04_probability_distributions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability distributions and random variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Throughout the course we will work with probability distributions. Hence, it is important to master the basic principles of probability distributions, and learn to manipulate probabilities in code.\n",
    "\n",
    "Probability distributions and random variables serve as tools for describing and performing calculations related to <font color='orange'>random events</font>, specifically those whose outcomes are uncertain. \n",
    "\n",
    "An illustrative example of such an uncertain event would be the act of flipping a coin or rolling a dice. In the former case, the potential outcomes are heads or tails. \n",
    "\n",
    "An example of a random event from epidemiology, is the number of disease cases $y(t)$ on a given day $t$.\n",
    "\n",
    "In the context of epidemiological modelling, we will encounter <font color='orange'>data of different types</font> and origin. It is crucial to grasp the suitability of different probability distributions for modeling specific types of data.\n",
    "\n",
    "Since the probabilistic programming language that we will be using for this course is  <font color='orange'>Numpyro</font>, also in this section we will use the implementations of distributions from this library available via `import numpyro.distributions as dist`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# uncomment this line on Colab\n",
    "# !pip install numpyro"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{margin}\n",
    "In all notebooks of this course, we will always import all necessary libraries in the first code cell.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/esemenov/miniconda3/envs/aims/lib/python3.9/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html\n",
      "  from .autonotebook import tqdm as notebook_tqdm\n"
     ]
    }
   ],
   "source": [
    "import jax\n",
    "import jax.numpy as jnp\n",
    "\n",
    "# distributions\n",
    "import numpyro.distributions as dist\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# since we are using jax, we will need a random key:\n",
    "rng = jax.random.PRNGKey(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discrete distributions\n",
    "\n",
    "Discrete probability distributions represent the probabilities of distinct outcomes in a <span style=\"color:orange\">finite</span> or <span style=\"color:orange\">countably infinite</span> sample space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Bernoulli distribution\n",
    "\n",
    "A Bernoulli distribution is used to describe random events with <span style=\"color:orange\">two possible outcomes</span> e.g. when we have a random variable $X$ that takes on one of the two values $x \\in \\{0, 1\\}$ with probabilities $1-p$ and $p, 0 \\le p \\le 1$ respectively:\n",
    "\n",
    "\\begin{align*}\n",
    "p(X = 1) &= p, \\\\\n",
    "p(X = 0) &= 1 - p.\n",
    "\\end{align*}\n",
    "\n",
    "Here $p$ is the probability of the 'positive' outcome. For example, in the case of a *fair* coin toss, $p = 0.5$ so that both outcomes have a 50\\% chance of occurring.\n",
    "\n",
    "We will be denoting this distribution as \n",
    "\n",
    "$$\n",
    "X \\sim \\mathcal{Bern}(p).\n",
    "$$\n",
    "\n",
    "or, equivalently, \n",
    "\n",
    "$$\n",
    "\\mathcal{Bern}(X | p).\n",
    "$$\n",
    "\n",
    "#### Probability mass function\n",
    "A *discrete* probability distribution can be uniquely defined by its <span style=\"color:orange\">probability mass function (PMF)</span>.\n",
    "\n",
    "```{margin}\n",
    "The term 'mass' is used to underline that the support of the distribution is discrete, and each possible value carries a certain `mass` (probability).\n",
    "For continuous distributions, the analogous is  <span style=\"color:orange\">probability density function (PDF)</span>, we will see those later.\n",
    "```\n",
    "For the Bernoulli distribution, we write the PMF as\n",
    "\n",
    "\\begin{align*}\n",
    "p(X = x) = \\mathcal{Bern}(X\\mid p) &= \\begin{cases}\n",
    "p\\, & \\text{if } x = 1 \\\\\n",
    "1 - p\\, & \\text{if } x = 0\n",
    "\\end{cases} \\\\\n",
    "&= p^x(1-p)^{1-x}.\n",
    "\\end{align*}\n",
    "\n",
    "`````{admonition} Task 02\n",
    ":class: tip\n",
    "Convince yourself that the two definitions of the Bernoulli distribution shown above are equivalent.\n",
    "`````\n",
    "\n",
    "Now let's construct a Bernoulli distribution in code."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Drawing a sample\n",
    "\n",
