0a373d0f-601d-4fce-a1d5-e02732ece9fe
n_samples = 500001

# ╔═╡ b5e23f3d-b35a-40f1-8bd4-58d6128f2e29
count(in_rectangle, eachcol(rand(generative_dist, n_samples))) / n_samples

# ╔═╡ 51fb9204-b480-47f8-bb1e-515b5a0af730
md"""
### Using the `pdf`
"""

# ╔═╡ 042bab9b-e162-436e-894d-70e7ea98fb08
function ∫(f, S)
	prob = IntegralProblem(
		(x, p) -> f(x), 
		S
	)
	solve(prob, HCubatureJL(); abstol=1e-3).u
end

# ╔═╡ 3ca80762-1e7b-46c8-a101-9206de278077
∫(S) do x
	Distributions.pdf(generative_dist, x)
end

# ╔═╡ 0dd835ca-b7c6-11ef-0e33-1329e4ba13d8
md"""
### The Gaussian Distribution

Consider a random (vector) variable ``x \in \mathbb{R}^M`` that is "normally" (i.e., Gaussian) distributed. The *moment* parameterization of the Gaussian distribution is completely specified by its *mean* ``\mu`` and *variance* ``\Sigma`` and given by

```math
p(x | \mu, \Sigma) = \mathcal{N}(x|\mu,\Sigma) \triangleq \frac{1}{\sqrt{(2\pi)^M |\Sigma|}} \,\exp\left\{-\frac{1}{2}(x-\mu)^T \Sigma^{-1} (x-\mu) \right\}\,.
```

where ``|\Sigma| \triangleq \mathrm{det}(\Sigma)`` is the determinant of ``\Sigma``.  

For the scalar real variable ``x \in \mathbb{R}``, this works out to 

```math
p(x | \mu, \sigma^2) =  \frac{1}{\sqrt{2\pi\sigma^2 }} \,\exp\left\{-\frac{(x-\mu)^2}{2 \sigma^2} \right\}\,.
```

"""

# ╔═╡ 0dd84542-b7c6-11ef-3115-0f8b26aeaa5d
md"""
Alternatively, the <a id="natural-parameterization">*canonical* (a.k.a. *natural*  or *information* ) parameterization</a> of the Gaussian distribution is given by

```math
\begin{equation*}
p(x | \eta, \Lambda) = \mathcal{N}_c(x|\eta,\Lambda)  = \exp\left\{ a + \eta^T x - \frac{1}{2}x^T \Lambda x \right\}\,.
\end{equation*}
```

```math
a = -\frac{1}{2} \left( M \log(2 \pi) - \log |\Lambda| + \eta^T \Lambda \eta\right)
```

is the normalizing constant that ensures that ``\int p(x)\mathrm{d}x = 1``.

```math
\Lambda = \Sigma^{-1}
```

is called the *precision matrix*.

```math
\eta = \Sigma^{-1} \mu
```

is the *natural* mean or for clarity often called the *precision-weighted* mean.

"""

# ╔═╡ 0dd8528a-b7c6-11ef-3bc9-eb09c0c530d8
md"""
### Why the Gaussian?

Why is the Gaussian distribution so ubiquitously used in science and engineering? (see also [Jaynes, section 7.14](http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf#page=250), and the whole chapter 7 in his book).

"""

# ╔═╡ 0dd85c94-b7c6-11ef-06dc-7b8797c13fda
md"""
(1) Operations on probability distributions tend to lead to Gaussian distributions:

  * Any smooth function with single rounded maximum, if raised to higher and higher powers, goes into a Gaussian function. (useful in sequential Bayesian inference).
  * The [Gaussian distribution has higher entropy](https://en.wikipedia.org/wiki/Differential_entropy#Maximization_in_the_normal_distribution) than any other with the same variance. 

      * Therefore any operation on a probability distribution that discards information but preserves variance gets us closer to a Gaussian.
      * As an example, see [Jaynes, section 7.1.4](http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf#page=250) for how this leads to the [Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem), which results from performing convolution operations on distributions.

"""

# ╔═╡ 0dd8677a-b7c6-11ef-357f-2328b10f5274
md"""
(2) Once the Gaussian has been attained, this form tends to be preserved. e.g.,   

  * The convolution of two Gaussian functions is another Gaussian function (useful in sum of 2 variables and linear transformations)
  * The product of two Gaussian functions is another Gaussian function (useful in Bayes rule).
  * The Fourier transform of a Gaussian function is another Gaussian function.

"""

# ╔═╡ 0dd86f40-b7c6-11ef-2ae8-a3954469bcee
md"""
### Transformations and Sums of Gaussian Variables

A **linear transformation** ``z=Ax+b`` of a Gaussian variable ``x \sim \mathcal{N}(\mu_x,\Sigma_x)`` is Gaussian distributed as

```math
p(z) = \mathcal{N} \left(z \,|\, A\mu_x+b, A\Sigma_x A^T \right) \tag{SRG-4a}
```

In fact, after a linear transformation ``z=Ax+b``, no matter how ``x`` is distributed, the mean and variance of ``z`` are always given by ``\mu_z = A\mu_x + b``  and ``\Sigma_z = A\Sigma_x A^T``, respectively (see   [probability theory review lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Probability-Theory-Review.ipynb#linear-transformation)). In case ``x`` is not Gaussian, higher order moments may be needed to specify the distribution for ``z``. 

"""

# ╔═╡ 0dd87a3a-b7c6-11ef-2bc2-bf2b4969537c
md"""
The **sum of two independent Gaussian variables** is also Gaussian distributed. Specifically, if ``x \sim \mathcal{N} \left(\mu_x, \Sigma_x \right)`` and ``y \sim \mathcal{N} \left(\mu_y, \Sigma_y \right)``, then the PDF for ``z=x+y`` is given by

```math
\begin{align*}
p(z) &= \mathcal{N}(x\,|\,\mu_x,\Sigma_x) \ast \mathcal{N}(y\,|\,\mu_y,\Sigma_y) \\
  &= \mathcal{N} \left(z\,|\,\mu_x+\mu_y, \Sigma_x +\Sigma_y \right) \tag{SRG-8}
\end{align*}
```

The sum of two Gaussian *distributions* is NOT a Gaussian distribution. Why not?

"""

# ╔═╡ 5978ce29-3fd1-44c1-a3cf-37fd336e2a35
a = rand(Normal(4.3, 2), 51000)

# ╔═╡ 2c31df74-726b-48ff-a402-5a9f67392060
b = rand(Normal(3.0, 2.0), 51000)

# ╔═╡ e90979ad-cfd8-4fe8-9bd3-a434e394a8f2
md"""
The product of two Guassian distributed variables is **not** Gaussian distributed!
"""

# ╔═╡ 272ae2be-7784-4100-a03f-4b3aa400dd18
hist(data; kwargs...) = histogram(data; bins=-30.5:30, xlim=(-30,30), size=(650,90), legend=nothing, kwargs...)

