{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Generative Classification" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Preliminaries\n", "\n", "- Goal \n", " - Introduction to linear generative classification with a Gaussian-categorical generative model\n", " \n", "- Materials \n", " - Mandatory\n", " - These lecture notes\n", " - Optional\n", " - Bishop pp. 196-202 (section 4.2 focusses on binary classification, whereas in these lecture notes we describe generative classification for multiple classes). " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Challenge: an apple or a peach?\n", "\n", "- **Problem**: You're given numerical values for the skin features roughness and color for 200 pieces of fruit, where for each piece of fruit you also know if it is an apple or a peach. Now you receive the roughness and color values for a new piece of fruit but you don't get its class label (apple or peach). What is the probability that the new piece is an apple?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- **Solution**: To be solved later in this lesson.\n", "\n", "- Let's first generate a data set (see next slide)." ] }, { "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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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\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, Distributions\n", "\n", "N = 250; p_apple = 0.7; Σ = [0.2 0.1; 0.1 0.3]\n", "p_given_apple = MvNormal([1.0, 1.0], Σ) # p(X|y=apple)\n", "p_given_peach = MvNormal([1.7, 2.5], Σ) # p(X|y=peach)\n", "X = Matrix{Float64}(undef,2,N); y = Vector{Bool}(undef,N) # true corresponds to apple\n", "for n=1:N\n", " y[n] = (rand() < p_apple) # Apple or peach?\n", " X[:,n] = y[n] ? rand(p_given_apple) : rand(p_given_peach) # Sample features\n", "end\n", "X_apples = X[:,findall(y)]'; X_peaches = X[:,findall(.!y)]' # Sort features on class\n", "x_test = [2.3; 1.5] # Features of 'new' data point\n", "\n", "scatter(X_apples[:,1], X_apples[:,2], label=\"apples\", marker=:x, markerstrokewidth=3) # apples\n", "scatter!(X_peaches[:,1], X_peaches[:,2], label=\"peaches\", marker=:+, markerstrokewidth=3) # peaches\n", "scatter!([x_test[1]], [x_test[2]], label=\"unknown\") # 'new' unlabelled data\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Generative Classification Problem Statement\n", "\n", "- Given is a data set $D = \\{(x_1,y_1),\\dotsc,(x_N,y_N)\\}$\n", " - inputs $x_n \\in \\mathbb{R}^M$ are called **features**.\n", " - outputs $y_n \\in \\mathcal{C}_k$, with $k=1,\\ldots,K$; The **discrete** targets $\\mathcal{C}_k$ are called **classes**." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- We will again use the 1-of-$K$ notation for the discrete classes. Define the binary **class selection variable**\n", "$$\n", "y_{nk} = \\begin{cases} 1 & \\text{if } \\, y_n \\in \\mathcal{C}_k\\\\\n", "0 & \\text{otherwise} \\end{cases}\n", "$$\n", " - (Hence, the notations $y_{nk}=1$ and $y_n \\in \\mathcal{C}_k$ mean the same thing.)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- The plan for generative classification: build a model for the joint pdf $p(x,y)= p(x|y)p(y)$ and use Bayes to infer the posterior class probabilities \n", "\n", "$$\n", "p(y|x) = \\frac{p(x|y) p(y)}{\\sum_{y^\\prime} p(x|y^\\prime) p(y^\\prime)} \\propto p(x|y)\\,p(y)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 1 - Model specification \n", "\n", "##### Likelihood\n", "\n", "- Assume Gaussian **class-conditional distributions** with **equal covariance matrix** across the classes,\n", " $$\n", " p(x_n|\\mathcal{C}_{k}) = \\mathcal{N}(x_n|\\mu_k,\\Sigma)\n", " $$\n", "with