    "We construct the distribution with a certain value of the parameter `p`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = jnp.array(0.5)\n",
    "bernoulli = dist.Bernoulli(probs=p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{margin}\n",
    "We can think of a sample as a <span style=\"color:orange\">realisation</span> of a random variable. \n",
    "```\n",
    "Now that we have constructed the distribution we can get a  <span style=\"color:orange\">sample</span> from it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "sample = bernoulli.sample(key=rng)\n",
    "print(sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can evaluate the probability of observing a sample.\n",
    "\n",
    "**Note:** the distribution objects in `numpyro` (and most other libraries for probability distributions) return <span style=\"color:orange\">log-probabilities</span> rather than raw probabilities. This means that we need to take the exponent if we want to know the probability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "log p(X = 1) = -0.6931471824645996\n",
      "p(X = 1) = 0.5\n"
     ]
    }
   ],
   "source": [
    "log_prob = bernoulli.log_prob(sample)\n",
    "print(f\"log p(X = {sample}) = {log_prob}\")\n",
    "print(f\"p(X = {sample}) = {jnp.exp(log_prob)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, we get a probability of 0.5."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multiple samples\n",
    "\n",
    "We can also easily get multiple samples in one command by including `sample_shape`:\n",
    "\n",
    "```{margin}\n",
    "Multiple samples are different realisations of the <span style=\"color:orange\">same</span> random variable.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0 1 1 0 1 1]\n"
     ]
    }
   ],
   "source": [
    "n_samps = 7\n",
    "samples = bernoulli.sample(key=rng, sample_shape=(n_samps,))\n",
    "print(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if we wanted to evaluate the probability of observing all of our samples?\n",
    "\n",
    "The `bernoulli` object we created earlier treats each sample individually and returns the probabilities of observing each sample on its own:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5 0.5 0.5 0.5 0.5 0.5 0.5]\n"
     ]
    }
   ],
   "source": [
    "individual_sample_probs = jnp.exp(bernoulli.log_prob(samples))\n",
    "print(individual_sample_probs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But, we can use one of the laws of probability to compute the probability of observing all of the samples together, i.e. <span style=\"color:orange\">jointly</span>:\n",
    "\n",
    "\\begin{align*}\n",
    "p(X_1=x_1, X_2=x_2, \\dots, X_N=x_n) = \\prod_{n=1}^N p(X_n=x_n).\n",
    "\\end{align*}\n",
    "\n",
    "This is called the <span style=\"color:orange\">product rule of probability</span>, and it says that for <span style=\"color:orange\">independent</span> random variables, the joint probability (i.e., the probability of observing them all together) is equal to the product of the individual probabilities.\n",
    "\n",
    "Now, let's calculate the joint probability of our samples.\n",
    "\n",
    "```{margin}\n",
    "Working with log-probabilities is preferable due to numerical stability, computational efficiency, and ease of handling multiplicative operations.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0078125\n"
     ]
    }
   ],
   "source": [
    "joint_prob = jnp.prod(individual_sample_probs)\n",
    "print(joint_prob)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualise PMF\n",
    "\n",
    "Now let's visualise the PMF:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Bernouilli_vis(rng, p, n_samps):\n",
    "\n",
    "    # define distribution\n",
    "    bernoulli = dist.Bernoulli(probs=p)\n",
    "\n",
    "    # collect samples\n",
    "    samples = bernoulli.sample(key=rng, sample_shape=(n_samps,))\n",
    "\n",
    "    # how many ones\n",
    "    num_ones = (samples == 1.).sum()\n",
    "\n",
    "    # how many zeros\n",
    "    num_zeros = (samples == 0.).sum()\n",
    "\n",
    "    # plot\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    ax = fig.add_subplot(1, 1, 1)\n",
    "    ax.bar([0, 1], [num_zeros/n_samps, num_ones/n_samps], alpha=0.7, color='teal')\n",
    " \n",
    "    ax.set_xticks([0, 1])\n",
    "    ax.set_xlabel('Outcome (x)')\n",
    "    ax.set_ylabel('Probability Mass p(X=x)')\n",
    "    ax.set_title(f'Bernoulli Distribution (p={p})')\n",
    "    ax.grid(True)\n",
    "    \n",
    "    plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bernouilli_vis(rng, p=0.2, n_samps=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 03\n",
    ":class: tip\n",