# ╔═╡ 0837d757-9beb-4956-836c-a2c487bea9e3
hist(a; color="blue")

# ╔═╡ 09eb45b0-4117-46ae-adaf-fe1e82578478
hist(b; color="red")

# ╔═╡ 8554d2d0-0acf-47a1-92e1-9b025031a5d6
let
	sum = a .+ b

	hist(sum; color="purple", normalize=:pdf)
	plot!(x -> Distributions.pdf(Normal(mean(sum),std(sum)),x))
end

# ╔═╡ 3c454c77-2320-49b8-b95c-e869f9e9dcd9
let
	prod = a .* b

	hist(prod; color="purple", normalize=:pdf)
	plot!(x -> Distributions.pdf(Normal(mean(prod),std(prod)),x))
end

# ╔═╡ 0dd88110-b7c6-11ef-0b82-2ffe13a68cad
md"""
### Example: Gaussian Signals in a Linear System

<p style="text-align:center;"><img src="./figures/fig-linear-system.png" width="400px"></p>

Given independent variables

```math
x \sim \mathcal{N}(\mu_x,\sigma_x^2)
```

and ``y \sim \mathcal{N}(\mu_y,\sigma_y^2)``, what is the PDF for ``z = A\cdot(x -y) + b`` ? (for answer, see [Exercises](http://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/exercises/Exercises-The-Gaussian-Distribution.ipynb))

"""

# ╔═╡ 0dd88a84-b7c6-11ef-133c-3d85f0703c19
md"""
Think about the role of the Gaussian distribution for stochastic linear systems in relation to what sinusoidals mean for deterministic linear system analysis.

"""

# ╔═╡ c50f6bf0-a1e5-4bd6-a6a2-7201a4302c2f
TODO(md"Think about? ")

# ╔═╡ d93fd4bb-7bf2-453b-97be-602ee494a786


# ╔═╡ 9612754a-7ed3-4d4d-b4a9-9db3c2059271


# ╔═╡ 5dd56dd1-3db0-4b5e-bf74-1ba47f04b58a
TODO(md"""
What is defined here? Is ``\theta`` a stochastic var?
""")

# ╔═╡ 0dd890ee-b7c6-11ef-04b7-e7671227d8cb
md"""
### Bayesian Inference for the Gaussian

Let's estimate a constant ``\theta`` from one 'noisy' measurement ``x`` about that constant. 

We assume the following measurement equations (the tilde ``\sim`` means: 'is distributed as'):

```math
\begin{align*}
x &= \theta + \epsilon \\
\epsilon &\sim \mathcal{N}(0,\sigma^2)
\end{align*}
```

Also, let's assume a Gaussian prior for ``\theta``

```math
\begin{align*}
\theta &\sim \mathcal{N}(\mu_0,\sigma_0^2) \\
\end{align*}
```

"""

# ╔═╡ 0dd89b6e-b7c6-11ef-2525-73ee0242eb91
md"""
##### Model specification

Note that you can rewrite these specifications in probabilistic notation as follows:

```math
\begin{align*}
    p(x|\theta) &=  \mathcal{N}(x|\theta,\sigma^2) \\
    p(\theta) &=\mathcal{N}(\theta|\mu_0,\sigma_0^2)
\end{align*}
```

"""

# ╔═╡ 0dd8b5d6-b7c6-11ef-1eb9-4f4289261e79
md"""
(**Notational convention**). Note that we write ``\epsilon \sim \mathcal{N}(0,\sigma^2)`` but not ``\epsilon \sim \mathcal{N}(\epsilon | 0,\sigma^2)``, and we write  ``p(\theta) =\mathcal{N}(\theta|\mu_0,\sigma_0^2)`` but not ``p(\theta) =\mathcal{N}(\mu_0,\sigma_0^2)``. 

"""

# ╔═╡ 0dd8c024-b7c6-11ef-3ca4-f9e8286cbb64
md"""
##### Inference

For simplicity, we assume that the variance ``\sigma^2`` is given and will proceed to derive a Bayesian posterior for the mean ``\theta``. The case for Bayesian inference of ``\sigma^2`` with a given mean is [discussed in the optional slides](#inference-for-precision).

"""

# ╔═╡ 0dd8d976-b7c6-11ef-051f-4f6cb3db3d1b
md"""
Let's do Bayes rule for the posterior PDF ``p(\theta|x)``. 

```math
\begin{align*}
p(\theta|x)  &= \frac{p(x|\theta) p(\theta)}{p(x)} \propto p(x|\theta) p(\theta)  \\
    &= \mathcal{N}(x|\theta,\sigma^2) \mathcal{N}(\theta|\mu_0,\sigma_0^2)   \\
    &\propto \exp \left\{   -\frac{(x-\theta)^2}{2\sigma^2} - \frac{(\theta-\mu_0)^2}{2\sigma_0^2} \right\}  \\
    &\propto \exp \left\{ \theta^2 \cdot \left( -\frac{1}{2 \sigma_0^2} - \frac{1}{2\sigma^2}  \right)  + \theta \cdot  \left( \frac{\mu_0}{\sigma_0^2} + \frac{x}{\sigma^2}\right)   \right\} \\
    &= \exp\left\{ -\frac{\sigma_0^2 + \sigma^2}{2 \sigma_0^2 \sigma^2} \left( \theta - \frac{\sigma_0^2 x +  \sigma^2 \mu_0}{\sigma^2 + \sigma_0^2}\right)^2  \right\} 
\end{align*}
```

which we recognize as a Gaussian distribution w.r.t. ``\theta``. 

