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   "source": [
    "# Likelihood Function and Loss Function\n",
    "\n",
    "Feng Li\n",
    "\n",
    "School of Statistics and Mathematics\n",
    "\n",
    "Central University of Finance and Economics\n",
    "\n",
    "[feng.li@cufe.edu.cn](mailto:feng.li@cufe.edu.cn)\n",
    "\n",
    "[https://feng.li/statcomp](https://feng.li/statcomp)\n"
   ]
  },
  {
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   "id": "055df123",
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   "source": [
    "## Likelihood Function\n",
    "\n",
    "-   Given that $x_i\\sim N(\\mu,\\sigma)$ for $i=1,...,n$, the **likelihood\n",
    "    function** is\n",
    "\n",
    "    $$\\prod_{i=1}^n f(x_i,\\mu,\\sigma)$$\n",
    "\n",
    "-   However the **log likelihood function** is more often used\n",
    "    $$\\sum_{i=1}^n \\log f(x_i,\\mu,\\sigma)$$\n",
    "\n",
    "-   Do you know why?"
   ]
  },
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   "id": "42b9aabc",
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   "source": [
    "NormLike <- function(mu, sigma, data)\n",
    "  {\n",
    "    out = prod(dnorm(x = data, mean = mu, sd = sigma))\n",
    "    return(out)\n",
    "  }\n",
    "\n",
    "logNormLike <- function(mu, sigma, data)\n",
    "  {\n",
    "    out = sum(dnorm(\n",
    "        x = data, mean = mu, sd = sigma,\n",
    "        log = TRUE))\n",
    "    return(out)\n",
    "  }\n",
    "\n",
    "set.seed(123)\n",
    "data = rnorm(1000, mean=2, sd=1)\n",
    "NormLike(mu = 1, sigma = 1, data=data) \n",
    "logNormLike(mu = 1, sigma = 1, data=data) "
   ]
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   "source": [
    "\n",
    "### Walking APP example: How long do you walk every day?\n",
    "\n",
    "-   Here is a list about my past six days walking statistics. Can you\n",
    "    estimate how long do I walk everyday? and what is the variation?\n",
    "\n",
    "\n",
    "![walking](figures/walking.png)"
   ]
  },
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   "execution_count": 7,
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YVY4Fjw1RU/DMxuSrD82wGd7zjBT68W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\nULDkoGDJ+R9mRYU1spn4owAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title “Histogram of c(294, 262, 196, 79, 191, 677)”"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(c(294,262,196,79,191,677))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9bd295e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The likelihood function\n",
    "\n",
