happiness.ipynb

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    "## Maximum Likelihood Estimation\n",
    "\n",
    "The noise component $\\epsilon$ is often modeled as a random variable drawn from a Gaussian distribution (also called the normal distribution).\n",
    "\n",
    "The Gaussian distribution is described by its probability density function\n",
    "\\begin{align}\n",
    "\\mathcal{N}(x; \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{1}{2\\sigma^2}(x-\\mu)^2}\n",
    "\\end{align}\n",
    "\n",
    "and is dependent on the mean $\\mu$ and the variance $\\sigma^2$. If the noise is considered to be Gaussian \"white noise\", the mean is taken as zero and the variance as 1:\n",
    "\n",
    "\\begin{align}\n",
    "\\epsilon \\sim \\mathcal{N}(0, 1).\n",
    "\\end{align}\n",
    "\n",
    "We can also treat $y$ as a random variable drawn from a Gaussian distribution where $\\mu = \\theta x$ and $\\sigma^2 = 1$:\n",
    "\n",
    "\\begin{align}\n",
    "y \\sim \\mathcal{N}(\\theta x, 1)\n",
    "\\end{align}\n",
    "\n",
    "The probability of observing $y$ given $x$ and $\\theta$ is\n",
    "\\begin{align}\n",
    "p(y|x,\\theta) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{1}{2}(y-\\theta x)^2}\n",
    "\\end{align}\n",
    "\n",
    "The likelihood function $\\mathcal{L(\\theta)}$ is equal to the probability density function parameterized by that $\\theta$:\n",
    "\n",
    "\\begin{align}\n",
    "\\mathcal{L}(\\theta|x,y) = p(y|x,\\theta) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{1}{2\\sigma^2}(y-\\theta x)^2}\n",
    "\\end{align}"
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