    "Recreate this plot using `bernoulli.log_prob(sample)` functionality (see examples below).\n",
    "`````"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bernouilli_vis(rng, p=0.7, n_samps=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 04\n",
    ":class: tip\n",
    "Plot a panel of histograms where you vary probability $p$ horizontally and number of samples $n$ vertically. What do you observe?\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Common usage\n",
    "\n",
    "Bernoulli distribution is commonly used as a likelihood in models with <font color='orange'>binary outcomes</font>, such as presence or absence of a disease."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Binomial distribution\n",
    "\n",
    "A binomial distribution is a discrete probability distribution that models the <font color='orange'>number of successes</font> $x$ in a fixed number $n$ of independent and identical Bernoulli trials, where each trial has only two possible outcomes: \"success\" (represented as \"1\") with probability $p$ or \"failure\" (represented as \"0\") with probability $1-p$.\n",
    "\n",
    "We will use the notation \n",
    "\n",
    "$$\n",
    "X \\sim \\mathcal{Binom}(n,p)\n",
    "$$\n",
    "\n",
    "#### Probability mass function\n",
    "\n",
    "The PMF of the Binomial distribution is\n",
    "\n",
    "$$P(X = x) = \\binom{n}{x} p^x (1 - p)^{n - x},$$\n",
    "\n",
    "where\n",
    "\n",
    "- $P(X = x)$ is the probability of getting exactly $x$ successes,\n",
    "\n",
    "- $\\binom{n}{x}$ is the binomial coefficient, representing the number of ways to choose $x$ successes out of $n$ trials,\n",
    "\n",
    "- $p$ is the probability of success on a single trial,\n",
    "\n",
    "- $1-p$ is the probability of failure on a single trial,\n",
    "\n",
    "- the number of successes $x$ ranges from $0$ to $n$ inclusive.\n",
    "\n",
    "`````{admonition} Task 05\n",
    ":class: tip\n",
    "Compute $\\sum_{x=0}^n P(X=x)$.\n",
    "`````\n",
    "\n",
    "#### Drawing a sample"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As before, we begin by constructing the distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 0.3\n",
    "n = 10\n",
    "binomial = dist.Binomial(total_count=n, probs=p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can draw a sample from this distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n"
     ]
    }
   ],
   "source": [
    "sample = binomial.sample(key=rng)\n",
    "print(sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 06\n",
    ":class: tip\n",
    "Draw several samples from this distribution using different keys. And draw repeatedly several samples with the same key. What do you conclude about the role of `key` in reproducibility of numerical experiments?\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the probability to observe this sample?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "log p(X = 4) = -1.608832836151123\n",
      "p(X = 4) = 0.20012104511260986\n"
     ]
    }
   ],
   "source": [
    "log_prob = binomial.log_prob(sample)\n",
    "print(f\"log p(X = {sample}) = {log_prob}\")\n",
    "print(f\"p(X = {sample}) = {jnp.exp(log_prob)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multiple samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us generate several samples: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3 6 1 3 2 4 5]\n"
     ]
    }
   ],
   "source": [
    "n_samps = 7\n",
    "samples = binomial.sample(key=rng, sample_shape=(n_samps,))\n",
    "print(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Individual probabilities to observe the samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.26682812 0.03675695 0.12106086 0.26682812 0.23347507 0.20012105\n",
      " 0.10291947]\n"
     ]
    }
   ],
   "source": [
    "individual_sample_probs = jnp.exp(binomial.log_prob(samples))\n",
    "print(individual_sample_probs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming that the samples are independent, what is the joint probability of observeing them?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5234818e-06\n"
     ]
    }
   ],
   "source": [
    "joint_prob = jnp.prod(individual_sample_probs)\n",
    "print(joint_prob)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualise PMF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def binomial_vis(rng, p, n, x_ticks=True):\n",
    "    \n",
    "    # Binomial distribution with `n` trials and probability of success `p``\n",
    "    binomial = dist.Binomial(total_count=n, probs=p)\n",
    "    \n",
    "    # generate the possible outcomes (x values)\n",
    "    x_values = jnp.arange(0, n + 1)\n",
    "\n",
