04b_distributions.ipynb

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    "# Music Machine Learning - Probability distributions\n",
    "\n",
    "### Author: Philippe Esling (esling@ircam.fr)\n",
    "\n",
    "In this course we will cover\n",
    "1. A [quick recap](#recap) on simple probability concepts\n",
    "2. An introduction to [probability distributions](#distributions)\n",
    "3. An explanation on how to [sample](#sampling) from distributions ourselves"
   ]
  },
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    "<a id=\"recap\"> </a>\n",
    "\n",
    "## Quick recap on probability\n",
    "\n",
    "The field of probability aims to model random or uncertain events through random variables $X$ denoting a quantity that is uncertain, which take values $\\omega$ in a sample space $\\omega \\in \\Omega$. If we observe several occurrences of this variable $\\{\\mathbf{x}_{i}\\}_{i=1}$, we might try to model the _probability distribution_ $p(\\mathbf{x})$ of that variable. We recall here that the probability of an event $a$ is a real number $p(a)$, with $0 \\leq p(a) \\leq 1$, knowing that $p(\\Omega)=1$ and $p(\\{\\})=0$. The probability of two events occuring simultaneously is defined as $p\\left(a \\cup b \\right)$. Therefore, the probability of one event **or** the other occuring is defined as\n",
    "$\n",
    "\\begin{equation}\n",
    "p\\left(a \\cap b \\right) = p(a) + p(b) - p\\left(a \\cup b \\right)\n",
    "\\end{equation}\n",
    "$\n",
    "\n",
    "The *conditional probability* of an event $a$ occuring *given* another event $b$ is denoted $p \\left(a \\mid b \\right)$ and is defined as\n",
    "$$\n",
    "\\begin{equation}\n",
    "p \\left(a \\mid b \\right) = \\frac{p \\left(a , b \\right)}{p \\left(b \\right)}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "This can be understood as the probability of event $a$ to occur if we restrict the world of possibilities to event $b$. The *chain rule* defines the probabilities of a set of events to co-occur simultaneously\n",
    "$$\n",
    "\\begin{equation}\n",
    "p \\left(x_{1},...,x_{n} \\right) = \\prod_{i=n}^{1}{p \\left(x_{i}\\mid x_{i-1},..., x_{1} \\right)}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Finally, we say that two events are independent if $p(a\\mid b) = p(a)$.\n",
    "\n",
    "To understand these concepts graphically, we will rely on `PyTorch` and specifically the `distributions` package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch.distributions as distribution\n",
    "import torch.distributions.transforms as transform\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "from helper_plot import hdr_plot_style\n",
    "hdr_plot_style()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"distributions\"> </a>\n",
    "### Probability distributions\n",
    "\n",
    "Let $X$ be a random variable associated with a *probability distribution function* that assigns probabilities to the different outcomes $X$ can take in the sample space. We can divide random variables into three different types:\n",
    "\n",
    "- **$X$ is discrete**: Discrete random variables may only assume values on a specified list. \n",
    "- **$X$ is continuous**: Continuous random variable can take on arbitrarily exact values. \n",
    "- **$X$ is mixed**: Mixed random variables assign probabilities to both discrete and continuous random variables, (i.e. a combination of the above two categories). \n",
    "\n",
    "**Expected Value**\n",
    "The expected value $\\mathbb{E}\\left[X\\right]$ for a given probability distribution can be described as _\"the mean expected value for many repeated samples from that distribution.\"_ As the number of repeated observation goes to infinity, the difference between the average outcome and the expected value becomes arbitrarily small. Formally, it is defined for discrete variables as\n",
    "\n",
    "$$\n",
    "\\mathbb{E}\\left[X\\right] = \\sum\\limits_{i}\\bx_i p(\\bx=\\bx_i)\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Discrete distributions\n",