"""

# ╔═╡ 0dd8df66-b7c6-11ef-011a-8d90bba8e2cd
md"""
(Just as an aside,) this computational 'trick' for multiplying two Gaussians is called **completing the square**. The procedure makes use of the equality 

```math
ax^2+bx+c_1 = a\left(x+\frac{b}{2a}\right)^2+c_2
```

"""

# ╔═╡ 0dd8ea56-b7c6-11ef-0116-691b99023eb5
md"""
In particular, it follows that the posterior for ``\theta`` is

```math
\begin{equation*}
    p(\theta|x) = \mathcal{N} (\theta |\, \mu_1, \sigma_1^2)
\end{equation*}
```

where

```math
\begin{align*}
  \frac{1}{\sigma_1^2}  &= \frac{\sigma_0^2 + \sigma^2}{\sigma^2 \sigma_0^2} = \frac{1}{\sigma_0^2} + \frac{1}{\sigma^2}  \\
  \mu_1   &= \frac{\sigma_0^2 x +  \sigma^2 \mu_0}{\sigma^2 + \sigma_0^2} = \sigma_1^2 \, \left(  \frac{1}{\sigma_0^2} \mu_0 + \frac{1}{\sigma^2} x \right) 
\end{align*}
```

"""

# ╔═╡ 36a731be-8795-4bb3-9d0c-6b9f5abe2a53
TODO(
	md"""
	Ik vind dit voorbeeld eigenlijk niet zo nuttig, misschien gelijk naar het voorbeeld met meerdere waarnemingen? Dat is nuttig voor data, deze niet
	"""
)

# ╔═╡ 0dd8f1fe-b7c6-11ef-3386-e37f33577577
md"""
### (Multivariate) Gaussian Multiplication

So, multiplication of two Gaussian distributions yields another (unnormalized) Gaussian with

  * posterior precision equals **sum of prior precisions**
  * posterior precision-weighted mean equals **sum of prior precision-weighted means**

"""

# ╔═╡ 0dd8fbe2-b7c6-11ef-1f78-63dfd48146fd
md"""
As we just saw, a Gaussian prior, combined with a Gaussian likelihood, make Bayesian inference analytically solvable (!):

```math
\begin{equation*}
\underbrace{\text{Gaussian}}_{\text{posterior}}
 \propto \underbrace{\text{Gaussian}}_{\text{likelihood}} \times \underbrace{\text{Gaussian}}_{\text{prior}}
\end{equation*}
```

"""

# ╔═╡ 0dd90644-b7c6-11ef-2fcf-2948d45f43bb
md"""
<a id="Gaussian-multiplication"></a>In general, the multiplication of two multi-variate Gaussians over ``x`` yields an (unnormalized) Gaussian over ``x``:

```math
\begin{equation*}
\boxed{\mathcal{N}(x|\mu_a,\Sigma_a) \cdot \mathcal{N}(x|\mu_b,\Sigma_b) = \underbrace{\mathcal{N}(\mu_a|\, \mu_b, \Sigma_a + \Sigma_b)}_{\text{normalization constant}} \cdot \mathcal{N}(x|\mu_c,\Sigma_c)} \tag{SRG-6}
\end{equation*}
```

where

```math
\begin{align*}
\Sigma_c^{-1} &= \Sigma_a^{-1} + \Sigma_b^{-1} \\
\Sigma_c^{-1} \mu_c &= \Sigma_a^{-1}\mu_a + \Sigma_b^{-1}\mu_b
\end{align*}
```

"""

# ╔═╡ 0dd91b7a-b7c6-11ef-1326-7bbfe5ac16bf
md"""
Check out that normalization constant ``\mathcal{N}(\mu_a|\, \mu_b, \Sigma_a + \Sigma_b)``. Amazingly, this constant can also be expressed by a Gaussian!

"""

# ╔═╡ 0dd9264e-b7c6-11ef-0fa9-d3e4e5053654
md"""
```math
\Rightarrow
```

Note that Bayesian inference is trivial in the [*canonical* parameterization of the Gaussian](#natural-parameterization), where we would get

```math
\begin{align*}
 \Lambda_c &= \Lambda_a + \Lambda_b  \quad &&\text{(precisions add)}\\
 \eta_c &= \eta_a + \eta_b \quad &&\text{(precision-weighted means add)}
\end{align*}
```

This property is an important reason why the canonical parameterization of the Gaussian distribution is useful in Bayesian data processing. 

"""

# ╔═╡ 0dd93204-b7c6-11ef-143e-2b7b182f8be1
md"""
### Code Example: Product of Two Gaussian PDFs

Let's plot the exact product of two Gaussian PDFs as well as the normalized product according to the above derivation.

"""

# ╔═╡ 0dd93236-b7c6-11ef-2656-b914f13c4ecd
let
	d1 = Normal(0, 1) # μ=0, σ^2=1
	d2 = Normal(3, 2) # μ=3, σ^2=4
	
	# Calculate the parameters of the product d1*d2
	s2_prod = (d1.σ^-2 + d2.σ^-2)^-1
	m_prod = s2_prod * ((d1.σ^-2)*d1.μ + (d2.σ^-2)*d2.μ)
	d_prod = Normal(m_prod, sqrt(s2_prod)) # Note that we neglect the normalization constant.
	
	# Plot stuff
	x = range(-4, stop=8, length=100)
	plot(x, pdf.(d1,x), label=L"\mathcal{N}(0,1)", fill=(0, 0.1))                                   # Plot the first Gaussian
	plot!(x, pdf.(d2,x), label=L"\mathcal{N}(3,4)", fill=(0, 0.1))                                  # Plot the second Gaussian
	plot!(x, pdf.(d1,x) .* pdf.(d2,x), label=L"\mathcal{N}(0,1) \mathcal{N}(3,4)", fill=(0, 0.1))   # Plot the exact product
	plot!(x, pdf.(d_prod,x), label=L"Z^{-1} \mathcal{N}(0,1) \mathcal{N}(3,4)", fill=(0, 0.1))      # Plot the normalized Gaussian product
end

# ╔═╡ 0dd93f08-b7c6-11ef-3ad5-97d01baafa7c
md"""
### Bayesian Inference with multiple Observations

Now consider that we measure a data set ``D = \{x_1, x_2, \ldots, x_N\}``, with measurements

```math
\begin{aligned}
x_n &= \theta + \epsilon_n \\
\epsilon_n &\sim \mathcal{N}(0,\sigma^2)
\end{aligned}
```

and the same prior for ``\theta``:

```math
\theta \sim \mathcal{N}(\mu_0,\sigma_0^2) \\
```

Let's derive the distribution ``p(x_{N+1}|D)`` for the next sample . 

"""

# ╔═╡ 0dd94cb4-b7c6-11ef-0d42-5f5f3b071afa
md"""
##### inference

First, we derive the posterior for ``\theta``:

```math
\begin{align*}
p(\theta|D) \propto  \underbrace{\mathcal{N}(\theta|\mu_0,\sigma_0^2)}_{\text{prior}} \cdot \underbrace{\prod_{n=1}^N \mathcal{N}(x_n|\theta,\sigma^2)}_{\text{likelihood}}
\end{align*}
```

which is a multiplication of ``N+1`` Gaussians and is therefore also Gaussian-distributed.