    "    pmf_values = jnp.exp(binomial.log_prob(x_values))\n",
    "\n",
    "    # create a bar plot (PMF plot)\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    plt.bar(x_values, pmf_values, align='center', alpha=0.7, color='teal')\n",
    "    plt.xlabel('Number of successes (x)')\n",
    "    plt.ylabel('Probability mass p(X=x)')\n",
    "    plt.title(f'Binomial distribution (n={n}, p={p})')\n",
    "    if x_ticks:\n",
    "        plt.xticks(x_values)\n",
    "    plt.grid( linestyle='--', alpha=0.7)\n",
    "\n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binomial_vis(rng, p=0.2, n=6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binomial_vis(rng, p=0.2, n=100, x_ticks=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 07\n",
    ":class: tip\n",
    "What is qualitatively different between the shapes of distributions `Binomial(p=0.2, n=6)` and `Binomial(p=0.2, n=100)`?\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Common usage\n",
    "\n",
    "Binomial distribution is commonly used as a likelihood in models with binary outcomes with multiple experiments. For example, to model disease prevalence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Categorical distribution\n",
    "\n",
    "A categorical distribution is used to model random events with <font color='orange'>multiple discrete unordered outcomes</font>, such as the die-rolling event from above. \n",
    "\n",
    "We can characterise the categorical distribution with its PMF:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align*}\n",
    "p(X = x) = \\mathcal{Categorical}(X \\mid p) = \\prod_{k=1}^K p_k^{I_{x=k}},\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $K$ is the number of possible outcomes, $p_k$ is the probability of the $k$th outcome, and $I_{x=k}$ is the indicator function which evaluates to 1 if $x = k$ and 0 otherwise. All probabilities $p_k$ form a vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "p = \n",
    "\\begin{pmatrix}\n",
    "p_1\\\\ \n",
    "p_2\\\\\n",
    "\\dots \\\\ \n",
    "p_K\n",
    "\\end{pmatrix},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "such that \n",
    "\n",
    "$$\n",
    "\\sum_k p_k = 1.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 08\n",
    ":class: tip\n",
    "Explain why a categorical distribution with $K = 2$ is equivalent to a Bernoulli distribution.\n",
    "`````"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "ps = jnp.array([0.1, 0.2, 0.3, 0.4])\n",
    "categorical = dist.Categorical(probs=ps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As before we can take some samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3 2 1 3 3 1 2 1 3 3]\n"
     ]
    }
   ],
   "source": [
    "samples = categorical.sample(key=rng, sample_shape=(10,))\n",
    "print(samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p(X=0) = 0.1\n",
      "p(X=1) = 0.2\n",
      "p(X=2) = 0.3\n",
      "p(X=3) = 0.4\n"
     ]
    }
   ],
   "source": [
    "print(f\"p(X=0) = {jnp.exp(categorical.log_prob(0)):.1f}\")\n",
    "print(f\"p(X=1) = {jnp.exp(categorical.log_prob(1)):.1f}\")\n",
    "print(f\"p(X=2) = {jnp.exp(categorical.log_prob(2)):.1f}\")\n",
    "print(f\"p(X=3) = {jnp.exp(categorical.log_prob(3)):.1f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use an alternative way to represent the categorical distribution. Instead of specifying the probabilities $p_k$, we specify <font color='orange'>logits</font> $l_k$. Each $p_k$ is then computed as\n",
    "\n",
    "$$\n",
    "p_k = \\frac{\\exp(l_k)}{\\sum_{k'}\\exp(l_{k'})},\n",
    "$$\n",
    "\n",
    "i.e., using the <font color='orange'>softmax</font> function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "l_0 = 0.7 \n",
    "l_1 = 0.3 \n",
    "l_2 = 2 \n",
    "l_3 = 1.6\n",
    "\n",
    "logits = jnp.array([l_0, l_1, l_2, l_3], dtype=jnp.float32)\n",
    "categorical = dist.Categorical(logits=logits)\n",
    "samples = categorical.sample(key=rng, sample_shape=(1000,))\n",
    "\n",
    "values =[0, 1, 2, 3]\n",
    "num_bins = len(values)\n",
    "\n",
    "hist, _ = jnp.histogram(samples, bins=num_bins, density=True)\n",
    "\n",
    "fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "plt.bar(values, hist, color='teal', alpha=0.7)\n",
    "plt.xticks(values)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('p(X=x)')\n",
    "plt.grid(linestyle='--', alpha=0.7)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Ordinal distribution\n",
    "\n",
    "The ordinal distribution is a probability distribution used to model outcomes that are ranked or ordered, often encountered in scenarios where data points lack clear numerical interpretation but possess a defined order of precedence. It assigns probabilities to different rank orders, capturing the relative likelihood of each outcome's position within the ordered set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define study hours for each student\n",