    "-   We assume everyday’s walking steps ($x_i$) are independent, and\n",
    "    $x_i$ follows standard normal distribution $\\sim  N(\\mu,\\sigma)$, the corresponding likelihood function is\n",
    "\n",
    "    $$\\prod_{i=1}^n f(x_i,\\mu,\\sigma)$$ which can be easily written in R\n",
    "\n",
    "-   **The scope** Find a proper combination of $\\mu$ and $\\sigma$ that\n",
    "    maximizes the loglikelihood function."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5944b92",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Conditional likelihood function\n",
    "\n",
    "-   Fix other parameters\n",
    "\n",
    "![conddensity](figures/conddensity.png)\n",
    "\n",
    "Left: fix variance to allow $\\mu$ to change with likelihood function. Right: fix mean to allow $\\sigma$ to change with likelihood function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f9071724",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "x <- c(294, 262, 196, 79, 191, 677)\n",
    "mu = 260:300\n",
    "sigma = 180:220\n",
    "\n",
    "parMat <- expand.grid(mu, sigma)\n",
    "muALL <- parMat[, 1]\n",
    "sigmaALL <- parMat[, 2]\n",
    "\n",
    "myLogLike <- matrix(NA, 1, length(sigma))\n",
    "for(i in 1:length(sigmaALL))\n",
    "{\n",
    "  myLogLike[i] <- logNormLike(mu = muALL[i], sigma = sigmaALL[i], data = x)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1916648",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "```R\n",
    "persp(as.vector(mu), as.vector(sigma), \n",
    "      matrix(myLogLike, length(mu),), \n",
    "      theta = 90, phi = 30, expand = 0.5, \n",
    "      col = \"lightblue\", xlab = \"mu\", \n",
    "      ylab = \"sigma\", zlab = \"log likelihood\", \n",
    "      ticktype = \"detailed\")\n",
    "\n",
    "filled.contour(as.vector(mu), as.vector(sigma), \n",
    "               matrix(myLogLike, length(mu),), \n",
    "               xlab = \"mu\", ylab = \"sigma\")\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "876dcc3b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- 2D and 3D loglikelihood function\n",
    "\n",
    " ![3dlike](figures/3dlike.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd211cfe",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "-   Are $\\mu$ and $\\sigma$ we obtained the best combination?\n",
    "\n",
    " ![2dlike](figures/2dlike.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f7f59d6",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Likelihood function for linear regression\n",
    "\n",
    "-   Assume you want to make a regression model\n",
    "    $$y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