"""

# ╔═╡ 0dd96092-b7c6-11ef-08b6-99348eca8529
md"""
Using the property that precisions and precision-weighted means add when Gaussians are multiplied, we can immediately write the posterior 

```math
p(\theta|D) = \mathcal{N} (\theta |\, \mu_N, \sigma_N^2)
```

as 

```math
\begin{align*}
  \frac{1}{\sigma_N^2}  &= \frac{1}{\sigma_0^2} + \sum_n  \frac{1}{\sigma^2} \qquad &\text{(B-2.142)} \\
  \mu_N   &= \sigma_N^2 \, \left( \frac{1}{\sigma_0^2} \mu_0 + \sum_n \frac{1}{\sigma^2} x_n  \right) \qquad &\text{(B-2.141)}
\end{align*}
```

"""

# ╔═╡ 0dd992ee-b7c6-11ef-3add-cdf7452bc514
md"""
##### application: prediction of future sample

We now have a posterior for the model parameters. Let's write down what we know about the next sample ``x_{N+1}``.

```math
\begin{align*}
p(x_{N+1}|D) &= \int p(x_{N+1}|\theta) p(\theta|D)\mathrm{d}\theta \\
  &= \int \mathcal{N}(x_{N+1}|\theta,\sigma^2) \mathcal{N}(\theta|\mu_N,\sigma^2_N) \mathrm{d}\theta \\
  &= \int \mathcal{N}(\theta|x_{N+1},\sigma^2) \mathcal{N}(\theta|\mu_N,\sigma^2_N) \mathrm{d}\theta \\
  &= \int  \mathcal{N}(x_{N+1}|\mu_N, \sigma^2_N +\sigma^2 ) \mathcal{N}(\theta|\cdot,\cdot)\mathrm{d}\theta \tag{use SRG-6} \\
  &= \mathcal{N}(x_{N+1}|\mu_N, \sigma^2_N +\sigma^2 ) \underbrace{\int \mathcal{N}(\theta|\cdot,\cdot)\mathrm{d}\theta}_{=1} \\
  &=\mathcal{N}(x_{N+1}|\mu_N, \sigma^2_N +\sigma^2 )
\end{align*}
```

"""

# ╔═╡ 0dd9a40a-b7c6-11ef-2864-8318d8f3d827
md"""
Uncertainty about ``x_{N+1}`` involved both uncertainty about the parameter (``\sigma_N^2``) and observation noise ``\sigma^2``.

"""

# ╔═╡ 14169498-8a04-446a-bf4d-41f5026a2d4c
TODO(md"``\mu_N`` is not the same as ``\mu_N``")

# ╔═╡ 0dd9b71a-b7c6-11ef-2c4a-a3f9e7f2bc87
md"""
### Maximum Likelihood Estimation for the Gaussian

In order to determine the *maximum likelihood* estimate of ``\theta``, we let ``\sigma_0^2 \rightarrow \infty`` (leads to uniform prior for ``\theta``), yielding $ \frac{1}{\sigma_N^2} = \frac{N}{\sigma^2}$ and consequently

```math
\begin{align*}
  \mu_{\text{ML}}  = \left.\mu_N\right\vert_{\sigma_0^2 \rightarrow \infty} = \sigma_N^2 \, \left(   \frac{1}{\sigma^2}\sum_n  x_n  \right) = \frac{1}{N} \sum_{n=1}^N x_n 
  \end{align*}
```

"""

# ╔═╡ 0dd9ccfa-b7c6-11ef-2379-2967a0b4ad07
md"""
As expected, having an expression for the maximum likelihood estimate, it is now possible to rewrite the (Bayesian) posterior mean for ``\theta`` as 

```math
\begin{align*}
  \underbrace{\mu_N}_{\text{posterior}}   &= \sigma_N^2 \, \left( \frac{1}{\sigma_0^2} \mu_0 + \sum_n \frac{1}{\sigma^2} x_n  \right) \\
  &= \frac{\sigma_0^2 \sigma^2}{N\sigma_0^2 + \sigma^2} \, \left( \frac{1}{\sigma_0^2} \mu_0 + \sum_n \frac{1}{\sigma^2} x_n  \right) \\
  &= \frac{ \sigma^2}{N\sigma_0^2 + \sigma^2}   \mu_0 + \frac{N \sigma_0^2}{N\sigma_0^2 + \sigma^2} \mu_{\text{ML}}   \\
  &= \underbrace{\mu_0}_{\text{prior}} + \underbrace{\underbrace{\frac{N \sigma_0^2}{N \sigma_0^2 + \sigma^2}}_{\text{gain}}\cdot \underbrace{\left(\mu_{\text{ML}} - \mu_0 \right)}_{\text{prediction error}}}_{\text{correction}}\tag{B-2.141}
\end{align*}
```

"""

# ╔═╡ 0dd9db78-b7c6-11ef-1005-73e5d7a4fc4b
md"""
Hence, the posterior mean always lies somewhere between the prior mean ``\mu_0`` and the maximum likelihood estimate (the "data" mean) ``\mu_{\text{ML}}``.

"""

# ╔═╡ 0dd9ed22-b7c6-11ef-19e5-038711d75259
md"""
### Conditioning and Marginalization of a Gaussian

Let ``z = \begin{bmatrix} x \\ y \end{bmatrix}`` be jointly normal distributed as

```math
\begin{align*}
p(z) &= \mathcal{N}(z | \mu, \Sigma) 
  =\mathcal{N} \left( \begin{bmatrix} x \\ y \end{bmatrix} \left| \begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}, 
  \begin{bmatrix} \Sigma_x & \Sigma_{xy} \\ \Sigma_{yx} & \Sigma_y \end{bmatrix} \right. \right)
\end{align*}
```

"""

# ╔═╡ 0dd9fb08-b7c6-11ef-0350-c529776149da
md"""
Since covariance matrices are by definition symmetric, it follows that ``\Sigma_x`` and ``\Sigma_y`` are symmetric and ``\Sigma_{xy} = \Sigma_{yx}^T``.

"""

# ╔═╡ 0dda09f4-b7c6-11ef-2429-377131c95b8e
md"""
Let's factorize ``p(z) = p(x,y)`` as ``p(x,y) = p(y|x) p(x)`` through conditioning and marginalization.