    "study_hours = jnp.array([5, 7, 3, 10, 8, 6, 4, 9, 2, 7])\n",
    "\n",
    "# define cutpoints for the ordered categories\n",
    "cutpoints = jnp.array([0., 5., 7., 10.])  # Ordered categories: (0, 5], (5, 7], (7, 10]\n",
    "\n",
    "# sample logits (unnormalized probabilities) based on study hours\n",
    "logits = 0.5 * study_hours\n",
    "\n",
    "ordinal = dist.OrderedLogistic(logits, cutpoints)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# sample from the OrderedLogistic distribution\n",
    "ordinal = dist.OrderedLogistic(logits, cutpoints)\n",
    "samples = ordinal.sample(rng, (1000,))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous distributions\n",
    "\n",
    "Continuous probability distributions describe the probabilities of outcomes within a continuous range of values.\n",
    "\n",
    "### The Beta distribution\n",
    "\n",
    "The Beta distribution is a continuous distribution with support in $[0,1] \\subset \\mathbb{R}$.\n",
    "The Beta distribution can be used to describe a continuous random variable between 0 and 1, for example, percentages and ratios. It has the following form\n",
    "```{margin}\n",
    "Note that in this distribution more than one parameter defines its shape.\n",
    "```\n",
    "\n",
    "$$\n",
    "p(X = x) = \\mathcal{Beta}(x|\\alpha,\\beta) = \\frac{1}{\\mathrm{B}(\\alpha,\\beta)}x^{\\alpha-1}(1 - x)^{\\beta - 1},\n",
    "$$\n",
    "\n",
    "where $\\alpha > 0$ and $\\beta > 0$ are the two <font color='orange'>shape</font> parameters of the distribution, and $\\mathrm{B}$ is called the beta function. \n",
    "\n",
    "Let's visualise the distribution.\n",
    "\n",
    "`````{admonition} Task 09\n",
    ":class: tip\n",
    "* Try making each parameter big or small while leaving the other at the same value.\n",
    "* Then try make them both big or small.\n",
    "`````"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def beta_vis(a, b, x_ticks=True):\n",
    "    \n",
    "    beta = dist.Beta(a, b)\n",
    "    \n",
    "    x_values = jnp.linspace(0, 1, 1000)\n",
    "\n",
    "    pmf_values = jnp.exp(beta.log_prob(x_values))\n",
    "\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    plt.plot(x_values, pmf_values, alpha=0.7, color='teal')\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('p(X=x)')\n",
    "    plt.title(f'Beta distribution (a={a}, b={b})')\n",
    "    if x_ticks:\n",
    "        plt.xticks(x_values)\n",
    "    plt.grid( linestyle='--', alpha=0.7)\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "beta_vis(a=4.3, b=3.2, x_ticks=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Gamma distribution\n",
    "\n",
    "The Gamma distribution is a continuous distribution with support in $\\mathbb{R}^+$.\n",
    "Its PDF has the form \n",
    "\n",
    "$$p(X=x) = \\mathcal{Gamma}(x; \\alpha, \\beta) = \\frac{ \\beta^{\\alpha}}{\\Gamma(\\alpha)} x^{\\alpha - 1} e^{-\\beta x},$$\n",
    "\n",
    "where $\\alpha>0$ is the <font color='orange'>shape parameter</font>, which determines the shape of the distribution, and $\\beta>0$ is the <font color='orange'>scale</font> parameter. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def gamma_vis(a, b, x_ticks=True):\n",
    "    \n",
    "    gamma = dist.Gamma(a, b)\n",
    "    \n",
    "    x_values = jnp.linspace(0, 20, 1000)\n",
    "\n",
    "    pmf_values = jnp.exp(gamma.log_prob(x_values))\n",
    "\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    plt.plot(x_values, pmf_values, alpha=0.7, color='teal')\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('p(X=x)')\n",
    "    plt.title(f'Gamma distribution (a={a}, b={b})')\n",
    "    if x_ticks:\n",
    "        plt.xticks(x_values)\n",
    "    plt.grid( linestyle='--', alpha=0.7)\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "gamma_vis(a=2, b=0.5, x_ticks=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Uniform distribution\n",
    "\n",
    "Perhaps, the simplest, but still a very important distribution among continuous distributions is the uniform distribution. Under this distribution, all possible values are equally likely. The uniform distribution has the following form\n",
    "\n",
    "$$\n",
    "p(X = x) = \\mathrm{Uniform}(x\\mid a, b) = \\begin{cases}\n",
    "\\frac{1}{b - a}\\, & \\text{if } a \\le x \\le b \\\\\n",
    "0\\, & \\text{otherwise},\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "where $a$ and $b$ are the <font color='orange'>upper</font> and <font color='orange'>lower bound</font> parameters, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def uniform_vis(a, b, x_ticks=True):\n",