    "    where $\\epsilon_i \\sim N(0, \\sigma^2)$\n",
    "\n",
    "-   What is the (log) likelihood function?\n",
    "\n",
    "-   What are the unknown parameters?\n",
    "\n",
    "-   How do we estimate the parameters?\n",
    "\n",
    "    -   Write down a likelihood function with respect to the unknown\n",
    "        parameters.\n",
    "\n",
    "    -   Use an optimization algorithm to find the estimates of the\n",
    "        unknown parameters.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "60704413",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
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5E2e/AAvr7IzBRz9KAgmJpi1Sq4uZV+\n65tv4OUlIhPJ7eUFPXTo0Hr16vXt23fz5s0ZwmfpiF7Bgwfo3x+WlqhdGzY2aNMGPXvC2xv2\n9nj4UFiqHj0AwMEB0dFYuRKtW6NBA7Rvj59/xrRpwlKRzF4+B71kyZJNmzZt375969atJiYm\nvXv3HjBgQNeuXfV51yyST3IyfvgBc+ciN1c1kp6Oe/eEZgJ0dfHNN+jevejl8OEYPlxoJtIG\nLz+DHjVq1OHDhxMTExcvXtyyZctff/21R48etra2I0aMOHToUL7waytE/+/QIbi6YtasonaW\nwZtvIiYGU6eKzkFa6FUvEtra2o4ePbqwqT09PVetWtWxY0cHB4cJEyZERkZWaUqi8u3di4AA\ndOmCtDTRUYrR0UFICE6e5B30qZIqvIrD1tbWz8+vY8eOTk5OAO7cubNw4UIfHx83N7ctW7ZU\nQUKi0nJz8cMPeOst2NqiQwe0aYPAQPz5J/LyxOQxNlYtXi5kYoKvv8b161i6lHfQp8p71XXQ\nubm5R48e/f3337dv356QkADAzs4uJCSkT58+VlZW4eHhK1as6Nev38mTJ1u1alWVgdXs3r17\n48ePzy33n8TXr18HoJTn4cw1zP37OHgQjx/D21t1/7YvvsD8+ap3U1IERlPJyMCUKdi4EbGx\nANC2LVasgKen6Fik/V5e0Fu2bPn999937dr16NEjAM7Ozp999lmfPn18fHwU/3/vlhYtWnz4\n4YctWrTYsmWLdhW0oaFh48aNyy/oglWGCt6oRoRduzB4MB49Ur20tYWfHyT8p5qeHmJikJIC\nfX1YWopOQ9XFyws6ODgYQPPmzcePHx8UFNTsBfegdXZ2rlu3rpWVlZoDVjEzM7MZM2aUv83y\n5cv37dunmTxU3L17+OgjpKYWjSQn47ffxAV6sYIbZdSrJzoHVS8vL+h58+YFBQU1bty4/M3M\nzMzuCV/NRNVLRESJdpbEO+/g0CFkZxeNNGiAwEBxgaj6evlFwkmTJr20nYmqgmztXK8e5s7F\nnj3YsweurqrBtm2xezcsLIQmo2qKN0siST15gsREkQEUCqxejdhYODigTx+YmhYt1ejUiTPO\npAksaJLRihUYMwY5OSIzBAdj6NDyNuCMM1U1FjRJRKnEjh04cABLlkDsssZ+/bBihcgARGBB\nkzyystClC44eFRyjY0ds3Ii6dQXHIALvB03C7dsHX1+YmsLERHw729lhzx62M8mCZ9CkUWvX\nYuFC3L8PJyc4OODECcTHi8yzciWsrfHvfyM9Hf36ISREZBiiUljQpCEZGWjfHqdOqV7evCk0\nDQDAzAzDhkGhwHvviY5CVBYWNFWJnBz85z+4cQMuLujSBQoFOnUqamdJLF8OfoCfZMaCJvXI\nzsbatYiKgqUl2rXDuHG4ckX1lqUlJk/G8eOajlSnTtFNPAwN8e676N0b+/bh2jU0aIAxY+Dv\nr+lIRBXCgiY1ePoUb7+N8+dVLxWKEovkHj7EP/8pINW8eXB2xvnzsLVF9+4wMQGAjz4SkISo\ncljQpAaff17UzoDgJcwFmjfHxx9DoUD79qKjEFUWl9nR68rNxbp1gjOUuim+hQWOH+f8Mmk9\nFjS9rrNn8fSpyAC6uti3D/37o1492Nqif3/Ex8PYWGQkIrXgFAeVISMD+/bhzh14eKBduxJv\n5eTg2bMSN28Te4Nma2ssXIgOHdChg8gYRFWBZ9BU2oULcHdHUBBGjUL79ujcGenpAJCUhOBg\n1K6NOnXg5obduwFg1izMmSMs6po1SEzEBx8IC0BUpVjQVIJSiQ8+wI0bRSP//S++/hq5uejT\nB1u2qO4wFxuLoCAcOIBvvhEUFLC2Rp8+fCQrVWcsaCrhxg1ER5ce3L0bkZGIjCwxmJODjz9G\nfr7GokGn2E+