"""

# ╔═╡ 0dda16ce-b7c6-11ef-3b84-056673f08e89
md"""
```math
\begin{equation*}
\text{conditioning: }\boxed{ p(y|x) = \mathcal{N}\left(y\,|\,\mu_y + \Sigma_{yx}\Sigma_x^{-1}(x-\mu_x),\, \Sigma_y - \Sigma_{yx}\Sigma_x^{-1}\Sigma_{xy} \right)}
\end{equation*}
```

"""

# ╔═╡ 0dda22f4-b7c6-11ef-05ec-ef5e23c533a1
md"""
```math
\begin{equation*}
\text{marginalization: } \boxed{ p(x) = \mathcal{N}\left( x|\mu_x, \Sigma_x \right)}
\end{equation*}
```

"""

# ╔═╡ 0dda301e-b7c6-11ef-0188-0d6a9782abfa
md"""
**proof**: in Bishop pp.87-89

"""

# ╔═╡ 0dda3d8e-b7c6-11ef-0e2e-9942afc06c32
md"""
Hence, conditioning and marginalization in Gaussians leads to Gaussians again. This is very useful for applications to Bayesian inference in jointly Gaussian systems.

"""

# ╔═╡ 0dda4b3a-b7c6-11ef-17c2-5f5ccd912eee
md"""
With a natural parameterization of the Gaussian ``p(z) = \mathcal{N}_c(z|\eta,\Lambda)`` with precision matrix ``\Lambda = \Sigma^{-1} = \begin{bmatrix} \Lambda_x & \Lambda_{xy} \\ \Lambda_{yx} & \Lambda_y \end{bmatrix}``,  the conditioning operation results in a simpler result, see Bishop pg.90, eqs. 2.96 and 2.97. 

"""

# ╔═╡ 0dda6b2e-b7c6-11ef-14ee-25d9a3acaf11
md"""
As an exercise, interpret the formula for the conditional mean (``\mathbb{E}[y|x]=\mu_y + \Sigma_{yx}\Sigma_x^{-1}(x-\mu_x)``) as a prediction-correction operation.

"""

# ╔═╡ 0dda770e-b7c6-11ef-2988-397f0085c3a3
md"""
### Code Example: Joint, Marginal, and Conditional Gaussian Distributions

Let's plot of the joint, marginal, and conditional distributions.

"""

# ╔═╡ 0dda774a-b7c6-11ef-2750-4960eef0932b
using Plots, LaTeXStrings, Distributions

# Define the joint distribution p(x,y)
μ = [1.0; 2.0]
Σ = [0.3 0.7;
     0.7 2.0]
joint = MvNormal(μ,Σ)

# Define the marginal distribution p(x)
marginal_x = Normal(μ[1], sqrt(Σ[1,1]))

# Plot p(x,y)
x_range = y_range = range(-2,stop=5,length=1000)
joint_pdf = [ pdf(joint, [x_range[i];y_range[j]]) for  j=1:length(y_range), i=1:length(x_range)]
plot_1 = heatmap(x_range, y_range, joint_pdf, title = L"p(x, y)")

# Plot p(x)
plot_2 = plot(range(-2,stop=5,length=1000), pdf.(marginal_x, range(-2,stop=5,length=1000)), title = L"p(x)", label="", fill=(0, 0.1))

# Plot p(y|x = 0.1)
x = 0.1
conditional_y_m = μ[2]+Σ[2,1]*inv(Σ[1,1])*(x-μ[1])
conditional_y_s2 = Σ[2,2] - Σ[2,1]*inv(Σ[1,1])*Σ[1,2]
conditional_y = Normal(conditional_y_m, sqrt.(conditional_y_s2))
plot_3 = plot(range(-2,stop=5,length=1000), pdf.(conditional_y, range(-2,stop=5,length=1000)), title = L"p(y|x = %$x)", label="", fill=(0, 0.1))
plot(plot_1, plot_2, plot_3, layout=(1,3), size=(1200,300))


# ╔═╡ 0dda842e-b7c6-11ef-24b6-19e2fad91333
md"""
As is clear from the plots, the conditional distribution is a renormalized slice from the joint distribution.

"""

# ╔═╡ 0dda9086-b7c6-11ef-2455-732cd6d69407
md"""
### Example: Conditioning of Gaussian

Consider (again) the system 

```math
\begin{align*}
p(x\,|\,\theta) &= \mathcal{N}(x\,|\,\theta,\sigma^2) \\
p(\theta) &= \mathcal{N}(\theta\,|\,\mu_0,\sigma_0^2)
\end{align*}
```

"""

# ╔═╡ 0dda9d36-b7c6-11ef-1ab4-7b341b8cfcdf
md"""
Let ``z = \begin{bmatrix} x \\ \theta \end{bmatrix}``. The distribution for ``z`` is then given by (Exercise)

```math
p(z) = p\left(\begin{bmatrix} x \\ \theta \end{bmatrix}\right) = \mathcal{N} \left( \begin{bmatrix} x\\ 
  \theta  \end{bmatrix} 
  \,\left|\, \begin{bmatrix} \mu_0\\ 
  \mu_0\end{bmatrix}, 
         \begin{bmatrix} \sigma_0^2+\sigma^2  & \sigma_0^2\\ 
         \sigma_0^2 &\sigma_0^2 
  \end{bmatrix} 
  \right. \right)
```

"""

# ╔═╡ 0ddaa9f4-b7c6-11ef-01a0-a78e551e6414
md"""
Direct substitution of the rule for Gaussian conditioning leads to the <a id="precision-weighted-update">posterior</a> (derivation as an Exercise):

```math
\begin{align*}
p(\theta|x) &= \mathcal{N} \left( \theta\,|\,\mu_1, \sigma_1^2 \right)\,,
\end{align*}
```

with

```math
\begin{align*}
K &= \frac{\sigma_0^2}{\sigma_0^2+\sigma^2} \qquad \text{($K$ is called: Kalman gain)}\\
\mu_1 &= \mu_0 + K \cdot (x-\mu_0)\\
\sigma_1^2 &= \left( 1-K \right) \sigma_0^2  
\end{align*}
```

"""

# ╔═╡ 0ddab62e-b7c6-11ef-1b65-df9e3d1087d6
md"""
```math
\Rightarrow
```

Moral: For jointly Gaussian systems, we can do inference simply in one step by using the formulas for conditioning and marginalization.

"""

# ╔═╡ 0ddae00e-b7c6-11ef-33f2-b565ce8fc3ba
md"""
### Recursive Bayesian Estimation for Adaptive Signal Processing

Consider the signal ``x_t=\theta+\epsilon_t``, where ``D_t= \left\{x_1,\ldots,x_t\right\}`` is observed *sequentially* (over time).

**Problem**: Derive a recursive algorithm for ``p(\theta|D_t)``, i.e., an update rule for (posterior) ``p(\theta|D_t)`` based on (prior) ``p(\theta|D_{t-1})`` and (new observation) ``x_t``.