    "    \n",
    "    uniform = dist.Uniform(low=a, high=b)\n",
    "    x_values = jnp.linspace(-5, 5, 1000)\n",
    "\n",
    "    pmf_values = jnp.exp(uniform.log_prob(x_values))\n",
    "\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    plt.plot(x_values, pmf_values, alpha=0.7, color='teal')\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('p(X=x)')\n",
    "    plt.title(f'Uniform distribution (a={a}, b={b})')\n",
    "    if x_ticks:\n",
    "        plt.xticks(x_values)\n",
    "    plt.grid(axis='y', linestyle='--', alpha=0.7)\n",
    "    plt.grid(axis='x', linestyle='--', alpha=0.7)\n",
    "    plt.xlim(-5, 5)\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "uniform_vis(a=-1, b=3, x_ticks=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Normal distribution\n",
    "\n",
    "The normal – also known as *Gaussian* – distribution is one of the most common distributions for modeling *continuous* random variables, i.e., corresponding to events with an uncountable number of outcomes. Its probability density function is\n",
    "\n",
    "$$\n",
    "p(X = x) = \\mathcal{N}(x\\mid \\mu, \\sigma) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left(-\\frac{(\\mu - x)^2}{2\\sigma^2}\\right),\n",
    "$$\n",
    "\n",
    "where $\\mu$ and $\\sigma$ are the <font color='orange'>mean</font> and <font color='orange'>standard deviation</font> (also called the location, and scale or square-root of the <font color='orange'>variance</font> $\\sigma^2$, respectively).\n",
    "\n",
    "`````{admonition} Task 10\n",
    ":class: tip\n",
    "How do the mean and standard deviation affect the samples?\n",
    "`````"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def norm_vis(mu, sigma, x_ticks=False):\n",
    "    \n",
    "    normal = dist.Normal(loc=mu, scale=sigma)\n",
    "    \n",
    "    x_values = jnp.linspace(-5, 5, 1000)\n",
    "\n",
    "    pmf_values = jnp.exp(normal.log_prob(x_values))\n",
    "\n",
    "    fig = plt.figure(dpi=100, figsize=(5, 3))\n",
    "    plt.plot(x_values, pmf_values, alpha=0.7, color='teal')\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('p(X=x)')\n",
    "    plt.title(f'Normal distribution ($\\mu$={mu}, $\\sigma$={sigma})')\n",
    "    if x_ticks:\n",
    "        plt.xticks(x_values)\n",
    "    plt.grid(axis='y', linestyle='--', alpha=0.7)\n",
    "    plt.grid(axis='x', linestyle='--', alpha=0.7)\n",
    "    plt.xlim(-5, 5)\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "norm_vis(mu = 0, sigma = 1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "`````{admonition} Task 11\n",
    ":class: tip\n",
    "Implement the PDF of the Normal distribution and test it using the function provided below.\n",
    "`````"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_normal_pdf(pdf_fn, run=False):\n",
    "  if not run:\n",
    "    return\n",
    "  assert pdf_fn(0, 1, 0) == jnp.exp(dist.Normal(loc=0, scale=1).log_prob(0)), \"Normal(X=0|0, 1) is incorrect.\"\n",
    "  assert pdf_fn(0, 2, 0) == jnp.exp(dist.Normal(loc=0, scale=2).log_prob(0)), \"Normal(X=0|0, 2) is incorrect.\"\n",
    "  assert pdf_fn(0, 1, 1) == jnp.exp(dist.Normal(loc=0, scale=1).log_prob(1)), \"Normal(X=0|1, 1) is incorrect.\"\n",
    "  assert pdf_fn(2, 3, 1) == jnp.exp(dist.Normal(loc=2, scale=3).log_prob(1)), \"Normal(X=1|2, 3) is incorrect.\"\n",
    "  print(\"Nice! Your answer looks correct.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Multivariate Normal distribution\n",
    "\n",
    "```{margin}\n",
    "This distribution will be very useful in chapters where we will talk about Gaussian Processes and spatial modelling.\n",
    "```\n",
    "\n",
    "The *multivariate normal distribution* is a generalisation of the *univariate normal distribution* to consider multiple random variables that have a <font color='orange'>jointly normal distribution</font>. In other words, it lets us model variables that are not independent – if we know the value of one variable, that tells us something about the other variables!\n",
    "More concretely, the multivariate normal distribution lets us consider multiple random variables such that when we condition on some of these variables the remaining variables have a normal distribution. These variables are distributed in a kind of stretched fuzzy ball in higher dimensional space.\n",
    "\n",
    "As a rule of thumb, the more one variable tells us about another, the larger the *covariance* or *correlation* between the two. \n",
    "\n",