rhQV++UW1co6ouuIcNJVQ5kNM7t1DeHgZ47dvq/noffrA2Rk3b8LfH/n5mDCh\n6C8AQ0McOoTMTNVnYd59F9bWaj46kWxY0FSChweMjZGRUWIwNRVLl1btcXV18fPP+PDDEoOd\nOmHxYty4gTfewLhxqgez8mIg1RwsaCohJwdmZqULuioYGSEzU/V1y5Y4dAimpqW3cXfHjz9W\neRIiaXEOukb7z3/w7rvw8kK/fjh9GgAOHUJKSpUft107JCVh0yZ8/z327cOpU2W0MxHxDLrG\nyc9XXW1bvRrDh6sGL17Etm2YOxf792sig5kZ6tRBv36aOBaR9tLuM+j8/Py4uLhLly7l5uaK\nziK71FSMGYO6dWFsDF9fHD6MCRNKbJCXh9BQ7NuniTAXLmjiKETaTmsKesqUKatXry58mZub\nGxYWZm5u7urq6uHhYWJiMnLkyMePHwtMKLOC1c1LluDBA2Rn4/hxdO6MtLQytlTvsjl9/bLX\nWnBCg+hVaE1Bz5gxY12xW/KEhoZ+8cUX+vr6wcHBI0eOfOutt1asWOHr65uVlSUwpLSiovCf\n/5QYycur8oMGBCA1FTt2lPFW//5VfnSiakBrCrq46OjoxYsXt2nTJi4ubvPmzcuWLTt+/Piq\nVasuXbo0c+ZM0emkk5SEr77S6BH19LB3Lw4dQq1a8PHBqFEl3m3VCl9+qdE8RFpKKy8SHjt2\nTKlULliwwMrKqnBw2LBhq1ev3rt377Rp0159V3l5ebt27crOzi5nmzNnzlQ+q2g3bqBFi6IH\ni1QpHR34+sLXF99+i9q1i8aXLEHfvvj9dzx9ioAADBwIXV1N5CHSdlpZ0Ldv3wbg5eVVatzL\nyyu8zE+8lburTz/9NLNwRW5ZCqZNlDLchb6CcnIwdqyG2tnYGKtWvfC+RR07omNHTcQgqk60\nsqBdXFwAJCQkeHh4FB9PTk5u2LBhhXbl5OSUlJRU/jbLly8PCQlRaMPt33NzEROD9HR4eODI\nEfzjH0hM1NChf/0VvXtr6FhENYQ2zUHHxsbOmDHjt99+c3Nzs7a2njFjRvF3T506tXv37tat\nW4uKJ9yJE/DwgKcn2rSBvT2CgqqkncucnejQAe+9p/5jEdV0Si3h6Oj4/DnswYMHC9798ssv\njY2Nraysbt68qfZDL1u2DEBaWpra96xG9+8r7eyUQNX+Z2mpXLdO+e23yoYNlYaGSiMjpaOj\n8rPPlI8fi/7/J6qsgjnMiIgI0UHKoDVTHDdv3szIyLh69WpsbGxcXFxcXFxsbKyenir/9u3b\n69atu27dOkdHR7E5NSwzE8OGYcsWlHuZs8L09VX3fS40YgSCg9GmDczNAaAiF2KJqJK0pqAB\nGBsbe3l5PX9tEMCWLVuaNGmio6NNMzaVEBWFK1fg6AhfX9XHtbt2xdGjaj6KuTlu3sTp05g1\nC5cvo0EDjB6NgQP5DFYiTdOmgi6Hu7u76AhV69kz9OuHvXtVL1u2xLZtOHhQ/e0MoE4dmJlx\n3QWReNWkoKu9zz8vamcAZ87ggw9w9myVHKtbtyrZLRFVVDWfE6g2tm0rPRIRgWfPKraT3r1h\nZvaSbdq2RVhYxXZLRFWEZ9BaIDsb9++/7k4UCsyahTNnSj+1pMDQofDygrs7unZFdZ/JJ9Ia\nLGjZPXyITp1Kr6moBKUS+voYNAh2dvjyS5w5U3Tjur59sWoVe5lIOixo2YWGqm2u+fRpODuj\nY0ecPIlbt7BnD1JT0aYNn/JHJCkWtFwSEnDrFlxdYWOjGvnjj8rsR6HA8/cOuXu36GtHR4wc\nWamIRKQp/GeteHfu4M8/ceYMevZEw4bw94edHUJCkJsLpRLp6S/fg4EBateGuTn8/LB2Lf78\nE0FBZWxWr57asxNRFeIZtEi5uRg9GitXln6OSX4+li+HjQ2++w4+Pi9/DFVODq5fR/36RSOP\nH2Pr1hLbWFoiIEA9sYlIM3gGLdL06Vix4oVPmfrlFwBYsODla+OUSkRFlRjp2RPTp8PAQPWy\nXj2sX180bUJEWoFn0GKcO4clS1D+zasTEtCyJRo1wk8/4dQpnDmDiIgXzng8X77ffIOhQ3Hy\nJExM0LYtHwNIpH1Y0ALs2YP33nv5UwHz8/H33/j7b2zZgg0bMGsWUlOxfDm2bMGpUyW2bNwY\nzZuXsQcHBzg4qC02EWkYpzgEGDWqws9snTABACws8MUXiIxESEjRsmVnZ2zeDENDNYckIuF4\nBq1pyclISHjJNs8vkktORmKi6jKgQoGlSxEainPnYGUFX1+2M1H1xILWNBMT6Oi88MIgAAsL\nAEhNLTGoq6saL+TiAheXKshHRNLgFIemmZjA3/+F71pYYN26Mh7u16VLiedkE1FNwDNoAX76\nCd26ITZW9fLNNzF3LuLiYGGBzp1hbQ1/f9y4gcOHVRu0bIlVqwRlJSJxWNACNGqECxewdy9u\n3ICLC955B7q66Ny5aANzcxw6hMhIxMbCyQl+fryTEVFNxIIWw8AAvXq9ZBtvb3h7ayQNEUmJ\nJ2ZERJLS1jPoJ0+epKWl6ejo1KtXr9o/K5aIaiYtq7aLFy8OGTLEzs7O3NzcwcHB3t7ewMDA\nwcFh4MCBERERAoPl5SE+vsLPoCIiKoc2FfTYsWObNWu2du1ahULh7e0dGBgYGBjYunVrhUKx\nfv16Pz+/Tz75REiwBQtgaYnGjWFqioED8fChkBREVN1ozRTHjz/+uHjx4nfeeWfWrFlvvfVW\nqXejo6OnT5++cuXKpk2bhoaGajLY2rWYOFH1tVKJ9euRloadOzUZgYiqJ605gw4PD3dzc9u1\na9fz7QzAw8Nj/fr1/v7+W0vdBbnqrVhRemTXLiQmajgFEVVDWlPQFy9e9PHx0dN74Sm/QqHw\n9/e/ePFihXZ7/fp1Y2NjRblCQkIK9l/mHq5de9VBIqIK0ZopDk9Pz8jIyLy8PF1d3Rdtc/z4\ncU9PzwrttlGjRvv27cvOzi5nm+jo6AkTJujr65f5bpMmSE4uY5CI6DVpTUEPGjRo9OjRPXv2\nnDNnjpeXV6l3Y2Njp02bdujQobCwsArtVqFQtGvXrvxtatWqVc67n31W9JnsAkOG8NklRKQG\nWlPQo0aNunDhwrJly/bu3evo6Ojk5GRpaalQKB49enTr1q34+HgAQ4cO/eyzzzQc7N13sWkT\npkxBbCzq1MHw4Zg6VcMRiKh6UihL3XhYbmfPng0LC9u/f//9+/cLRnR1dW1sbAICAkaOHNm+\nffuqOOixY8fefvvtrKwsg8Jn/JUlK4v3ZSbSPtnZ2YaGhhEREb6+vqKzlKY1Z9AF3nzzzV9/\n/RVAampqWlqavr6+jY1NVX+SsKCXDdm+RNVX+adfomjZGbQo586dy83NfZUtJ06caGxs/NFH\nH1V1pFeRmZn5ySeffPfdd40aNRKdBQAOHDhw4MCB2bNniw6iMm3atBYtWvTs2VN0EABITk7+\n5z//+cMPP9SpU0d0FgDYvHnz9evXv/jiC9FBVMaPHz927Ngq+s3S09NrXuZjPYVTklr16dNn\n3LhxolOopKWlATh16pToICpLlixxd3cXnaKIj4/PrFmzRKdQiYmJAZCYmCg6iMrXX3/dpUsX\n0SmKODo6rl27VnQKTdOaddBERDUNC5qISFIsaCIiSbGgiYgkxYImIpIUC5qISFIsaCIiSbGg\niYgkxYImIpIUC1rNDAwM5PlQv56eno6Ojjx5pPrFgWR5DAwMFAqFPHn09fXlCQPJfrM0hvfi\nULP79+8bGBiYmZmJDqJy/fr1xo0bi06hkp2dfffuXQcHB9FBVO7cuWNhYWFsbCw6iIpUv1lP\nnz5NT0+3kebW5jdv3rS3ty/nmUrVEguaiEhSnOIgIpIUC5qISFIsaCIiSbGgiYgkxYImIpIU\nC5qISFIsaCIiSbGgiYgkxYImIpIUC5qISFIsaCIiSbGgiYgkxYImIpIUC5qISFIs6Gru6tWr\nixcvFp2CXtXTp0/XrFlz+/Zt0UFICixotVm6dKmfn5+FhYWfn9/SpUtFx1FZtGjRlClTRKdA\nVlbW119/3a5dO3Nzc2dn54EDB167dk1UmPj4+IEDB7q4uNSuXdvLy+vzzz9//PixqDCljB07\ndujQoefOnROYwdHRUfEcgT9FR48e7dy5s7m5ub29ff/+/QX+5AigJHUICQkB4ObmNnjwYFdX\nVwBjxowRHUr5xx9/GBoaWlhYiI2Rmprq7+8PwN3d/R//+EfXrl0VCoWxsXFUVH1BzMUAAAmO\nSURBVJTmw8TFxdWuXVtPT69jx44hISHe3t4APDw8MjIyNB+mlM2bNxf8qdy1a5eoDOnp6QqF\nwt7ePqCkVatWCcmzYcMGAwMDe3v7gQMH9urVS1dX18rKKiEhQUgYzWNBq0FUVBSAbt265eTk\nKJXKnJycgg66cOGCqEiDBg1yc3Mr+NMuvKAnT54MYPTo0YUju3fv1tHRad68uebD9O3bV6FQ\n7Nixo3Bk4sSJABYtWqT5MMXdvn3b0tLSxMREbEGfP38ewPTp00UFKC4hIUFPT8/b2zs1NbVg\n5N///jeAIUOGCM2lOZziUIOwsDAAc+bMKXhgmp6e3qxZs5RK5dy5c0VFSk9Pd3Fx6dGjh6mp\nqagMhbZt22Zqavqvf/2rcCQwMLBjx47nzp27e/euhsP89ddfLVq06NmzZ+HIxx9/DODvv//W\ncJLilErl4MGDzc3Nx40bJzAGgNjYWABNmjQRG6PAwoULc3NzFyxYYG5uXjAyfPjw77//3sfH\nR2wwjalZT2CsIvv373dwcGjWrFnhSIsWLezs7P744w9RkbZu3VrwhZeXl/ArTjo6Ou3btzc0\nNCw+WPCE5kePHmnysaT5+flTpkxxcnIqPpiSkgLgjTfe0FiM5/3rX/86fPjwn3/+GRERITAG\ngLi4OABOTk7h4eFxcXEODg6+vr7u7u5CwmzYsMHR0bF4HSsUigkTJggJI4boU3it9+jRIwBv\nv/12qfGCyc0nT54ISVXI09NT+BTH8+7evWtkZFSvXr2CSSEh0tPTExMT9+zZ4+LiUq9evdjY\nWFFJoqKiDAwMJk+erFQqZ8+eDaFTHMOGDQNgbW1dWBE6Ojpjx47V/O9UWloaAH9//7Nnz/bs\n2dPGxsbR0TE4ODguLk