"""

# ╔═╡ 0ddafb7a-b7c6-11ef-3c3f-c9fa7af39c92
md"""
##### Model specification

Let's define the estimate after ``t`` observations (i.e., our *solution* ) as ``p(\theta|D_t) = \mathcal{N}(\theta\,|\,\mu_t,\sigma_t^2)``.

We define the joint distribution for ``\theta`` and ``x_t``, given background ``D_{t-1}``, by

```math
\begin{align*} p(x_t,\theta \,|\, D_{t-1}) &= p(x_t|\theta) \, p(\theta|D_{t-1}) \\
  &= \underbrace{\mathcal{N}(x_t\,|\, \theta,\sigma^2)}_{\text{likelihood}} \, \underbrace{\mathcal{N}(\theta\,|\,\mu_{t-1},\sigma_{t-1}^2)}_{\text{prior}}
\end{align*}
```

"""

# ╔═╡ 0ddb085c-b7c6-11ef-34fd-6b1b18a95ff1
md"""
##### Inference

Use Bayes rule,

```math
\begin{align*}
p(\theta|D_t) &= p(\theta|x_t,D_{t-1}) \\
  &\propto p(x_t,\theta | D_{t-1}) \\
  &= p(x_t|\theta) \, p(\theta|D_{t-1}) \\
  &= \mathcal{N}(x_t|\theta,\sigma^2) \, \mathcal{N}(\theta\,|\,\mu_{t-1},\sigma_{t-1}^2) \\
  &= \mathcal{N}(\theta|x_t,\sigma^2) \, \mathcal{N}(\theta\,|\,\mu_{t-1},\sigma_{t-1}^2) \;\;\text{(note this trick)}\\
  &= \mathcal{N}(\theta|\mu_t,\sigma_t^2) \;\;\text{(use Gaussian multiplication formula SRG-6)}
\end{align*}
```

with

```math
\begin{align*}
K_t &= \frac{\sigma_{t-1}^2}{\sigma_{t-1}^2+\sigma^2} \qquad \text{(Kalman gain)}\\
\mu_t &= \mu_{t-1} + K_t \cdot (x_t-\mu_{t-1})\\
\sigma_t^2 &= \left( 1-K_t \right) \sigma_{t-1}^2 
\end{align*}
```

"""

# ╔═╡ 0ddb163a-b7c6-11ef-2b06-a1d6677b7191
md"""
This linear *sequential* estimator of mean and variance in Gaussian observations is called a **Kalman Filter**.

<!–- - The new observation ``x_t`` 'updates' the old estimate ``\mu_{t-1}`` by a quantity that is proportional to the *innovation* (or *residual*)  ``\left( x_t - \mu_{t-1} \right)``. –-> 

"""

# ╔═╡ 0ddb2302-b7c6-11ef-1f50-27711dbe4d33
md"""
The so-called Kalman gain ``K_t`` serves as a "learning rate" (step size) in the parameter update equation ``\mu_t = \mu_{t-1} + K_t \cdot (x_t-\mu_{t-1})``. Note that *you* don't need to choose the learning rate. Bayesian inference computes its own (optimal) learning rates.  

"""

# ╔═╡ 0ddb2fa0-b7c6-11ef-3ac5-8979f2a0a00c
md"""
Note that the uncertainty about ``\theta`` decreases over time (since ``0<(1-K_t)<1``). If we assume that the statistics of the system do not change (stationarity), each new sample provides new information about the process, so the uncertainty decreases. 

"""

# ╔═╡ 0ddb3c34-b7c6-11ef-2a77-895cbc5796f3
md"""
Recursive Bayesian estimation as discussed here is the basis for **adaptive signal processing** algorithms such as Least Mean Squares (LMS) and Recursive Least Squares (RLS). Both RLS and LMS are special cases of Recursive Bayesian estimation.

"""

# ╔═╡ 0ddb4b54-b7c6-11ef-121d-5d00e547debd
md"""
### Code Example: Kalman Filter

Let's implement the Kalman filter described above. We'll use it to recursively estimate the value of ``\theta`` based on noisy observations.

"""

# ╔═╡ 0ddb4bb4-b7c6-11ef-373a-ab345190363a
using Plots, Distributions

n = 100         # specify number of observations
θ = 2.0         # true value of the parameter we would like to estimate
noise_σ2 = 0.3  # variance of observation noise

observations = noise_σ2 * randn(n) .+ θ

function perform_kalman_step(prior :: Normal, x :: Float64, noise_σ2 :: Float64)
    K = prior.σ / (noise_σ2 + prior.σ)          # compute the Kalman gain
    posterior_μ = prior.μ + K*(x - prior.μ)     # update the posterior mean
    posterior_σ = prior.σ * (1.0 - K)           # update the posterior standard deviation
    return Normal(posterior_μ, posterior_σ)     # return the posterior distribution
end

post_μ = fill!(Vector{Float64}(undef,n + 1), NaN)     # means of p(θ|D) over time
post_σ2 = fill!(Vector{Float64}(undef,n + 1), NaN)    # variances of p(θ|D) over time

prior = Normal(0, 1)    # specify the prior distribution (you can play with the parameterization of this to get a feeling of how the Kalman filter converges)

post_μ[1] = prior.μ     # save prior mean and variance to show these in plot
post_σ2[1] = prior.σ

for (i, x) in enumerate(observations)                           # note that this loop demonstrates Bayesian learning on streaming data; we update the prior distribution using observation(s), after which this posterior becomes the new prior for future observations
    posterior = perform_kalman_step(prior, x, noise_σ2)         # compute the posterior distribution given the observation
    post_μ[i + 1] = posterior.μ                                 # save the mean of the posterior distribution
    post_σ2[i + 1] = posterior.σ                                # save the variance of the posterior distribution
    prior = posterior                                           # the posterior becomes the prior for future observations
end

obs_scale = collect(2:n+1)
scatter(obs_scale, observations, label=L"D", )  
post_scale = collect(1:n+1)                                                         # scatter the observations
plot!(post_scale, post_μ, ribbon=sqrt.(post_σ2), linewidth=3, label=L"p(θ | D_t)")  # lineplot our estimated means of intermediate posterior distributions
plot!(post_scale, θ*ones(n + 1), linewidth=2, label=L"θ")                           # plot the true value of θ



# ╔═╡ 0ddb7294-b7c6-11ef-0585-3f1a218aeb42
md"""
The shaded area represents 2 standard deviations of posterior ``p(\theta|D)``. The variance of the posterior is guaranteed to decrease monotonically for the standard Kalman filter.

"""

# ╔═╡ 0ddb9904-b7c6-11ef-3808-35b8ee37dd04
md"""
### <a id="product-of-gaussians">Product of Normally Distributed Variables</a>

(We've seen that) the sum of two Gausssian distributed variables is also Gaussian distributed.