    "We will be using the same notation for multivatiate and univariate normals $\\mathcal{N}$. Which one to use, should be clear from the context throughout this course.\n",
    "\n",
    "The PDF for an $D$-dimensional random variable $X$:\n",
    "\n",
    "$$\n",
    "p(X = x) = \\mathcal{N}(x\\mid \\mu, \\Sigma) = \\frac{1}{\\sqrt{(2\\pi)^D|\\Sigma|}}\\exp\\left(-\\frac{1}{2}(x - \\mu)^\\intercal\\Sigma^{-1}(x - \\mu)\\right),\n",
    "$$\n",
    "\n",
    "where $x$ and $\\mu$ are now vectors of numbers rather than single numbers, $\\Sigma$ is a <font color='orange'>covariance matrix</font> that replaces $\\sigma$ from our univariate definition above, and $|\\Sigma|$ is its determinant. The covariance matrix looks like this:\n",
    "\\begin{equation*}\n",
    "\\Sigma = \\begin{bmatrix}\\sigma_1^2 & \\rho_{12} \\sigma_1 \\sigma_2 & \\cdots &  \\rho_{1D} \\sigma_1 \\sigma_D  \\\\\n",
    "    \\rho_{21} \\sigma_2\\sigma_1 & \\sigma_2^2 & \\cdots &  \\rho_{2D} \\sigma_2 \\sigma_D \\\\\n",
    "    \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    \\rho_{D1} \\sigma_D \\sigma_1 & \\rho_{D2} \\sigma_D \\sigma_2 & \\cdots & \\sigma_D^2 \\end{bmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "where $\\sigma_i^2$ is the variance for the $i$-th dimension, and $\\rho_{ij} = \\rho_{ji}$ is the *correlation* between the $i$-th and $j$-th dimensions. The covariance matrix tells us how the \"ball\" of random variables is stretched and rotated in space.\n",
    "\n",
    "`````{admonition} Task 12\n",
    ":class: tip\n",
    "Show that the equation above is equivalent to the univariate case when $D = 1$\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Now let's look at how the equation above simplifies in the two-dimensional case\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "&p(X_1 = x_1, X_2 = x_2) = \\\\\n",
    "&\\mathcal{N}\\left( \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} \\middle| \\begin{bmatrix} \\mu_1 \\\\ \\mu_2 \\end{bmatrix}, \\begin{bmatrix} \\sigma_1^2 & \\rho \\sigma_1 \\sigma_2 \\\\ \\rho \\sigma_1 \\sigma_2  &\\sigma_2^2 \\end{bmatrix}\\right)= \\\\\n",
    "&\\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1 - \\rho^2}}\\exp\\left(-\\frac{1}{2(1 - \\rho^2)}\\left[\\left(\\frac{x_1 - \\mu_1}{\\sigma_1}\\right)^2 -2\\rho\\left(\\frac{x_1 - \\mu_1}{\\sigma_1}\\right)\\left(\\frac{x_2 - \\mu_2}{\\sigma_2}\\right)  + \\left(\\frac{x_2 - \\mu_2}{\\sigma_2}\\right)^2 \\right] \\right)\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "`````{admonition} Group task\n",
    "Try to understand what this equation means. Discuss the following questions with your neighbors. \n",
    "\n",
    "1. If $\\rho = 0$, how does this two-dimensional case relate to the one-dimensional case above?\n",
    "2. Now, think about what happens as $\\rho$ becomes larger? What if it becomes negative?\n",
    "\n",
    "`````\n",
    "\n",
    "We will come back to this distribution later in the course."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Batch and event shapes\n",
    "\n",
    "All distributions in `numpyro` have an `event_shape` which describes how many dimensions the random variable is, e.g., for a 2-dimensional normal distribution this would be 2, and a `batch_shape` which describes how many sets of parameters the distribution has – it is probably easier to show what this means with the following examples rather than tell.\n",
    "\n",
    "Let's first look at a simple univariate normal $\\mathcal{N}(x|0, 1)$. We will evaluate the PDFs at $X = 1$ and $X = 2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "event_shape = ()\n",
      "batch_shape = ()\n",
      "p(X = [1. 2.]) = [0.24197073 0.05399096]\n"
     ]
    }
   ],
   "source": [
    "values = jnp.array([1., 2.])\n",
    "normal = dist.Normal(0., 1.)\n",
    "print(f\"event_shape = {normal.event_shape}\")\n",
    "print(f\"batch_shape = {normal.batch_shape}\")\n",
    "print(f\"p(X = {values}) = {jnp.exp(normal.log_prob(values))}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that this distribution has an empty event shape, which you can think of as being the same as an event shape of 1 (like how a scalar is the same as a vector of length 1). The batch shape is also empty, since we only specified one set of parameters ($\\mu = 0, \\sigma = 1$).  \n",
    "\n",
    "Now since we tried to evaluate the probability of two values at once, and neither the event shape nor the batch shape are 2, this is equivalent to calling `dist.log_prob(1.)` and `dist.log_prob(2.)` separately. `numpyro` is just making our lives easier by *broadcasting* the `log_prob` calculation to do both $\\mathcal{N}(X=1|0, 1)$ and $\\mathcal{N}(X=2|0, 1)$ at the same time.\n",
    "\n",