7DSQTiFMfrKvgxsrKyKjVeMPLkyRMBmeQWGxvr6+ubmZk5e/bsgkkh\nIUJDQ+vXrx8YGJiUlFRQ00JiZGRkDBo0yN3dferUqUIClFIwxdGpU6fz58+npaX99ddfLVu2\nXLRo0fz58zWcJDU1FUBSUpKfn9+NGzd69Ojh4eGxdevW5s2bnz59WsNhhBH9N4TWu3PnDoBe\nvXqVGg8MDASQlJQkJFUhqc6gnz59+u233xobGxsZGS1evFhsmLNnz27cuPF///d/GzRoYGho\nuH37diExRo8ebWRkdPHixYKXws+gjxw5cvDgweIj9+7dq1OnjomJSV5eniaTXL9+vaCjvvzy\ny/z8/ILB/fv3KxSKt956S5NJBGJBv668vDxdXd127dqVGvfx8dHV1dXwz/Tz5CnoPXv2NGjQ\nAECPHj2uXLkiOk6RxMREU1PT+vXra/7QBw4cAPD9998Xjggv6DIFBwcD0PAsUHJyMgArK6vc\n3Nzi4127dgWQkpKiyTCicIrjdeno6NjY2Dx/IS4xMdHW1lZHh7/CAPA///M/gYGBpqamf/75\n586dOwuXAGrYtWvXli9ffvHixeKD9vb2rVq1SkxMLLicoElnz54FMHHixMLPg3z55ZcAevTo\noVAoVq1apeE8L1IwX5eTk6PJg1pbWxsZGTVq1EhXV7f4eOPGjQEIv/StGVzFoQYBAQHr16+P\njY0t+IgKgOjo6Fu3bn3wwQdig0lizZo133333YABA9asWVOweEOUlJSUkJCQcePGLVy4sPj4\nvXv3TExMCtdyaUzz5s0LPuJUKCoqKjIysnv37k5OTppf63bp0qW+ffsGBQXNnDmz+Pi5c+cM\nDQ0Lf7w1Q0dHJyAg4NixY5mZmUZGRoXjly9f1tHREfV3vKaJPoWvDg4fPgzgww8/LHiZn5/f\nv39/AEePHhUbTCnBFEd+fr6bm1v9+vVl+Khedna2jY2Nubn5tWvXCgc3bNiAsq4iCCF2iiMv\nL8/R0dHY2PjkyZOFgwUn8iNGjNB8nn379gEYPXp04VThxo0bAfTo0UPzYYTgGbQatG/ffujQ\noT///HNSUpKPj89ff/115MiR4cOH+/n5iY4mXkJCQkxMjLW1dVBQ0PPvrlu3rm7duhoLo6+v\nv2jRogEDBnh5eQUGBtrY2Fy+fPnQoUP16tVbsmSJxmJIS0dHZ926dX369Hn77bd79Ohha2t7\n/vz5iIiIpk2bzpkzR/N5unbtOnTo0CVLlhw5cqRt27bx8fH79++3s7OT5143VU703xDVRH5+\n/pw5c3x9fc3MzHx9fefOnSs6kYrwM+j//ve/5fz43b59W/ORDh482K1bNysrq1q1ajVv3jw0\nNPThw4eaj1EmGS4SJiQkfPzxx56eniYmJq1atZoyZYrYf/3MmzfPz8/P1NTU3d19zJgx8vxm\naYBCqVRq6u8CIiKqAK4xICKSFAuaiEhSLGgiIkmxoImIJMWCJiKSFAuaiEhSLGgiIkmxoImI\nJMWCJiKSFAuaiEhSLGgiIkmxoImIJMWCJiKSFAuaiEhSLGgiIkmxoImIJMWCJiKSFAuaiEhS\nLGgiIkmxoImIJMWCJiKSFAuaiEhSLGgiIkmxoImIJMWCJiKSFAuaiEhSLGgiIkmxoImIJMWC\nJiKSFAuaiEhSLGgiIkmxoImIJMWCJiKSFAuaiEhSLGgiIkmxoImIJMWCJgKA6OhoQ0PDDh06\nFI7k5OR4eXlZWVklJycLDEY1GQuaCAA8PDy++uqrw4cP//TTTwUjc+fOvXjx4g8//GBrays2\nG9VYCqVSKToDkRSys7NbtmyZlJR05cqVx48fe3l5denSZceOHaJzUc3FgiYqEhkZ6evrO2DA\ngOTk5KioqOjoaDs7O9GhqObSEx2ASCLe3t7jx4///vvvAaxdu5btTGLxDJqohKtXr7q4uNSu\nXTspKcnMzEx0HKrReJGQqIRJkyYZGBg8e/Zs8uTJorNQTceCJioSHh6+Y8eO2bNnBwcHL126\n9NixY6ITUY3GKQ4ilZSUFA8Pj4YNG0ZGRqakpDRt2tTBwSEqKsrAwEB0NKqheAZNpDJq1KjU\n1NQVK1bo6ura29vPnDnz0qVLs2fPFp2Lai6eQRMBwMaNGwcMGDBp0qR58+YVjOTn57dt2/bc\nuXNRUVFNmzYVG49qJhY0EZGkOMVBRCQpFjQRkaRY0EREkmJBExFJigVNRCQpFjQRkaRY0ERE\nkmJBExFJigVNRCQpFjQRkaRY0EREkmJBExFJigVNRCQpFjQRkaRY0EREkmJBExFJigVNRCQp\nFjQRkaRY0EREkmJBExFJigVNRCQpFjQRkaRY0EREkmJBExFJigVNRCQpFjQRkaRY0EREkmJB\nExFJigVNRCQpFjQRkaT+DyPjRYuycX2IAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta0 <- 1\n",
    "beta1 <- 3\n",
    "sigma <- 0.5\n",
    "\n",
    "n <- 1000\n",
    "x <- rnorm(n, 3, 1)\n",
    "\n",