"""

# ╔═╡ 0ddba9ee-b7c6-11ef-3148-9db5fbb13d77
md"""
Has the *product* of two Gaussian distributed variables also a Gaussian distribution?

"""

# ╔═╡ 0ddbba2e-b7c6-11ef-04cf-1119024af1d1
md"""
**No**! In general this is a difficult computation. As an example, let's compute ``p(z)`` for ``Z=XY`` for the special case that ``X\sim \mathcal{N}(0,1)`` and ``Y\sim \mathcal{N}(0,1)``.

```math
\begin{align*}
p(z) &= \int_{X,Y} p(z|x,y)\,p(x,y)\,\mathrm{d}x\mathrm{d}y \\
  &= \frac{1}{2 \pi}\int  \delta(z-xy) \, e^{-(x^2+y^2)/2} \, \mathrm{d}x\mathrm{d}y \\
  &=  \frac{1}{\pi} \int_0^\infty \frac{1}{x} e^{-(x^2+z^2/x^2)/2} \, \mathrm{d}x \\
  &= \frac{1}{\pi} \mathrm{K}_0( \lvert z\rvert )\,.
\end{align*}
```

where  ``\mathrm{K}_n(z)`` is a [modified Bessel function of the second kind](http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html).

"""

# ╔═╡ 0ddbc78a-b7c6-11ef-2ce4-f76fa4153e4b
md"""
### Code Example: Product of Gaussian Distributions

We plot ``p(Z=XY)`` and ``p(X)p(Y)`` for ``X\sim\mathcal{N}(0,1)`` and ``Y \sim \mathcal{N}(0,1)`` to give an idea of how these distributions differ.

"""

# ╔═╡ 0ddbc7c8-b7c6-11ef-004f-8bfaa5f29eba
using Plots, Distributions, SpecialFunctions, LaTeXStrings
X = Normal(0,1)
Y = Normal(0,1)
pdf_product_std_normals(z::Vector) = (besselk.(0, abs.(z))./π)
range1 = collect(range(-4,stop=4,length=100))
plot(range1, pdf.(X, range1), label=L"p(X)=p(Y)=\mathcal{N}(0,1)", fill=(0, 0.1))
plot!(range1, pdf.(X,range1).*pdf.(Y,range1), label=L"p(X)*p(Y)", fill=(0, 0.1))
plot!(range1, pdf_product_std_normals(range1), label=L"p(Z=X*Y)", fill=(0, 0.1))

# ╔═╡ 0ddbd3ce-b7c6-11ef-20e1-070d736f7b95
md"""
In short, Gaussian-distributed variables remain Gaussian in linear systems, but this is not the case in non-linear systems. 

"""

# ╔═╡ 0ddbf246-b7c6-11ef-16a5-bbf396f80915
md"""
### Solution to Example Problem

We apply maximum likelihood estimation to fit a 2-dimensional Gaussian model (``m``) to data set ``D``. Next, we evaluate ``p(x_\bullet \in S | m)`` by (numerical) integration of the Gaussian pdf over ``S``: ``p(x_\bullet \in S | m) = \int_S p(x|m) \mathrm{d}x``.

"""

# ╔═╡ 0ddbf278-b7c6-11ef-20f5-7ffd3163b14f
using HCubature, LinearAlgebra, Plots, Distributions# Numerical integration package
# Maximum likelihood estimation of 2D Gaussian
N = length(sum(D,dims=1))
μ = 1/N * sum(D,dims=2)[:,1]
D_min_μ = D - repeat(μ, 1, N)
Σ = Hermitian(1/N * D_min_μ*D_min_μ')
m = MvNormal(μ, convert(Matrix, Σ));

contour(range(-3, 4, length=100), range(-3, 4, length=100), (x, y) -> pdf(m, [x, y]))

# Numerical integration of p(x|m) over S:
(val,err) = hcubature((x)->pdf(m,x), [0., 1.], [2., 2.])
println("p(x⋅∈S|m) ≈ $(val)")

scatter!(D[1,:], D[2,:], marker=:x, markerstrokewidth=3, label=L"D")
scatter!([x_dot[1]], [x_dot[2]], label=L"x_\bullet")
plot!(range(0, 2), [1., 1., 1.], fillrange=2, alpha=0.4, color=:gray, label=L"S")

# ╔═╡ 0ddc02d6-b7c6-11ef-284e-018c7895536e
md"""
### Summary

A **linear transformation** ``z=Ax+b`` of a Gaussian variable ``x \sim \mathcal{N}(\mu_x,\Sigma_x)`` is Gaussian distributed as

```math
p(z) = \mathcal{N} \left(z \,|\, A\mu_x+b, A\Sigma_x A^T \right) 
```

Bayesian inference with a Gaussian prior and Gaussian likelihood leads to an analytically computable Gaussian posterior, because of the **multiplication rule for Gaussians**:

```math
\begin{equation*}
\mathcal{N}(x|\mu_a,\Sigma_a) \cdot \mathcal{N}(x|\mu_b,\Sigma_b) = \underbrace{\mathcal{N}(\mu_a|\, \mu_b, \Sigma_a + \Sigma_b)}_{\text{normalization constant}} \cdot \mathcal{N}(x|\mu_c,\Sigma_c)
\end{equation*}
```

where

```math
\begin{align*}
\Sigma_c^{-1} &= \Sigma_a^{-1} + \Sigma_b^{-1} \\
\Sigma_c^{-1} \mu_c &= \Sigma_a^{-1}\mu_a + \Sigma_b^{-1}\mu_b
\end{align*}
```

**Conditioning and marginalization** of a multivariate Gaussian distribution yields Gaussian distributions. In particular, the joint distribution

```math
\mathcal{N} \left( \begin{bmatrix} x \\ y \end{bmatrix} \left| \begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}, 
  \begin{bmatrix} \Sigma_x & \Sigma_{xy} \\ \Sigma_{yx} & \Sigma_y \end{bmatrix} \right. \right)
```

can be decomposed as

```math
\begin{align*}
 p(y|x) &= \mathcal{N}\left(y\,|\,\mu_y + \Sigma_{yx}\Sigma_x^{-1}(x-\mu_x),\, \Sigma_y - \Sigma_{yx}\Sigma_x^{-1}\Sigma_{xy} \right) \\
p(x) &= \mathcal{N}\left( x|\mu_x, \Sigma_x \right)
\end{align*}
```

Here's a nice [summary of Gaussian calculations](https://github.com/bertdv/AIP-5SSB0/raw/master/lessons/notebooks/files/RoweisS-gaussian_formulas.pdf) by Sam Roweis. 

