{ "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": 2, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m " ] }, { "name": "stderr", "output_type": "stream", "text": [ "project at `~/github/bertdv/BMLIP/lessons`\n" ] } ], "source": [ "using Pkg; Pkg.activate(\"../.\"); Pkg.instantiate();\n", "using IJulia; try IJulia.clear_output(); catch _ end" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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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": 4, "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": 5, "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 (Exercise)\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": 6, "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": 7, "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": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p(x⋅∈S|m) ≈ 0.23491845066574196\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": 9, "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 1.9.3", "language": "julia", "name": "julia-1.9" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.9.3" }, "widgets": { "state": { "3e8b6f2f-6500-4ec7-963f-2db519e88817": { "views": [ { "cell_index": 11 } ] } }, "version": "1.2.0" } }, "nbformat": 4, "nbformat_minor": 4 }