notational shorthand: $\\mathcal{C}_{k} \\triangleq (y_n \\in \\mathcal{C}_{k})$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "##### Prior\n", "\n", "- We use a categorical distribution for the class labels $y_{nk}$: \n", "$$p(\\mathcal{C}_{k}) = \\pi_k$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Hence, using the one-hot coding formulation for $y_{nk}$, the generative model $p(x_n,y_n)$ can be written as\n", "\n", "$$\\begin{align*}\n", " p(x_n,y_n) &= \\prod_{k=1}^K p(x_n,y_{nk}=1)^{y_{nk}} \\\\\n", " &= \\prod_{k=1}^K \\left( \\pi_k \\cdot\\mathcal{N}(x_n|\\mu_k,\\Sigma)\\right)^{y_{nk}}\n", "\\end{align*}$$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- We will refer to this model as the **Gaussian-Categorical Model** (GCM). \n", " - N.B. In the literature, this model (with possibly unequal $\\Sigma_k$ across classes) is often called the Gaussian Discriminant Analysis model and the special case with equal covariance matrices $\\Sigma_k=\\Sigma$ is also called Linear Discriminant Analysis. We think these names are a bit unfortunate as it may lead to confusion with the [discriminative method for classification](https://nbviewer.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Discriminative-Classification.ipynb)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- As usual, once the model has been specified, the rest (inference for parameters and model prediction) through straight probability theory." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Computing the log-likelihood\n", "\n", "- The log-likelihood given the full data set $D=\\{(x_n,y_n), n=1,2,\\ldots,N\\}$ is then\n", "$$\\begin{align*}\n", "\\log\\, p(D|\\theta) &\\stackrel{\\text{IID}}{=} \\sum_n \\log \\prod_k p(x_n,y_{nk}=1\\,|\\,\\theta)^{y_{nk}} \\\\\n", " &= \\sum_{n,k} y_{nk} \\log p(x_n,y_{nk}=1\\,|\\,\\theta) \\\\\n", " &= \\sum_{n,k} y_{nk} \\log p(x_n|y_{nk}=1) + \\sum_{n,k} y_{nk} \\log p(y_{nk}=1) \\\\\n", " &= \\sum_{n,k} y_{nk} \\log\\mathcal{N}(x_n|\\mu_k,\\Sigma) + \\sum_{n,k} y_{nk} \\log \\pi_k \\\\\n", " &= \\sum_{n,k} y_{nk} \\underbrace{ \\log\\mathcal{N}(x_n|\\mu_k,\\Sigma) }_{ \\text{see Gaussian lecture} } + \\underbrace{ \\sum_k m_k \\log \\pi_k }_{ \\text{see multinomial lecture} } \n", "\\end{align*}$$\n", "where we used $m_k \\triangleq \\sum_n y_{nk}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 2 - Parameter Inference for Classification\n", "\n", "- We'll do Maximum Likelihood estimation for $\\theta = \\{ \\pi_k, \\mu_k, \\Sigma \\}$ from data $D$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Recall (from the previous slide) the log-likelihood (LLH)\n", "\n", "$$\n", "\\log\\, p(D|\\theta) = \\sum_{n,k} y_{nk} \\underbrace{ \\log\\mathcal{N}(x_n|\\mu_k,\\Sigma) }_{ \\text{Gaussian} } + \\underbrace{ \\sum_k m_k \\log \\pi_k }_{ \\text{multinomial} } \n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Maximization of the LLH for the GDA model breaks down into\n", " - **Gaussian density estimation** for parameters $\\mu_k, \\Sigma$, since the first term contains exactly the log-likelihood for MVG density estimation. We've already done this, see the [Gaussian distribution lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/The-Gaussian-Distribution.ipynb#ML-for-Gaussian).\n", " - **Multinomial density estimation** for class priors $\\pi_k$, since the second term holds exactly the log-likelihood for multinomial density estimation, see the [Multinomial distribution lesson](https://nbviewer.jupyter.