    "y <- beta0 +x*beta1 + rnorm(n, mean = 0, sd = sigma)\n",
    "plot(x, y, col = \"blue\", pch = 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "706f5887",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in dnorm(x = y, mean = mean, sd = sigma, log = TRUE):\n",
      "“NaNs produced”\n",
      "Warning message in dnorm(x = y, mean = mean, sd = sigma, log = TRUE):\n",
      "“NaNs produced”\n",
      "Warning message in dnorm(x = y, mean = mean, sd = sigma, log = TRUE):\n",
      "“NaNs produced”\n"
     ]
    }
   ],
   "source": [
    "logNormLikelihood <- function(par, y, x)\n",
    "    {\n",
    "        beta0 <- par[1]\n",
    "        beta1 <- par[2]\n",
    "        sigma <- par[3]\n",
    "        mean <- beta0 + x*beta1\n",
    "        logDens <- dnorm(x = y, mean = mean, sd = sigma, log = TRUE)\n",
    "        loglikelihood <- sum(logDens)\n",
    "        return(loglikelihood)\n",
    "    }\n",
    "\n",
    "optimOut <- optim(c(0.2, 0.3, 0.5), logNormLikelihood,\n",
    "                  control = list(fnscale = -1),\n",
    "                  x = x, y = y)\n",
    "\n",
    "beta0Hat <- optimOut$par[1]\n",
    "beta1Hat <- optimOut$par[2]\n",
    "sigmaHat <- optimOut$par[3]\n",
    "yHat <- beta0Hat + beta1Hat*x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "ea97b1ad",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jv0h5YwHyIBEvCez6AwgrfSooAlNW3KqFFYLLRvX8A9RExHAS03KjWViIgC9byL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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "myLM <- lm(y~x)\n",
    "myLMCoef <- myLM$coefficients\n",
    "yHatOLS <- myLMCoef[1] + myLMCoef[2]*x\n",
    "\n",
    "plot(x, y, pch = 20, col = \"blue\")\n",
    "points(sort(x), yHat[order(x)], type = \"l\", col = \"red\", lwd = 5)\n",
    "points(sort(x), yHatOLS[order(x)], type = \"l\", lty = \"dashed\",\n",
    "       col = \"yellow\", lwd = 2, pch = 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c51a5cb6",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Loss function\n",
    "\n",
    "- Every statistical learning algorithm is trained to learn a prediction. \n",
    "\n",
    "- These predictions should be as close as possible to the label value (ground-truth value). \n",
    "\n",
    "- The loss function measures how near or far are these predicted values compared to original label values.\n",
    "\n",
    "- Loss functions are also referred to as error functions as it gives an idea about prediction error.\n",
    "\n",
