{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### MidTerm Assignment: notebook 1: Revision" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
Given date : March 30
\n", "\n", "Due date : April 17\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Total : 10 pts " ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "#### Question 1.1. Statistical learning: Maximum likelihood (Total 5pts)\n", "\n", "This exercise contains a pen and paper part and a coding part. You should submit the pen and paper either in lateX or take a picture of your written solution and join it to the Assignment folder. \n", "\n", "We consider the dataset given below. This dataset was generated from a Gaussian distribution with a given mean $\\mathbf{\\mu} = (\\mu_1, \\mu_2)$ and covariance matrix $\\mathbf{\\Sigma} = \\left[\\begin{array}{cc}\n", "\\sigma_1^2 & 0 \\\\\n", "0 & \\sigma_2^2\n", "\\end{array}\\right]$. We would like to recover the mean and variance from the data. In order to do this, use the following steps:\n", "\n", "1. Write the general expression for the probability (multivariate (2D) Gaussian with diagonal covariance matrix) to observe a single sample \n", "2. We will assume that the samples are independent and identically distributed so that the probability of observing the whole dataset is the product of the probabilties of observing each one of the samples $\\left\\{\\mathbf{x}^{(i)} = (x_1^{(i)}, x_2^{(i)})\\right\\}_{i=1}^N$. Write down this probability\n", "3. Take the negative logarithm of this probability\n", "4. Once you have taken the logarithm, find the expression for $mu_1, \\mu_2$, $\\sigma_1$ and $\\sigma_2$ by maximizing the probability. " ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "image/png": 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