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/The-Multinomial-Distribution.ipynb#ML-for-multinomial). \n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ " - The ML for multinomial class prior (we've done this before!)\n", "$$\\begin{align*} \n", "\\hat \\pi_k = \\frac{m_k}{N} \n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Now group the data into separate classes and do MVG ML estimation for class-conditional parameters (we've done this before as well):\n", "$$\\begin{align*}\n", " \\hat \\mu_k &= \\frac{ \\sum_n y_{nk} x_n} { \\sum_n y_{nk} } = \\frac{1}{m_k} \\sum_n y_{nk} x_n \\\\\n", " \\hat \\Sigma &= \\frac{1}{N} \\sum_{n,k} y_{nk} (x_n-\\hat \\mu_k)(x_n-\\hat \\mu_k)^T \\\\\n", " &= \\sum_k \\hat \\pi_k \\cdot \\underbrace{ \\left( \\frac{1}{m_k} \\sum_{n} y_{nk} (x_n-\\hat \\mu_k)(x_n-\\hat \\mu_k)^T \\right) }_{ \\text{class-cond. variance} } \\\\\n", " &= \\sum_k \\hat \\pi_k \\cdot \\hat \\Sigma_k\n", "\\end{align*}$$\n", "where $\\hat \\pi_k$, $\\hat{\\mu}_k$ and $\\hat{\\Sigma}_k$ are the sample proportion, sample mean and sample variance for the $k$th class, respectively." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Note that the binary class selection variable $y_{nk}$ groups data from the same class." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 3 - Application: Class prediction for new Data\n", "\n", "- Let's apply the trained model to predict the class for given a 'new' input $x_\\bullet$:\n", "$$\\begin{align*}\n", "p(\\mathcal{C}_k|x_\\bullet,D ) &= \\int p(\\mathcal{C}_k|x_\\bullet,\\theta ) \\underbrace{p(\\theta|D)}_{\\text{ML: }\\delta(\\theta - \\hat{\\theta})} \\mathrm{d}\\theta \\\\\n", "&= p(\\mathcal{C}_k|x_\\bullet,\\hat{\\theta} ) \\\\\n", "&\\propto p(\\mathcal{C}_k)\\,p(x_\\bullet|\\mathcal{C}_k) \\\\\n", "&= \\hat{\\pi}_k \\cdot \\mathcal{N}(x_\\bullet | \\hat{\\mu}_k, \\hat{\\Sigma}) \\\\\n", " &\\propto \\hat{\\pi}_k \\exp \\left\\{ { - {\\frac{1}{2}}(x_\\bullet - \\hat{\\mu}_k )^T \\hat{\\Sigma}^{ - 1} (x_\\bullet - \\hat{\\mu}_k )} \\right\\}\\\\\n", " &=\\exp \\Big\\{ \\underbrace{-\\frac{1}{2}x_\\bullet^T \\hat{\\Sigma}^{ - 1} x_\\bullet}_{\\text{not a function of }k} + \\underbrace{\\hat{\\mu}_k^T \\hat{\\Sigma}^{-1}}_{\\beta_k^T} x_\\bullet \\underbrace{- {\\frac{1}{2}}\\hat{\\mu}_k^T \\hat{\\Sigma}^{ - 1} \\hat{\\mu}_k + \\log \\hat{\\pi}_k }_{\\gamma_k} \\Big\\} \\\\\n", " &\\propto \\frac{1}{Z}\\exp\\{\\beta_k^T x_\\bullet + \\gamma_k\\} \\\\\n", " &\\triangleq \\sigma\\left( \\beta_k^T x_\\bullet + \\gamma_k\\right)\n", "\\end{align*}$$\n", "where \n", "$\\sigma(a_k) \\triangleq \\frac{\\exp(a_k)}{\\sum_{k^\\prime}\\exp(a_{k^\\prime})}$ is called a [**softmax**](https://en.wikipedia.org/wiki/Softmax_function) (a.k.a. **normalized exponential**) function, and\n", "$$\\begin{align*}\n", "\\beta_k &= \\hat{\\Sigma}^{-1} \\hat{\\mu}_k \\\\\n", "\\gamma_k &= - \\frac{1}{2} \\hat{\\mu}_k^T \\hat{\\Sigma}^{-1} \\hat{\\mu}_k + \\log \\hat{\\pi}_k \\\\\n", "Z &= \\sum_{k^\\prime}\\exp\\{\\beta_{k^\\prime}^T x_\\bullet + \\gamma_{k^\\prime}\\}\\,. \\quad \\text{(normalization constant)} \n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- The softmax function is a smooth approximation to the max-function. Note that we did not a priori specify a softmax posterior, but rather it followed from applying Bayes rule to the prior and likelihood assumptions. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Note the following properties of the softmax function $\\sigma(a_k)$:\n", " - $\\sigma(a_k)$ is monotonicaly ascending function and hence it preserves the order of $a_k$. That is, if $a_j>a_k$ then $\\sigma(a_j)>\\sigma(a_k)$. \n", " - $\\sigma(a_k)$ is always a proper probability distribution, since $\\sigma(a_k)>0$ and $\\sum_k \\sigma(a_k) = 1$. \n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Discrimination Boundaries\n", "\n", "- The class log-posterior $\\log p(\\mathcal{C}_k|x) \\propto \\beta_k^T x + \\gamma_k$ is a linear function of the input features." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Thus, the contours of equal probability (**discriminant functions**) are lines (hyperplanes) in the feature space\n", "$$\n", "\\log \\frac{{p(\\mathcal{C}_k|x,\\theta )}}{{p(\\mathcal{C}_j|x,\\theta )}} = \\beta_{kj}^T x + \\gamma_{kj} = 0\n", "$$\n", "where we defined $\\beta_{kj} \\triangleq \\beta_k - \\beta_j$ and similarly for $\\gamma_{kj}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- How to classify a new input $x_\\bullet$? The Bayesian answer is a posterior distribution $ p(\\mathcal{C}_k|x_\\bullet)$. If you must choose, then the class with maximum posterior class probability\n", "$$\\begin{align*}\n", "k^* &= \\arg\\max_k p(\\mathcal{C}_k|x_\\bullet) \\\\\n", " &= \\arg\\max_k \\left( \\beta _k^T x_\\bullet + \\gamma_k \\right)\n", "\\end{align*}$$\n", "is an appealing decision. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Code Example: Working out the \"apple or peach\" example problem\n", "\n", "We'll apply the above results to solve the \"apple or peach\" example problem." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p(apple|x=x∙) = 0.6614585032838237\n" ] }, { "data": { "image/png": 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Distributions, Plots\n", "# Multinomial (in this case binomial) density estimation\n", "p_apple_est = sum(y.==true) / length(y)\n", "π_hat = [p_apple_est; 1-p_apple_est]\n", "\n", "# Estimate class-conditional multivariate Gaussian densities\n", "d1 = fit_mle(FullNormal, X_apples') # MLE density estimation d1 = N(μ₁, Σ₁)\n", "d2 = fit_mle(FullNormal, X_peaches') # MLE density estimation d2 = N(μ₂, Σ₂)\n", "Σ = π_hat[1]*cov(d1) + π_hat[2]*cov(d2) # Combine Σ₁ and Σ₂ into Σ\n", "conditionals = [MvNormal(mean(d1), Σ); MvNormal(mean(d2), Σ)] # p(x|C)\n", "\n", "# Calculate posterior class probability of x∙ (prediction)\n", "function predict_class(k, X) # calculate p(Ck|X)\n", " norm = π_hat[1]*pdf(conditionals[1],X) + π_hat[2]*pdf(conditionals[2],X)\n", " return π_hat[k]*pdf(conditionals[k], X) ./ norm\n", "end\n", "println(\"p(apple|x=x∙) = $(predict_class(1,x_test))\")\n", "\n", "# Discrimination boundary of the posterior (p(apple|x;D) = p(peach|x;D) = 0.5)\n", "β(k) = inv(Σ)*mean(conditionals[k])\n", "γ(k) = -0.5 * mean(conditionals[k])' * inv(Σ) * mean(conditionals[k]) + log(π_hat[k])\n", "function discriminant_x2(x1)\n", " # Solve discriminant equation for x2\n", " β12 = β(1) .- β(2)\n", " γ12 = (γ(1) .- γ(2))[1,1]\n", " return -1*(β12[1]*x1 .+ γ12) ./ β12[2]\n", "end\n", "\n", "scatter(X_apples[:,1], X_apples[:,2], label=\"apples\", marker=:x, markerstrokewidth=3) # apples\n", "scatter!(X_peaches[:,1], X_peaches[:,2], label=\"peaches\", marker=:+, markerstrokewidth=3) # peaches\n", "scatter!([x_test[1]], [x_test[2]], label=\"unknown\") # 'new' unlabelled data point\n", "\n", "x1 = range(-1,length=10,stop=3)\n", "plot!(x1, discriminant_x2(x1), color=\"black\", label=\"\") # Plot discrimination boundary\n", "plot!(x1, discriminant_x2(x1), fillrange=-10, alpha=0.2, color=:blue, label=\"\")\n", "plot!(x1, discriminant_x2(x1), fillrange=10, alpha=0.2, color=:red, xlims=(-0.5, 3), ylims=(-1, 4), label=\"\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Why Be Bayesian?