{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "Continuous Data and the Gaussian Distribution\n", "=======" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Preliminaries\n", "\n", "- Goal \n", " - Review of information processing with Gaussian distributions in linear systems\n", "- Materials \n", " - Mandatory\n", " - These lecture notes\n", " - Optional\n", " - Bishop pp. 85-93 \n", " - [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!)\n", " - [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\n", " - References\n", " - [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)\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Example Problem\n", "\n", "Consider a set of observations $D=\\{x_1,…,x_N\\}$ in the 2-dimensional plane (see Figure). All observations were generated by the same process. We now draw an extra observation $x_\\bullet = (a,b)$ from the same data generating process. What is the probability that $x_\\bullet$ lies within the shaded rectangle $S$?\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "using Pkg; Pkg.activate(\"../.\"); Pkg.instantiate();\n", "using IJulia; try IJulia.clear_output(); catch _ end" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "using Distributions, Plots, LaTeXStrings\n", "\n", "N = 100\n", "generative_dist = MvNormal([0,1.], [0.8 0.5; 0.5 1.0])\n", "\n", "D = rand(generative_dist, N) # Generate observations from generative_dist\n", "scatter(D[1,:], D[2,:], marker=:x, markerstrokewidth=3, label=L\"D\")\n", "x_dot = rand(generative_dist) # Generate x∙\n", "scatter!([x_dot[1]], [x_dot[2]], label=L\"x_\\bullet\")\n", "plot!(range(0, 2), [1., 1., 1.], fillrange=2, alpha=0.4, color=:gray,label=L\"S\")\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### The Gaussian Distribution \n", "\n", "- 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\n", "$$\n", "p(x | \\mu, \\Sigma) = \\mathcal{N}(x|\\mu,\\Sigma) \\triangleq \\frac{1}{\\sqrt{(2\\pi)^M |\\Sigma|}} \\,\\exp\\left\\{-\\frac{1}{2}(x-\\mu)^T \\Sigma^{-1} (x-\\mu) \\right\\}\\,.\n", "$$\n", "where $|\\Sigma| \\triangleq \\mathrm{det}(\\Sigma)$ is the determinant of $\\Sigma$. \n", " - For the scalar real variable $x \\in \\mathbb{R}$, this works out to \n", "$$\n", "p(x | \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2 }} \\,\\exp\\left\\{-\\frac{(x-\\mu)^2}{2 \\sigma^2} \\right\\}\\,.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Alternatively, the _canonical_ (a.k.a. _natural_ or _information_ ) parameterization of the Gaussian distribution is given by\n", "$$\\begin{equation*}\n", "p(x | \\eta, \\Lambda) = \\mathcal{N}_c(x|\\eta,\\Lambda) = \\exp\\left\\{ a + \\eta^T x - \\frac{1}{2}x^T \\Lambda x \\right\\}\\,.\n", "\\end{equation*}$$\n", " - $a = -\\frac{1}{2} \\left( M \\log(2 \\pi) - \\log |\\Lambda| + \\eta^T \\Lambda \\eta\\right)$ is the normalizing constant that ensures that $\\int p(x)\\mathrm{d}x = 1$.\n", " - $\\Lambda = \\Sigma^{-1}$ is called the *precision matrix*.\n", " - $\\eta = \\Sigma^{-1} \\mu$ is the _natural_ mean or for clarity often called the *precision-weighted* mean." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Why the Gaussian?\n", "\n", "- 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)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- (1) Operations on probability distributions tend to lead to Gaussian distributions:\n", " - Any smooth function with single rounded maximum, if raised to higher and higher powers, goes into a Gaussian function. (useful in sequential Bayesian inference).\n", " - 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. \n", " - Therefore any operation on a probability distribution that discards information but preserves variance gets us closer to a Gaussian. \n", " - 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.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- (2) Once the Gaussian has been attained, this form tends to be preserved. e.g., \n", " - The convolution of two Gaussian functions is another Gaussian function (useful in sum of 2 variables and linear transformations)\n", " - The product of two Gaussian functions is another Gaussian function (useful in Bayes rule).\n", " - The Fourier transform of a Gaussian function is another Gaussian function." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Transformations and Sums of Gaussian Variables\n", "\n", "- A **linear transformation** $z=Ax+b$ of a Gaussian variable $x \\sim \\mathcal{N}(\\mu_x,\\Sigma_x)$ is Gaussian distributed as\n", "$$\n", "p(z) = \\mathcal{N} \\left(z \\,|\\, A\\mu_x+b, A\\Sigma_x A^T \\right) \\tag{SRG-4a}\n", "$$ \n", " - 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 $z$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- 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\n", "$$\\begin{align*}\n", "p(z) &= \\mathcal{N}(x\\,|\\,\\mu_x,\\Sigma_x) \\ast \\mathcal{N}(y\\,|\\,\\mu_y,\\Sigma_y) \\\\\n", " &= \\mathcal{N} \\left(z\\,|\\,\\mu_x+\\mu_y, \\Sigma_x +\\Sigma_y \\right) \\tag{SRG-8}\n", "\\end{align*}$$\n", " \n", " - The sum of two Gaussian _distributions_ is NOT a Gaussian distribution. Why not?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Example: Gaussian Signals in a Linear System\n", "\n", "

\n", "\n", "- Given independent variables\n", "$x \\sim \\mathcal{N}(\\mu_x,\\sigma_x^2)$ 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))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Think about the role of the Gaussian distribution for stochastic linear systems in relation to what sinusoidals mean for deterministic linear system analysis." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Bayesian Inference for the Gaussian\n", "\n", "\n", "- Let's estimate a constant $\\theta$ from one 'noisy' measurement $x$ about that constant. \n", "\n", "- We assume the following measurement equations (the tilde $\\sim$ means: 'is distributed as'):\n", "$$\\begin{align*}\n", "x &= \\theta + \\epsilon \\\\\n", "\\epsilon &\\sim \\mathcal{N}(0,\\sigma^2)\n", "\\end{align*}$$\n", "\n", "- Also, let's assume a Gaussian prior for $\\theta$\n", "$$\\begin{align*}\n", "\\theta &\\sim \\mathcal{N}(\\mu_0,\\sigma_0^2) \\\\\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "##### Model specification\n", " \n", "- Note that you can rewrite these specifications in probabilistic notation as follows:\n", "\n", "$$\\begin{align*}\n", " p(x|\\theta) &= \\mathcal{N}(x|\\theta,\\sigma^2) \\\\\n", " p(\\theta) &=\\mathcal{N}(\\theta|\\mu_0,\\sigma_0^2)\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- (**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 $p(\\theta) =\\mathcal{N}(\\mu_0,\\sigma_0^2)$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "##### Inference\n", "\n", "- 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)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Let's do Bayes rule for the posterior PDF $p(\\theta|x)$. \n", "$$\\begin{align*}\n", "p(\\theta|x) &= \\frac{p(x|\\theta) p(\\theta)}{p(x)} \\propto p(x|\\theta) p(\\theta) \\\\\n", " &= \\mathcal{N}(x|\\theta,\\sigma^2) \\mathcal{N}(\\theta|\\mu_0,\\sigma_0^2) \\\\\n", " &\\propto \\exp \\left\\{ -\\frac{(x-\\theta)^2}{2\\sigma^2} - \\frac{(\\theta-\\mu_0)^2}{2\\sigma_0^2} \\right\\} \\\\\n", " &\\propto \\exp \\left\\{ \\theta^2 \\cdot \\left( -\\frac{1}{2 \\sigma_0^2} - \\frac{1}{2\\sigma^2} \\right) + \\theta \\cdot \\left( \\frac{\\mu_0}{\\sigma_0^2} + \\frac{x}{\\sigma^2}\\right) \\right\\} \\\\\n", " &= \\exp\\left\\{ -\\frac{\\sigma_0^2 + \\sigma^2}{2 \\sigma_0^2 \\sigma^2} \\left( \\theta - \\frac{\\sigma_0^2 x + \\sigma^2 \\mu_0}{\\sigma^2 + \\sigma_0^2}\\right)^2 \\right\\} \n", "\\end{align*}$$\n", "which we recognize as a Gaussian distribution w.r.t. $\\theta$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "- (Just as an aside,) this computational 'trick' for multiplying two Gaussians is called **completing the square**. The procedure makes use of the equality $$ax^2+bx+c_1 = a\\left(x+\\frac{b}{2a}\\right)^2+c_2$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " \n", "- In particular, it follows that the posterior for $\\theta$ is\n", "$$\\begin{equation*}\n", " p(\\theta|x) = \\mathcal{N} (\\theta |\\, \\mu_1, \\sigma_1^2)\n", "\\end{equation*}$$\n", "where\n", "$$\\begin{align*}\n", " \\frac{1}{\\sigma_1^2} &= \\frac{\\sigma_0^2 + \\sigma^2}{\\sigma^2 \\sigma_0^2} = \\frac{1}{\\sigma_0^2} + \\frac{1}{\\sigma^2} \\\\\n", " \\mu_1 &= \\frac{\\sigma_0^2 x + \\sigma^2 \\mu_0}{\\sigma^2 + \\sigma_0^2} = \\sigma_1^2 \\, \\left( \\frac{1}{\\sigma_0^2} \\mu_0 + \\frac{1}{\\sigma^2} x \\right) \n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### (Multivariate) Gaussian Multiplication\n", "\n", "- So, multiplication of two Gaussian distributions yields another (unnormalized) Gaussian with\n", " - posterior precision equals **sum of prior precisions**\n", " - posterior precision-weighted mean equals **sum of prior precision-weighted means**" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- As we just saw, a Gaussian prior, combined with a Gaussian likelihood, make Bayesian inference analytically solvable (!):\n", "\n", "$$\\begin{equation*}\n", "\\underbrace{\\text{Gaussian}}_{\\text{posterior}}\n", " \\propto \\underbrace{\\text{Gaussian}}_{\\text{likelihood}} \\times \\underbrace{\\text{Gaussian}}_{\\text{prior}}\n", "\\end{equation*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- In general, the multiplication of two multi-variate Gaussians over $x$ yields an (unnormalized) Gaussian over $x$:\n", "$$\\begin{equation*}\n", "\\boxed{\\mathcal{N}(x|\\mu_a,\\Sigma_a) \\cdot \\mathcal{N}(x|\\mu_b,\\Sigma_b) = \\underbrace{\\mathcal{N}(\\mu_a|\\, \\mu_b, \\Sigma_a + \\Sigma_b)}_{\\text{normalization constant}} \\cdot \\mathcal{N}(x|\\mu_c,\\Sigma_c)} \\tag{SRG-6}\n", "\\end{equation*}$$\n", "where\n", "$$\\begin{align*}\n", "\\Sigma_c^{-1} &= \\Sigma_a^{-1} + \\Sigma_b^{-1} \\\\\n", "\\Sigma_c^{-1} \\mu_c &= \\Sigma_a^{-1}\\mu_a + \\Sigma_b^{-1}\\mu_b\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- 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!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- $\\Rightarrow$ Note that Bayesian inference is trivial in the [_canonical_ parameterization of the Gaussian](#natural-parameterization), where we would get\n", "$$\\begin{align*}\n", " \\Lambda_c &= \\Lambda_a + \\Lambda_b \\quad &&\\text{(precisions add)}\\\\\n", " \\eta_c &= \\eta_a + \\eta_b \\quad &&\\text{(precision-weighted means add)}\n", "\\end{align*}$$\n", " - This property is an important reason why the canonical parameterization of the Gaussian distribution is useful in Bayesian data processing. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Code Example: Product of Two Gaussian PDFs\n", "\n", "- Let's plot the exact product of two Gaussian PDFs as well as the normalized product according to the above derivation." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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product d1*d2\n", "s2_prod = (d1.σ^-2 + d2.σ^-2)^-1\n", "m_prod = s2_prod * ((d1.σ^-2)*d1.μ + (d2.σ^-2)*d2.μ)\n", "d_prod = Normal(m_prod, sqrt(s2_prod)) # Note that we neglect the normalization constant.\n", "\n", "# Plot stuff\n", "x = range(-4, stop=8, length=100)\n", "plot(x, pdf.(d1,x), label=L\"\\mathcal{N}(0,1)\", fill=(0, 0.1)) # Plot the first Gaussian\n", "plot!(x, pdf.(d2,x), label=L\"\\mathcal{N}(3,4)\", fill=(0, 0.1)) # Plot the second Gaussian\n", "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\n", "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\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Bayesian Inference with multiple Observations\n", "\n", "- Now consider that we measure a data set $D = \\{x_1, x_2, \\ldots, x_N\\}$, with measurements\n", "$$\\begin{aligned}\n", "x_n &= \\theta + \\epsilon_n \\\\\n", "\\epsilon_n &\\sim \\mathcal{N}(0,\\sigma^2)\n", "\\end{aligned}$$\n", "and the same prior for $\\theta$:\n", "$$\n", "\\theta \\sim \\mathcal{N}(\\mu_0,\\sigma_0^2) \\\\\n", "$$\n", "\n", "- Let's derive the distribution $p(x_{N+1}|D)$ for the next sample . " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "##### inference \n", "\n", "- First, we derive the posterior for $\\theta$:\n", "$$\\begin{align*}\n", "p(\\theta|D) \\propto \\underbrace{\\mathcal{N}(\\theta|\\mu_0,\\sigma_0^2)}_{\\text{prior}} \\cdot \\underbrace{\\prod_{n=1}^N \\mathcal{N}(x_n|\\theta,\\sigma^2)}_{\\text{likelihood}}\n", "\\end{align*}$$\n", "which is a multiplication of $N+1$ Gaussians and is therefore also Gaussian-distributed." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Using the property that precisions and precision-weighted means add when Gaussians are multiplied, we can immediately write the posterior $$p(\\theta|D) = \\mathcal{N} (\\theta |\\, \\mu_N, \\sigma_N^2)$$ as \n", "$$\\begin{align*}\n", " \\frac{1}{\\sigma_N^2} &= \\frac{1}{\\sigma_0^2} + \\sum_n \\frac{1}{\\sigma^2} \\qquad &\\text{(B-2.142)} \\\\\n", " \\mu_N &= \\sigma_N^2 \\, \\left( \\frac{1}{\\sigma_0^2} \\mu_0 + \\sum_n \\frac{1}{\\sigma^2} x_n \\right) \\qquad &\\text{(B-2.141)}\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "##### application: prediction of future sample\n", "\n", "- We now have a posterior for the model parameters. Let's write down what we know about the next sample $x_{N+1}$.\n", "\n", "\n", "$$\\begin{align*}\n", "p(x_{N+1}|D) &= \\int p(x_{N+1}|\\theta) p(\\theta|D)\\mathrm{d}\\theta \\\\\n", " &= \\int \\mathcal{N}(x_{N+1}|\\theta,\\sigma^2) \\mathcal{N}(\\theta|\\mu_N,\\sigma^2_N) \\mathrm{d}\\theta \\\\\n", " &= \\int \\mathcal{N}(\\theta|x_{N+1},\\sigma^2) \\mathcal{N}(\\theta|\\mu_N,\\sigma^2_N) \\mathrm{d}\\theta \\\\\n", " &= \\int \\mathcal{N}(x_{N+1}|\\mu_N, \\sigma^2_N +\\sigma^2 ) \\mathcal{N}(\\theta|\\cdot,\\cdot)\\mathrm{d}\\theta \\tag{use SRG-6} \\\\\n", " &= \\mathcal{N}(x_{N+1}|\\mu_N, \\sigma^2_N +\\sigma^2 ) \\underbrace{\\int \\mathcal{N}(\\theta|\\cdot,\\cdot)\\mathrm{d}\\theta}_{=1} \\\\\n", " &=\\mathcal{N}(x_{N+1}|\\mu_N, \\sigma^2_N +\\sigma^2 )\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Uncertainty about $x_{N+1}$ involved both uncertainty about the parameter ($\\sigma_N^2$) and observation noise $\\sigma^2$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Maximum Likelihood Estimation for the Gaussian\n", "\n", "- 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\n", "$$\\begin{align*}\n", " \\mu_{\\text{ML}} = \\left.\\mu_N\\right\\vert_{\\sigma_0^2 \\rightarrow \\infty} = \\sigma_N^2 \\, \\left( \\frac{1}{\\sigma^2}\\sum_n x_n \\right) = \\frac{1}{N} \\sum_{n=1}^N x_n \n", " \\end{align*}$$\n", " \n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- As expected, having an expression for the maximum likelihood estimate, it is now possible to rewrite the (Bayesian) posterior mean for $\\theta$ as \n", "$$\\begin{align*}\n", " \\underbrace{\\mu_N}_{\\text{posterior}} &= \\sigma_N^2 \\, \\left( \\frac{1}{\\sigma_0^2} \\mu_0 + \\sum_n \\frac{1}{\\sigma^2} x_n \\right) \\\\\n", " &= \\frac{\\sigma_0^2 \\sigma^2}{N\\sigma_0^2 + \\sigma^2} \\, \\left( \\frac{1}{\\sigma_0^2} \\mu_0 + \\sum_n \\frac{1}{\\sigma^2} x_n \\right) \\\\\n", " &= \\frac{ \\sigma^2}{N\\sigma_0^2 + \\sigma^2} \\mu_0 + \\frac{N \\sigma_0^2}{N\\sigma_0^2 + \\sigma^2} \\mu_{\\text{ML}} \\\\\n", " &= \\underbrace{\\mu_0}_{\\text{prior}} + \\underbrace{\\underbrace{\\frac{N \\sigma_0^2}{N \\sigma_0^2 + \\sigma^2}}_{\\text{gain}}\\cdot \\underbrace{\\left(\\mu_{\\text{ML}} - \\mu_0 \\right)}_{\\text{prediction error}}}_{\\text{correction}}\\tag{B-2.141}\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Hence, the posterior mean always lies somewhere between the prior mean $\\mu_0$ and the maximum likelihood estimate (the \"data\" mean) $\\mu_{\\text{ML}}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Conditioning and Marginalization of a Gaussian\n", "\n", "- Let $z = \\begin{bmatrix} x \\\\ y \\end{bmatrix}$ be jointly normal distributed as\n", "\n", "$$\\begin{align*}\n", "p(z) &= \\mathcal{N}(z | \\mu, \\Sigma) \n", " =\\mathcal{N} \\left( \\begin{bmatrix} x \\\\ y \\end{bmatrix} \\left| \\begin{bmatrix} \\mu_x \\\\ \\mu_y \\end{bmatrix}, \n", " \\begin{bmatrix} \\Sigma_x & \\Sigma_{xy} \\\\ \\Sigma_{yx} & \\Sigma_y \\end{bmatrix} \\right. \\right)\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Since covariance matrices are by definition symmetric, it follows that $\\Sigma_x$ and $\\Sigma_y$ are symmetric and $\\Sigma_{xy} = \\Sigma_{yx}^T$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Let's factorize $p(z) = p(x,y)$ as $p(x,y) = p(y|x) p(x)$ through conditioning and marginalization." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\\begin{equation*}\n", "\\text{conditioning: }\\boxed{ p(y|x) = \\mathcal{N}\\left(y\\,|\\,\\mu_y + \\Sigma_{yx}\\Sigma_x^{-1}(x-\\mu_x),\\, \\Sigma_y - \\Sigma_{yx}\\Sigma_x^{-1}\\Sigma_{xy} \\right)}\n", "\\end{equation*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\\begin{equation*}\n", "\\text{marginalization: } \\boxed{ p(x) = \\mathcal{N}\\left( x|\\mu_x, \\Sigma_x \\right)}\n", "\\end{equation*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- **proof**: in Bishop pp.87-89" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Hence, conditioning and marginalization in Gaussians leads to Gaussians again. This is very useful for applications to Bayesian inference in jointly Gaussian systems." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- 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. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- 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." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Code Example: Joint, Marginal, and Conditional Gaussian Distributions\n", "\n", "- Let's plot of the joint, marginal, and conditional distributions." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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"\n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "using Plots, LaTeXStrings, Distributions\n", "\n", "# Define the joint distribution p(x,y)\n", "μ = [1.0; 2.0]\n", "Σ = [0.3 0.7;\n", " 0.7 2.0]\n", "joint = MvNormal(μ,Σ)\n", "\n", "# Define the marginal distribution p(x)\n", "marginal_x = Normal(μ[1], sqrt(Σ[1,1]))\n", "\n", "# Plot p(x,y)\n", "x_range = y_range = range(-2,stop=5,length=1000)\n", "joint_pdf = [ pdf(joint, [x_range[i];y_range[j]]) for j=1:length(y_range), i=1:length(x_range)]\n", "plot_1 = heatmap(x_range, y_range, joint_pdf, title = L\"p(x, y)\")\n", "\n", "# Plot p(x)\n", "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))\n", "\n", "# Plot p(y|x = 0.1)\n", "x = 0.1\n", "conditional_y_m = μ[2]+Σ[2,1]*inv(Σ[1,1])*(x-μ[1])\n", "conditional_y_s2 = Σ[2,2] - Σ[2,1]*inv(Σ[1,1])*Σ[1,2]\n", "conditional_y = Normal(conditional_y_m, sqrt.(conditional_y_s2))\n", "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))\n", "plot(plot_1, plot_2, plot_3, layout=(1,3), size=(1200,300))\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "As is clear from the plots, the conditional distribution is a renormalized slice from the joint distribution.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Example: Conditioning of Gaussian\n", "\n", "- Consider (again) the system \n", "$$\\begin{align*}\n", "p(x\\,|\\,\\theta) &= \\mathcal{N}(x\\,|\\,\\theta,\\sigma^2) \\\\\n", "p(\\theta) &= \\mathcal{N}(\\theta\\,|\\,\\mu_0,\\sigma_0^2)\n", "\\end{align*}$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Let $z = \\begin{bmatrix} x \\\\ \\theta \\end{bmatrix}$. The distribution for $z$ is then given by (see Gaussian distribution Exercises)\n", "\n", "$$\n", "p(z) = p\\left(\\begin{bmatrix} x \\\\ \\theta \\end{bmatrix}\\right) = \\mathcal{N} \\left( \\begin{bmatrix} x\\\\ \n", " \\theta \\end{bmatrix} \n", " \\,\\left|\\, \\begin{bmatrix} \\mu_0\\\\ \n", " \\mu_0\\end{bmatrix}, \n", " \\begin{bmatrix} \\sigma_0^2+\\sigma^2 & \\sigma_0^2\\\\ \n", " \\sigma_0^2 &\\sigma_0^2 \n", " \\end{bmatrix} \n", " \\right. \\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Direct substitution of the rule for Gaussian conditioning leads to the posterior (derivation as an Exercise):\n", "$$\\begin{align*}\n", "p(\\theta|x) &= \\mathcal{N} \\left( \\theta\\,|\\,\\mu_1, \\sigma_1^2 \\right)\\,,\n", "\\end{align*}$$\n", "with\n", "$$\\begin{align*}\n", "K &= \\frac{\\sigma_0^2}{\\sigma_0^2+\\sigma^2} \\qquad \\text{($K$ is called: Kalman gain)}\\\\\n", "\\mu_1 &= \\mu_0 + K \\cdot (x-\\mu_0)\\\\\n", "\\sigma_1^2 &= \\left( 1-K \\right) \\sigma_0^2 \n", "\\end{align*}$$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "- $\\Rightarrow$ Moral: For jointly Gaussian systems, we can do inference simply in one step by using the formulas for conditioning and marginalization." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Recursive Bayesian Estimation for Adaptive Signal Processing \n", "\n", "- Consider the signal $x_t=\\theta+\\epsilon_t$, where $D_t= \\left\\{x_1,\\ldots,x_t\\right\\}$ is observed _sequentially_ (over time).\n", "\n", "- **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) $x_t$.