    "- Loss Functions, in simple terms, are nothing but an equation that gives the error between the actual value and the predicted value.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffc72b09",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Mean squared error loss\n",
    "\n",
    "- Mean squared error (MSE) can be computed by taking the actual value and predicted value as the inputs and returning the error via the below equation (mean squared error equation).\n",
    "$$Loss(\\beta_0, \\beta_1; y, x) = \\frac{1}{N}\\sum_{i=1}^{N}(y_i-(\\beta_0+\\beta_1 x_i))^2$$\n",
    "\n",
    "- The equation is very simple and thus can be implemented easily as a computer program. Besides that, it is powerful enough to solve complex problems.\n",
    "\n",
    "- The equation we have is differentiable hence the optimization becomes easy. This is one of the reasons for adopting MSE widely"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "3f6a8d35",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "loss <- function(par, y, x)\n",
    "    {\n",
    "        beta0 <- par[1]\n",
    "        beta1 <- par[2]\n",
    "        yHat <- beta0 + x*beta1\n",
    "        out <- mean((y- yHat)^2)\n",
    "        return(out)\n",
    "    }\n",
    "\n",
    "optimLossOut <- optim(c(0.2, 0.3), loss, x = x, y = y)\n",
    "\n",
    "beta0Hat <- optimLossOut$par[1]\n",
    "beta1Hat <- optimLossOut$par[2]\n",
    "yHatLoss <- beta0Hat + beta1Hat*x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "9b1191e7",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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F9bsYWw2XEPOzwFTWEuRAFcwudLBkUy8lfyOldiw4YMH47FQuvWedxDxHQU0HKn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\nZrMVdlkdOx6t1b7y9rUZi5ukNmjoOsrKoEH4+RV2MSJyFykyAf3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RMSkFtIiISSmgRURMSgEtImJSCmgREZNSQIuImJQCWkTEpBTQIiIm\npYAWETEpBbSIiEkpoEVETEoBLSJiUgpoERGTUkCLiJiUAlpExKT+H1CAwo807wVHAAAAAElF\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "myLM <- lm(y~x)\n",
    "myLMCoef <- myLM$coefficients\n",
    "yHatOLS <- myLMCoef[1] + myLMCoef[2]*x\n",
    "\n",
    "plot(x, y, pch = 20, col = \"blue\")\n",
    "points(sort(x), yHatLoss[order(x)], type = \"l\", col = \"red\", lwd = 5)\n",
    "points(sort(x), yHatOLS[order(x)], type = \"l\", lty = \"dashed\",\n",
    "       col = \"yellow\", lwd = 2, pch = 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bde63ca2",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Discussion\n",
    "\n",
    "- What are the advantages and disadvantages of using likelihood function and loss function?"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.3.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}