\n", "\n", "- A student in one of the previous years posed the following question at Piazza: \n", " \n", " > \" After re-reading topics regarding generative classification, this question popped into my mind: Besides the sole purpose of the lecture, which is getting to know the concepts of generative classification and how to implement them, are there any advantages of using this instead of using deep neural nets? DNNs seem simpler and more powerful.\"\n", "\n", "- The following answer was provided: \n", "\n", " - If you are only are interested in approximating a function, say $y=f_\\theta(x)$, and you have lots of examples $\\{(x_i,y_i)\\}$ of desired behavior, then often a non-probabilistic DNN is a fine approach.\n", "\n", " - However, if you are willing to formulate your models in a probabilistic framework, you can improve on the deterministic approach in many ways, eg,\n", "\n", " > 1. Bayesian evidence for model performance assessment. This means you can use the whole data set for training without an ad-hoc split into testing and training data sets.\n", "\n", " > 2. Uncertainty about parameters in the model is a measure that allows you to do _active learning_, ie, choose data that is most informative (see also the [lesson on intelligent agents](https://nbviewer.org/github/bertdv/BMLIP/blob/master/lessons/notebooks/Intelligent-Agents-and-Active-Inference.ipynb)). This will allow you to train on small data sets, whereas the deterministic DNNs generally require much larger data sets.\n", " \n", " > 3. Prediction with uncertainty/confidence bounds.\n", " \n", " > 4. Explicit specification and separation of your assumptions.\n", " \n", " > 5. A framework that supports scoring both accuracy and model complexity in the same currency (probability). How are you going to penalize the size of your network in a deterministic framework?\n", " \n", " > 6. Automatic learning rates, no tuning parameters. For instance, the Kalman gain is a data-dependent, optimal learning rate. How will _you_ choose your learning rates in a deterministic framework? Trial and error?\n", " \n", " > 7. Principled absorption of different sources of knowledge. Eg, outcome of one set of experiments can be captured by a posterior distribution that serves as a prior distribution for the next set of experiments.\n", " \n", " > Admittedly, it's not easy to understand the probabilistic approach, but it is worth the effort.\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Recap Generative Classification\n", "\n", "- Gaussian-Categorical Model specification: \n", "\n", "$$p(x,\\mathcal{C}_k|\\,\\theta) = \\pi_k \\cdot \\mathcal{N}(x|\\mu_k,\\Sigma)$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- If the class-conditional distributions are Gaussian with equal covariance matrices across classes ($\\Sigma_k = \\Sigma$), then\n", " the discriminant functions are hyperplanes in feature space." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- ML estimation for $\\{\\pi_k,\\mu_k,\\Sigma\\}$ in the GCM model breaks down to simple density estimation for Gaussian and multinomial/categorical distributions." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Posterior class probability is a softmax function\n", "$$ p(\\mathcal{C}_k|x,\\theta ) \\propto \\exp\\{\\beta_k^T x + \\gamma_k\\}$$\n", "where $\\beta _k= \\Sigma^{-1} \\mu_k$ and $\\gamma_k=- \\frac{1}{2} \\mu_k^T \\Sigma^{-1} \\mu_k + \\log \\pi_k$." ] }, { "cell_type": "code", "execution_count": 4, "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 1.10.5", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.5" } }, "nbformat": 4, "nbformat_minor": 4 }