"""

# ╔═╡ 0ddc1028-b7c6-11ef-1eec-6d72e52f4431
md"""
## <center> OPTIONAL SLIDES</center>

"""

# ╔═╡ 0ddc1c2e-b7c6-11ef-00b6-e98913a96420
md"""
### <a id="inference-for-precision">Inference for the Precision Parameter of the Gaussian</a>

Again, we consider an observed data set ``D = \{x_1, x_2, \ldots, x_N\}`` and try to explain these data by a Gaussian distribution.

"""

# ╔═╡ 0ddc287e-b7c6-11ef-1f72-910e6e7b06bb
md"""
We discussed earlier Bayesian inference for the mean with a given variance. Now we will derive a posterior for the variance if the mean is given. (Technically, we will do the derivation for a precision parameter ``\lambda = \sigma^{-2}``, since the discussion is a bit more straightforward for the precision parameter).

"""

# ╔═╡ 0ddc367a-b7c6-11ef-38f9-09fb462987dc
md"""
##### model specification

The likelihood for the precision parameter is 

```math
\begin{align*}
p(D|\lambda) &= \prod_{n=1}^N \mathcal{N}\left(x_n \,|\, \mu, \lambda^{-1} \right) \\
  &\propto \lambda^{N/2} \exp\left\{ -\frac{\lambda}{2}\sum_{n=1}^N \left(x_n - \mu \right)^2\right\} \tag{B-2.145}
\end{align*}
```

"""

# ╔═╡ 0ddc4796-b7c6-11ef-2156-8b3d6899a8c0
md"""
The conjugate distribution for this function of ``\lambda`` is the [*Gamma* distribution](https://en.wikipedia.org/wiki/Gamma_distribution), given by

```math
p(\lambda\,|\,a,b) = \mathrm{Gam}\left( \lambda\,|\,a,b \right) \triangleq \frac{1}{\Gamma(a)} b^{a} \lambda^{a-1} \exp\left\{ -b \lambda\right\}\,, \tag{B-2.146}
```

where ``a>0`` and ``b>0`` are known as the *shape* and *rate* parameters, respectively. 

<img src="./figures/B-fig-2.13.png" width="600px">

(Bishop fig.2.13). Plots of the Gamma distribution ``\mathrm{Gam}\left( \lambda\,|\,a,b \right) $ for different values of $a`` and ``b``.

"""

# ╔═╡ 0ddc55f6-b7c6-11ef-3975-9d92d7e3feca
md"""
The mean and variance of the Gamma distribution evaluate to ``\mathrm{E}\left( \lambda\right) = \frac{a}{b}`` and ``\mathrm{var}\left[\lambda\right] = \frac{a}{b^2}``. 

"""

# ╔═╡ 0ddc7284-b7c6-11ef-0c0c-bd949a5ef015
md"""
##### inference

We will consider a prior ``p(\lambda) = \mathrm{Gam}\left( \lambda\,|\,a_0, b_0\right)``, which leads by Bayes rule to the posterior

```math
\begin{align*}
p(\lambda\,|\,D) &\propto \underbrace{\lambda^{N/2} \exp\left\{ -\frac{\lambda}{2}\sum_{n=1}^N \left(x_n - \mu \right)^2\right\} }_{\text{likelihood}} \cdot \underbrace{\frac{1}{\Gamma(a_0)} b_0^{a_0} \lambda^{a_0-1} \exp\left\{ -b_0 \lambda\right\}}_{\text{prior}} \\
  &\propto \mathrm{Gam}\left( \lambda\,|\,a_N,b_N \right) 
\end{align*}
```

with

```math
\begin{align*}
a_N &= a_0 + \frac{N}{2} \qquad &&\text{(B-2.150)} \\
b_N &= b_0 + \frac{1}{2}\sum_n \left( x_n-\mu\right)^2 \qquad &&\text{(B-2.151)}
\end{align*}
```

"""

# ╔═╡ 0ddc7f40-b7c6-11ef-0253-6342085f708a
md"""
Hence the **posterior is again a Gamma distribution**. By inspection of B-2.150 and B-2.151, we deduce that we can interpret ``2a_0`` as the number of a priori (pseudo-)observations. 

"""

# ╔═╡ 0ddc8b70-b7c6-11ef-13cb-3daa72032cf9
md"""
Since the most uninformative prior is given by ``a_0=b_0 \rightarrow 0``, we can derive the **maximum likelihood estimate** for the precision as

```math
\lambda_{\text{ML}} = \left.\mathrm{E}\left[ \lambda\right]\right\vert_{a_0=b_0\rightarrow 0} = \left. \frac{a_N}{b_N}\right\vert_{a_0=b_0\rightarrow 0} = \frac{N}{\sum_{n=1}^N \left(x_n-\mu \right)^2}
```

"""

# ╔═╡ 0ddc9aac-b7c6-11ef-3d8e-f5a5d0e715f8
md"""
In short, if we do density estimation with a Gaussian distribution ``\mathcal{N}\left(x_n\,|\,\mu,\sigma^2 \right)`` for an observed data set ``D = \{x_1, x_2, \ldots, x_N\}``, the <a id="ML-for-Gaussian">maximum likelihood estimates</a> for ``\mu`` and ``\sigma^2`` are given by

```math
\begin{align*}
\mu_{\text{ML}} &= \frac{1}{N} \sum_{n=1}^N x_n \qquad &&\text{(B-2.121)} \\
\sigma^2_{\text{ML}} &= \frac{1}{N} \sum_{n=1}^N \left(x_n - \mu_{\text{ML}} \right)^2 \qquad &&\text{(B-2.122)}
\end{align*}
```

These estimates are also known as the *sample mean* and *sample variance* respectively. 

"""

# ╔═╡ 0ddc9ae8-b7c6-11ef-33a5-771f934e6ae8
md"""
open("../../styles/aipstyle.html") do f
    display("text/html", read(f, String))
end
""" |> TODO

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uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.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"

[[deps.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"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "47e45cd78224c53109495b3e324df0c37bb61fbe"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+0"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "fee57a273563e273f0f53275101cd41a8153517a"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+1"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "1a74296303b6524a0472a8cb12d3d87a78eb3612"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+1"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b9ead2d2bdb27330545eb14234a2e300da61232e"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+1"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "555d1076590a6cc2fdee2ef1469451f872d8b41b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6e50f145003024df4f5cb96c7fce79466741d601"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.56.3+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0ba42241cb6809f1a278d0bcb976e0483c3f1f2d"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+1"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b70c870239dc3d7bc094eb2d6be9b73d27bef280"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.44+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

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

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

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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