\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "##### Model specification \n", "- Let's define the estimate after $t$ observations (i.e., our _solution_ ) as $p(\\theta|D_t) = \\mathcal{N}(\\theta\\,|\\,\\mu_t,\\sigma_t^2)$.\n", "\n", "- We define the joint distribution for $\\theta$ and $x_t$, given background $D_{t-1}$, by\n", "\n", "$$\\begin{align*} p(x_t,\\theta \\,|\\, D_{t-1}) &= p(x_t|\\theta) \\, p(\\theta|D_{t-1}) \\\\\n", " &= \\underbrace{\\mathcal{N}(x_t\\,|\\, \\theta,\\sigma^2)}_{\\text{likelihood}} \\, \\underbrace{\\mathcal{N}(\\theta\\,|\\,\\mu_{t-1},\\sigma_{t-1}^2)}_{\\text{prior}}\n", "\\end{align*}$$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " \n", "##### Inference\n", "\n", "- Use Bayes rule,\n", "$$\\begin{align*}\n", "p(\\theta|D_t) &= p(\\theta|x_t,D_{t-1}) \\\\\n", " &\\propto p(x_t,\\theta | D_{t-1}) \\\\\n", " &= p(x_t|\\theta) \\, p(\\theta|D_{t-1}) \\\\\n", " &= \\mathcal{N}(x_t|\\theta,\\sigma^2) \\, \\mathcal{N}(\\theta\\,|\\,\\mu_{t-1},\\sigma_{t-1}^2) \\\\\n", " &= \\mathcal{N}(\\theta|x_t,\\sigma^2) \\, \\mathcal{N}(\\theta\\,|\\,\\mu_{t-1},\\sigma_{t-1}^2) \\;\\;\\text{(note this trick)}\\\\\n", " &= \\mathcal{N}(\\theta|\\mu_t,\\sigma_t^2) \\;\\;\\text{(use Gaussian multiplication formula SRG-6)}\n", "\\end{align*}$$\n", "with\n", "$$\\begin{align*}\n", "K_t &= \\frac{\\sigma_{t-1}^2}{\\sigma_{t-1}^2+\\sigma^2} \\qquad \\text{(Kalman gain)}\\\\\n", "\\mu_t &= \\mu_{t-1} + K_t \\cdot (x_t-\\mu_{t-1})\\\\\n", "\\sigma_t^2 &= \\left( 1-K_t \\right) \\sigma_{t-1}^2 \n", "\\end{align*}$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- This linear _sequential_ estimator of mean and variance in Gaussian observations is called a **Kalman Filter**.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- 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. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "- 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. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- 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." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Code Example: Kalman Filter\n", "\n", "- Let's implement the Kalman filter described above. We'll use it to recursively estimate the value of $\\theta$ based on noisy observations." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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value of the parameter we would like to estimate\n", "noise_σ2 = 0.3 # variance of observation noise\n", "\n", "observations = noise_σ2 * randn(n) .+ θ\n", "\n", "function perform_kalman_step(prior :: Normal, x :: Float64, noise_σ2 :: Float64)\n", " K = prior.σ / (noise_σ2 + prior.σ) # compute the Kalman gain\n", " posterior_μ = prior.μ + K*(x - prior.μ) # update the posterior mean\n", " posterior_σ = prior.σ * (1.0 - K) # update the posterior standard deviation\n", " return Normal(posterior_μ, posterior_σ) # return the posterior distribution\n", "end\n", "\n", "post_μ = fill!(Vector{Float64}(undef,n + 1), NaN) # means of p(θ|D) over time\n", "post_σ2 = fill!(Vector{Float64}(undef,n + 1), NaN) # variances of p(θ|D) over time\n", "\n", "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)\n", "\n", "post_μ[1] = prior.μ # save prior mean and variance to show these in plot\n", "post_σ2[1] = prior.σ\n", "\n", "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\n", " posterior = perform_kalman_step(prior, x, noise_σ2) # compute the posterior distribution given the observation\n", " post_μ[i + 1] = posterior.μ # save the mean of the posterior distribution\n", " post_σ2[i + 1] = posterior.σ # save the variance of the posterior distribution\n", " prior = posterior # the posterior becomes the prior for future observations\n", "end\n", "\n", "obs_scale = collect(2:n+1)\n", "scatter(obs_scale, observations, label=L\"D\", ) \n", "post_scale = collect(1:n+1) # scatter the observations\n", "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\n", "plot!(post_scale, θ*ones(n + 1), linewidth=2, label=L\"θ\") # plot the true value of θ\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "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.\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Product of Normally Distributed Variables\n", "- (We've seen that) the sum of two Gausssian distributed variables is also Gaussian distributed." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Has the _product_ of two Gaussian distributed variables also a Gaussian distribution?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- **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 $Y\\sim \\mathcal{N}(0,1)$.\n", "$$\\begin{align*}\n", "p(z) &= \\int_{X,Y} p(z|x,y)\\,p(x,y)\\,\\mathrm{d}x\\mathrm{d}y \\\\\n", " &= \\frac{1}{2 \\pi}\\int \\delta(z-xy) \\, e^{-(x^2+y^2)/2} \\, \\mathrm{d}x\\mathrm{d}y \\\\\n", " &= \\frac{1}{\\pi} \\int_0^\\infty \\frac{1}{x} e^{-(x^2+z^2/x^2)/2} \\, \\mathrm{d}x \\\\\n", " &= \\frac{1}{\\pi} \\mathrm{K}_0( \\lvert z\\rvert )\\,.\n", "\\end{align*}$$\n", "where $\\mathrm{K}_n(z)$ is a [modified Bessel function of the second kind](http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Code Example: Product of Gaussian Distributions\n", "\n", "- 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." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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collect(range(-4,stop=4,length=100))\n", "plot(range1, pdf.(X, range1), label=L\"p(X)=p(Y)=\\mathcal{N}(0,1)\", fill=(0, 0.1))\n", "plot!(range1, pdf.(X,range1).*pdf.(Y,range1), label=L\"p(X)*p(Y)\", fill=(0, 0.1))\n", "plot!(range1, pdf_product_std_normals(range1), label=L\"p(Z=X*Y)\", fill=(0, 0.1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- In short, Gaussian-distributed variables remain Gaussian in linear systems, but this is not the case in non-linear systems. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Solution to Example Problem\n", "\n", "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 $S$: $p(x_\\bullet \\in S | m) = \\int_S p(x|m) \\mathrm{d}x$." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p(x⋅∈S|m) ≈ 0.1877142306798326\n" ] }, { "data": { "image/png": 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LinearAlgebra, Plots, Distributions# Numerical integration package\n", "# Maximum likelihood estimation of 2D Gaussian\n", "N = length(sum(D,dims=1))\n", "μ = 1/N * sum(D,dims=2)[:,1]\n", "D_min_μ = D - repeat(μ, 1, N)\n", "Σ = Hermitian(1/N * D_min_μ*D_min_μ')\n", "m = MvNormal(μ, convert(Matrix, Σ));\n", "\n", "contour(range(-3, 4, length=100), range(-3, 4, length=100), (x, y) -> pdf(m, [x, y]))\n", "\n", "# Numerical integration of p(x|m) over S:\n", "(val,err) = hcubature((x)->pdf(m,x), [0., 1.], [2., 2.])\n", "println(\"p(x⋅∈S|m) ≈ $(val)\")\n", "\n", "scatter!(D[1,:], D[2,:], marker=:x, markerstrokewidth=3, label=L\"D\")\n", "scatter!([x_dot[1]], [x_dot[2]], label=L\"x_\\bullet\")\n", "plot!(range(0, 2), [1., 1., 1.], fillrange=2, alpha=0.4, color=:gray, label=L\"S\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Summary\n", "\n", "1. A **linear transformation** $z=Ax+b$ of a Gaussian variable $x \\sim \\mathcal{N}(\\mu_x,\\Sigma_x)$ is Gaussian distributed as\n", "$$\n", "p(z) = \\mathcal{N} \\left(z \\,|\\, A\\mu_x+b, A\\Sigma_x A^T \\right) \n", "$$ \n", "\n", "2. Bayesian inference with a Gaussian prior and Gaussian likelihood leads to an analytically computable Gaussian posterior, because of the **multiplication rule for Gaussians**:\n", "$$\\begin{equation*}\n", "\\mathcal{N}(x|\\mu_a,\\Sigma_a) \\cdot \\mathcal{N}(x|\\mu_b,\\Sigma_b) = \\underbrace{\\mathcal{N}(\\mu_a|\\, \\mu_b, \\Sigma_a + \\Sigma_b)}_{\\text{normalization constant}} \\cdot \\mathcal{N}(x|\\mu_c,\\Sigma_c)\n", "\\end{equation*}$$\n", "where\n", "$$\\begin{align*}\n", "\\Sigma_c^{-1} &= \\Sigma_a^{-1} + \\Sigma_b^{-1} \\\\\n", "\\Sigma_c^{-1} \\mu_c &= \\Sigma_a^{-1}\\mu_a + \\Sigma_b^{-1}\\mu_b\n", "\\end{align*}$$\n", "\n", "3. **Conditioning and marginalization** of a multivariate Gaussian distribution yields Gaussian distributions. In particular, the joint distribution\n", "$$\\mathcal{N} \\left( \\begin{bmatrix} x \\\\ y \\end{bmatrix} \\left| \\begin{bmatrix} \\mu_x \\\\ \\mu_y \\end{bmatrix}, \n", " \\begin{bmatrix} \\Sigma_x & \\Sigma_{xy} \\\\ \\Sigma_{yx} & \\Sigma_y \\end{bmatrix} \\right. \\right)\n", "$$\n", "can be decomposed as\n", "$$\\begin{align*}\n", " p(y|x) &= \\mathcal{N}\\left(y\\,|\\,\\mu_y + \\Sigma_{yx}\\Sigma_x^{-1}(x-\\mu_x),\\, \\Sigma_y - \\Sigma_{yx}\\Sigma_x^{-1}\\Sigma_{xy} \\right) \\\\\n", "p(x) &= \\mathcal{N}\\left( x|\\mu_x, \\Sigma_x \\right)\n", "\\end{align*}$$\n", "\n", "- 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. \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "##