    "\n",
    "If $X$ is discrete, then its distribution is called a *probability mass function* (pmf), which measures the probability $X$ takes on the value $x_{i}$, denoted $P(X=x_{i})$. Let $\\mathbf{x}$ be a discrete random variable with range $R_{X}=\\{x_1,\\cdots,x_n\\}$ (finite or countably infinite). The function\n",
    "\n",
    "$$\n",
    "p_{X}(x_{i})=p(X=x_{i}), \\forall i\\in\\{1,\\cdots,n\\}\n",
    "$$\n",
    "\n",
    "is called the probability mass function (PMF) of $X$.\n",
    "\n",
    "Hence, the PMF defines the probabilities of all possible values for a random variable. The above notation allows to express that the PMF is defined for the random variable $X$, so that $p_{X}(1)$ gives the probability that $X=1$. For discrete random variables, the PMF is also called the \\textit{probability distribution}. The PMF is a probability measure, therefore it satisfies all the corresponding properties\n",
    "- $0 \\leq p_{X}(x_i) < 1, \\forall x_i$\n",
    "- $\\sum_{x_i\\in R_{X}} p_{X}(x_i) = 1$\n",
    "- $\\forall A \\subset R_{X}, p(X \\in A)=\\sum_{x_a \\in A}p_{X}(x_a)$\n",
    "\n",
    "A very simple example of discrete distribution is the `Bernoulli` distribution. With this distribution, _we can model a coin flip_, if it has equal probability. More formally, a Bernoulli distribution is defined as\n",
    "\n",
    "$$ \n",
    "Bernoulli(x)= p^x (1-p)^{(1-x)} \n",
    "$$\n",
    "\n",
    "with $p$ controlling the probability of the two classes. Hence, a fair coin should have $p=0.5$, and if we throw the coin a very large number of times, we hope to see on average an equal amount of _heads_ and _tails_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bernoulli = distribution.Bernoulli(0.5)\n",
    "samples = bernoulli.sample((10000,))\n",
    "plt.figure(figsize=(10,8))\n",
    "sns.distplot(samples)\n",
    "plt.title(\"Samples from a Bernoulli (coin toss)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, we can also _sample_ from the distribution to have individual values of a single throw. In that case, we obtain a series of separate events that _follow_ the distribution "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Coin is tossed on tails\n",
      "Coin is tossed on tails\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on tails\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on tails\n"
     ]
    }
   ],
   "source": [
    "vals = ['heads', 'tails']\n",
    "samples = bernoulli.sample((10, ))\n",
    "for s in samples:\n",
    "    print('Coin is tossed on ' + vals[int(s)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can mess up our probability to model an unfair (loaded) coin, as shown in the following example (where we use a cheated coin that should give us a lot more of _heads_ than _tails_) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Coin is tossed on tails\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on tails\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n",
      "Coin is tossed on heads\n"
     ]
    }
   ],
   "source": [
    "bernoulli = distribution.Bernoulli(0.3)\n",
    "samples = bernoulli.sample((10000,))\n",
    "plt.figure(figsize=(10, 8)); sns.distplot(samples)\n",
    "plt.title(\"Samples from an unfair coin\"); plt.show()\n",
    "vals = ['heads', 'tails']\n",
    "samples = bernoulli.sample((10, ))\n",
    "for s in samples:\n",
    "    print('Coin is tossed on ' + vals[int(s)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Poisson distribution\n",
    "\n",