OPTIONAL SLIDES
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Inference for the Precision Parameter of the Gaussian\n", "\n", "- 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." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- 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)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "##### model specification\n", "- The likelihood for the precision parameter is \n", "$$\\begin{align*}\n", "p(D|\\lambda) &= \\prod_{n=1}^N \\mathcal{N}\\left(x_n \\,|\\, \\mu, \\lambda^{-1} \\right) \\\\\n", " &\\propto \\lambda^{N/2} \\exp\\left\\{ -\\frac{\\lambda}{2}\\sum_{n=1}^N \\left(x_n - \\mu \\right)^2\\right\\} \\tag{B-2.145}\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- The conjugate distribution for this function of $\\lambda$ is the [_Gamma_ distribution](https://en.wikipedia.org/wiki/Gamma_distribution), given by\n", "$$\n", "p(\\lambda\\,|\\,a,b) = \\mathrm{Gam}\\left( \\lambda\\,|\\,a,b \\right) \\triangleq \\frac{1}{\\Gamma(a)} b^{a} \\lambda^{a-1} \\exp\\left\\{ -b \\lambda\\right\\}\\,, \\tag{B-2.146}\n", "$$\n", "where $a>0$ and $b>0$ are known as the _shape_ and _rate_ parameters, respectively. \n", "\n", "\n", "\n", "- (Bishop fig.2.13). Plots of the Gamma distribution $\\mathrm{Gam}\\left( \\lambda\\,|\\,a,b \\right) $ for different values of $a$ and $b$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- The mean and variance of the Gamma distribution evaluate to $\\mathrm{E}\\left( \\lambda\\right) = \\frac{a}{b}$ and $\\mathrm{var}\\left[\\lambda\\right] = \\frac{a}{b^2}$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "##### inference\n", "\n", "- We will consider a prior $p(\\lambda) = \\mathrm{Gam}\\left( \\lambda\\,|\\,a_0, b_0\\right)$, which leads by Bayes rule to the posterior\n", "$$\\begin{align*}\n", "p(\\lambda\\,|\\,D) &\\propto \\underbrace{\\lambda^{N/2} \\exp\\left\\{ -\\frac{\\lambda}{2}\\sum_{n=1}^N \\left(x_n - \\mu \\right)^2\\right\\} }_{\\text{likelihood}} \\cdot \\underbrace{\\frac{1}{\\Gamma(a_0)} b_0^{a_0} \\lambda^{a_0-1} \\exp\\left\\{ -b_0 \\lambda\\right\\}}_{\\text{prior}} \\\\\n", " &\\propto \\mathrm{Gam}\\left( \\lambda\\,|\\,a_N,b_N \\right) \n", "\\end{align*}$$\n", "with\n", "$$\\begin{align*}\n", "a_N &= a_0 + \\frac{N}{2} \\qquad &&\\text{(B-2.150)} \\\\\n", "b_N &= b_0 + \\frac{1}{2}\\sum_n \\left( x_n-\\mu\\right)^2 \\qquad &&\\text{(B-2.151)}\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- 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. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- 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\n", "$$\n", "\\lambda_{\\text{ML}} = \\left.\\mathrm{E}\\left[ \\lambda\\right]\\right\\vert_{a_0=b_0\\rightarrow 0} = \\left. \\frac{a_N}{b_N}\\right\\vert_{a_0=b_0\\rightarrow 0} = \\frac{N}{\\sum_{n=1}^N \\left(x_n-\\mu \\right)^2}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- 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 maximum likelihood estimates for $\\mu$ and $\\sigma^2$ are given by\n", "$$\\begin{align*}\n", "\\mu_{\\text{ML}} &= \\frac{1}{N} \\sum_{n=1}^N x_n \\qquad &&\\text{(B-2.121)} \\\\\n", "\\sigma^2_{\\text{ML}} &= \\frac{1}{N} \\sum_{n=1}^N \\left(x_n - \\mu_{\\text{ML}} \\right)^2 \\qquad &&\\text{(B-2.122)}\n", "\\end{align*}$$\n", "\n", "- These estimates are also known as the _sample mean_ and _sample variance_ respectively. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "open(\"../../styles/aipstyle.html\") do f\n", " display(\"text/html\", read(f, String))\n", "end" ] } ], "metadata": { "@webio": { "lastCommId": null, "lastKernelId": null }, "anaconda-cloud": {}, "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Julia 1.10.4", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.5" }, "widgets": { "state": { "3e8b6f2f-6500-4ec7-963f-2db519e88817": { "views": [ { "cell_index": 11 } ] } }, "version": "1.2.0" } }, "nbformat": 4, "nbformat_minor": 4 }