    "Let's introduce one of the (many) useful probability mass functions. We say $Z$ is *Poisson*-distributed if:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "P\\left(Z = k\\right) =\\frac{ \\lambda^k e^{-\\lambda} }{k!}, \\; \\; k \\in \\mathbb{N^{+}} \n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "$\\lambda \\in \\mathbb{R}$ is a parameter of the distribution that controls its shape (usually termed the *intensity* of the Poisson distribution). By increasing $\\lambda$, we add more probability to larger values. One can describe $\\lambda$ as the *intensity* of the Poisson distribution. If a random variable $Z$ has a Poisson mass distribution, we denote it by\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "Z \\sim \\text{Poi}(\\lambda) \n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "One useful property of the Poisson distribution is that its expected value is equal to its parameter, i.e.:\n",
    "\n",
    "$$E\\large[ \\;Z\\; | \\; \\lambda \\;\\large] = \\lambda $$\n",
    "\n",
    "We will plot the probability mass distribution for different $\\lambda$ values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Continuous distributions\n",
    "\n",
    "The same ideas apply to _continuous_ random variables, which can model for instance the height of human beings. If we try to guess the height of someone that we do not know, there is a higher probability that this person will be around 1m70, instead of 20cm or 3m. For the rest of this course, we will use the shorthand notation $p(\\mathbf{x})$ for the distribution $p(\\mathbf{x}=x_{i})$,  which expresses for a real-valued random variable $\\mathbf{x}$, evaluated at $x_{i}$, the probability that $\\mathbf{x}$ takes the value $x_i$.\n",
    "\n",
    "One notorious example of such distributions is the Gaussian (or Normal) distribution, which is defined as \n",
    "\\begin{equation}\n",
    "    p(x)=\\mathcal{N}(\\mu,\\sigma)=\\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}e^{-\\frac{(x-\\mu)^{2}}{2\\sigma^{2}}}\n",
    "\\end{equation}\n",
    "\n",
    "Similarly as before, we can observe the behavior of this distribution with the following code (in our height example)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "normal = distribution.Normal(150., 30.)\n",
    "samples = normal.sample((10000, ))\n",
    "sns.distplot(samples)\n",
    "plt.title(\"Samples from a standard Normal\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we have access to this complete probability distribution (its exact parameterization and function), we can generate samples (in this case \"new humans\") that follow the correct distribution. You can experiment with this in the following code space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "######################\n",
    "# YOUR CODE GOES HERE\n",
    "######################\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An example of continuous random variable is a random variable with *exponential density*\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "f_Z(z | \\lambda) = \\lambda e^{-\\lambda z }, \\;\\; z\\ge 0\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "When a random variable $Z$ has an exponential distribution with parameter $\\lambda$, we say *$Z$ is exponential*\n",
    "\n",
    "$$Z \\sim \\text{Exp}(\\lambda)$$\n",
    "\n",
    "Given a specific $\\lambda$, the expected value of an exponential random variable is equal to the inverse of $\\lambda$, that is\n",
    "\n",
    "$$E[\\; Z \\;|\\; \\lambda \\;] = \\frac{1}{\\lambda}$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"distribs\"></a>\n",
    "### PyTorch distributions\n",
    "\n",
    "Here, we rely on the [PyTorch distributions module](https://pytorch.org/docs/stable/_modules/torch/distributions/), which is defined in `torch.distributions`. Most notably, we are going to rely both on the `Distribution` and `Transform` objects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports for plotting\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "# Define grids of points (for later plots)\n",
    "x = np.linspace(-4, 4, 1000)\n",
    "z = np.array(np.meshgrid(x, x)).transpose(1, 2, 0)\n",
    "z = np.reshape(z, [z.shape[0] * z.shape[1], -1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inside this toolbox, we can already find some of the major probability distributions that we are used to deal with\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = distrib.Normal(loc=0, scale=1)\n",
    "p = distrib.Bernoulli(probs=torch.tensor([0.5]))\n",
    "p = distrib.Beta(concentration1=torch.tensor([0.5]), concentration0=torch.tensor([0.5]))\n",
    "p = distrib.Gamma(concentration=torch.tensor([1.0]), rate=torch.tensor([1.0]))\n",
    "p = distrib.Pareto(alpha=torch.tensor([1.0]), scale=torch.tensor([1.0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The interesting aspect of these `Distribution` objects is that we can both obtain some samples from it through the `sample` (or `sample_n`) function, but we can also obtain the analytical density at any given point through the `log_prob` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Based on a normal\n",
    "n = distrib.Normal(0, 1)\n",
    "# Obtain some samples\n",
    "samples = n.sample((1000, ))\n",
    "# Evaluate true density at given points\n",
    "density = torch.exp(n.log_prob(torch.Tensor(x))).numpy()\n",
    "# Plot both samples and density\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, sharex=True, figsize=(15, 4))\n",
    "ax1.hist(samples, 50, alpha=0.8);\n",
    "ax1.set_title('Empirical samples', fontsize=18);\n",
    "ax2.plot(x, density); ax2.fill_between(x, density, 0, alpha=0.5)\n",
    "ax2.set_title('True density', fontsize=18);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here you can experiment with different distributions, and try to compare how they behave depending on their parameters and also on how much _samples_ you draw from these."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "######################\n",
    "# YOUR CODE GOES HERE\n",
    "######################\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"sampling\"></a>\n",
    "## Sampling from distributions\n",
    "\n",
    "The advantage of using probability distributions is that we can *sample* from these to obtain examples that follow the distribution. For instance, if we perform sampling repeatedly (up to infinity) from a Gaussian PDF, the different values will be distributed following the exact Gaussian distribution. However, although we know the PDF, we need to compute the *Cumulative Distribution Function* (CDF), and then its inverse to obtain the sampling function. Therefore, if we denote the PDF as $f_{X}(x)$, we need to compute the CDF \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "F_{X}\\left(x\\right)=\\intop_{\\infty}^{x}f_{X}\\left(t\\right)dt\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Then, the *inverse sampling method* consists in solving and applying the inverse CDF $F_{X}^{-1}\\left(x\\right)$. Here, we recall that the Exponential probability is defined with the following function.\n",
    "\n",
    "$$\n",
    "p_{\\lambda}(x) = \\lambda e^{-\\lambda x}\n",
    "$$\n",
    "\n",
    "with $\\lambda$ defining the _rate_ parameter. Therefore, to be able to define our own `sample` method, we need to solve for\n",
    "\n",
    "$$\n",
    "F^{-1}_Y(x) = \\left(\\int_0^x \\lambda e^{-\\lambda y} dy\\right)^{-1}\n",
    "$$\n",
    "\n",
    "***\n",
    "\n",
    "**Exercise**\n",
    "\n",
    "1. Code the exponential probability density functions \n",
    "2. Perform the inverse transform method on the exponential distribution PDF\n",
    "3. Code the `sample_exponential` function to sample from the exponential\n",
    "4. (optional) Perform the same operations for sampling from the Beta distribution\n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import expon\n",
    "nb_samples = 500\n",
    "nb_bins = 50\n",
    "\n",
    "def sample_exponential(mu, n, m):\n",
    "    \n",
    "    ######################\n",
    "    # YOUR CODE GOES HERE\n",
    "    ######################\n",
    "    \n",
    "    ######################\n",
    "    # Solution\n",
    "\n",
    "    samples = np.random.uniform(0, 1, size=(m,n))\n",
    "    samples = - np.log(1 - samples) \n",
    "    samples *= mu\n",
    "    \n",
    "    ######################\n",
    "    \n",
    "    return samples\n",
    "\n",
    "# Exponential distribution\n",
    "mu = 2\n",
    "samples = np.random.exponential(mu, nb_samples)\n",
    "samples_ex = sample_exponential(mu, nb_samples, 1)\n",
    "# Compute the PDF\n",
    "X = np.linspace(0, np.max(samples), int(np.max(samples)) * 100)\n",
    "y1 = expon.pdf(X) #* (nb_samples / nb_bins) * int(np.max(samples) * 1.5)\n",
    "# Display both\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.subplot(2,1,1)\n",
    "plt.hist(samples, 50, label='Samples')\n",
    "plt.plot(X,y1,ls='--',c='r',linewidth=2, label='Exponential PDF')\n",
    "plt.legend(loc=1)\n",
    "plt.subplot(2,1,2)\n",
    "plt.hist(samples_ex, 50, label='Samples')\n",
    "plt.plot(X,y1,ls='--',c='r',linewidth=2, label='Our exponential sampples')\n",
    "plt.legend(loc=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x16852a350>"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def sampleBeta(a, b, M, N):\n",
    "    \n",
    "    ######################\n",
    "    # YOUR CODE GOES HERE\n",
    "    ######################\n",
    "    \n",
    "    ######################\n",
    "    # Solution\n",
    "\n",
    "    samples = np.random.beta(a, b, size=(M, N))\n",
    "    \n",
    "    ######################\n",
    "    \n",
    "    return samples\n",
    "\n",
    "from scipy.stats import beta\n",
    "# Beta distribution\n",
    "a = 0.6\n",
    "b = 1.5\n",
    "samples = np.random.beta(a, b, nbSamples)\n",
    "samplesBeta = sampleBeta(a, b, nbSamples, 1)\n",
    "# Compute the PDF\n",
    "X = np.linspace(0, 1, 100)\n",
    "y1 = beta.pdf(X, a, b) * (nbSamples / nbBins) * (np.max(samples) * 1.5)\n",
    "# Display both\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.subplot(2,1,1)\n",
    "plt.hist(samples, 50, label='Samples')\n",
    "plt.plot(X,y1,ls='--',c='r',linewidth=2, label='Beta PDF')\n",
    "plt.legend(loc=1)\n",
    "plt.subplot(2,1,2)\n",
    "plt.hist(samplesBeta, 50, label='Samples')\n",
    "plt.plot(X,y1,ls='--',c='r',linewidth=2, label='Beta PDF')\n",
    "plt.legend(loc=1)"
   ]
  },
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   "source": [
    "### Comparing distributions (KL divergence)\n",
    "$\n",
    "\\newcommand{\\R}{\\mathbb{R}}\n",
    "\\newcommand{\\bb}[1]{\\mathbf{#1}}\n",
    "\\newcommand{\\bx}{\\bb{x}}\n",
    "\\newcommand{\\by}{\\bb{y}}\n",
    "\\newcommand{\\bz}{\\bb{z}}\n",
    "\\newcommand{\\KL}[2]{\\mathcal{D}_{\\text{KL}}\\left[#1 \\| #2\\right]}$\n",
    "Originally defined in the field of information theory, the _Kullback-Leibler (KL) divergence_ (usually noted $\\KL{p(\\bx)}{q(\\bx)}$) is a dissimilarity measure between two probability distributions $p(\\bx)$ and $q(\\bx)$. In the view of information theory, it can be understood as the cost in number of bits necessary for coding samples from $p(\\bx)$ by using a code optimized for $q(\\bx)$ rather than the code optimized for $p(\\bx)$. In the view of probability theory, it represents the amount of information lost when we use $q(\\bx)$ to approximate the true distribution $p(\\bx)$. %that explicit the cost incurred if events were generated by $p(\\bx)$ but charged under $q(\\bx)$\n",
    "\n",
    "\n",
    "Given two probability distributions $p(\\bx)$ and $q(\\bx)$, the Kullback-Leibler divergence of $q(\\bx)$ _from_ $p(\\bx)$ is defined to be\n",
    "\\begin{equation}\n",
    "    \\KL{p(\\bx)}{q(\\bx)}=\\int_{\\R} p(\\bx) \\log \\frac{p(\\bx)}{q(\\bx)}d\\bx\n",
    "\\end{equation}\n",
    "\n",
    "Note that this dissimilarity measure is _asymmetric_, therefore, we have\n",
    "\\begin{equation}\n",
    "    \\KL{p(\\bx)}{q(\\bx)}\\neq \\KL{q(\\bx)}{p(\\bx)}\n",
    "\\end{equation}\n",
    "This asymmetry also describes an interesting behavior of the KL divergence, depending on the order to which it is evaluated. The KL divergence can either be a _mode-seeking_ or _mode-coverage_ measure (we will come back to these notions in the _approximate inference_ course)"
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