    "We could also specify a *batch* of two sets of parameters so that we are essentially working with $\\mathcal{N}(x|0, 1)$ and $\\mathcal{N}(x|1, 2)$ at the same time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "event_shape = ()\n",
      "batch_shape = (2,)\n",
      "[p(X_1 = 1.0), p(X_2 = 2.0)] = [0.24197073 0.17603266]\n",
      "p(X_1 = 1.0, X_2 = 2.0) = 0.042594753205776215\n"
     ]
    }
   ],
   "source": [
    "batch_normal = dist.Normal(jnp.array([0., 1.]), jnp.array([1., 2.]))\n",
    "print(f\"event_shape = {batch_normal.event_shape}\")\n",
    "print(f\"batch_shape = {batch_normal.batch_shape}\")\n",
    "print(f\"[p(X_1 = {values[0]}), p(X_2 = {values[1]})] = {jnp.exp(batch_normal.log_prob(values))}\")\n",
    "print(f\"p(X_1 = {values[0]}, X_2 = {values[1]}) = {jnp.prod(jnp.exp(batch_normal.log_prob(values)))}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, notice that while the event shape is empty (as expected since we are still working with a univariate normal), the batch size is 2!\n",
    "\n",
    "As a result, the calculation we are doing is equivalent to separately calculating $p(X_1=1) = \\mathcal{N}(X_1=1|0, 1)$ and $p(X_2=2) = \\mathcal{N}(X_2=2|1, 2)$! Again, this is just `numpyro` making our lives easier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that `event_shape` and `batch_shape` correspond to the distribution we are working with, and they are related, but different from `sample_shape` as the examples below show."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = jax.random.PRNGKey(48)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "event_shape = ()\n",
      "batch_shape = ()\n",
      "sample_shape = (3,)\n"
     ]
    }
   ],
   "source": [
    "normal = dist.Normal(0., 1.)\n",
    "print(f\"event_shape = {normal.event_shape}\")\n",
    "print(f\"batch_shape = {normal.batch_shape}\")\n",
    "\n",
    "samples = normal.sample(key = rng, sample_shape=(3, ))\n",
    "print(f\"sample_shape = {samples.shape}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "event_shape = ()\n",
      "batch_shape = (3,)\n",
      "sample_shape = (4, 3)\n"
     ]
    }
   ],
   "source": [
    "batch_normal = dist.Normal(jnp.array([0., 1., 2.]), jnp.array([1., 2., 4.])) # array of means, array of vars\n",
    "print(f\"event_shape = {batch_normal.event_shape}\")\n",
    "print(f\"batch_shape = {batch_normal.batch_shape}\")\n",
    "\n",
    "samples  = batch_normal.sample(key = rng, sample_shape=(4, ))\n",
    "print(f\"sample_shape = {samples.shape}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "event_shape = (2,)\n",
      "batch_shape = ()\n",
      "sample_shape = (3, 2)\n"
     ]
    }
   ],
   "source": [
    "multivariate_full_normal = dist.MultivariateNormal(jnp.array([0., 1.]), jnp.array([[1., 1.], [1., 2.**2]]))\n",
    "print(f\"event_shape = {multivariate_full_normal.event_shape}\")\n",
    "print(f\"batch_shape = {multivariate_full_normal.batch_shape}\")\n",
    "\n",
    "samples = multivariate_full_normal.sample(key = rng, sample_shape=(3, ))\n",
    "print(f\"sample_shape = {samples.shape}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Measuring distances between distributions\n",
    "\n",
    "There are several ways to measure distances between two probability distributions with PDFs $p(x)$ and $q(x)$, each with its own characteristics and applications. \n",
    "\n",
    "- Total variation distance (TVD): This metric quantifies the difference between two probability distributions by measuring the total absolute difference between their probability mass functions (for discrete distributions) or probability density functions (for continuous distributions).\n",
    "\n",
    "$$\n",
    "\\text{TVD}(p, q) = \\frac{1}{2} \\int |p(x) - q(x)| \\, dx\n",
    "$$\n",
    "\n",
    "- Kullback-Leibler Divergence (KLD): Also known as relative entropy, measures the information lost when one probability distribution is used to approximate another. It is asymmetric, and, hence is not a 'distance' but a 'deviance'\n",
    "\n",
    "$$\n",
    "\\text{KLD}(p \\parallel q) = \\int p(x) \\log \\left( \\frac{p(x)}{q(x)} \\right) \\, dx\n",
    "$$\n",
    "\n",
    "- Jensen-Shannon Divergence (JSD): JSD is a symmetrized version of KLD. It measures the similarity between two probability distributions by averaging their KLD values.\n",
    "\n",
    "$$\n",
    "\\text{JSD}(p, q) = \\frac{1}{2} \\text{KLD}(p \\parallel m) + \\frac{1}{2} \\text{KLD}(q \\parallel m),\\\\\n",
    "m = \\frac{p+q}{2}\n",
    "$$\n",
    "\n",
    "- Hellinger Distance: This distance metric is used to measure the similarity between two probability distributions. It is based on the square root of the total variation distance and ranges between 0 and 1.\n",
    "\n",
    "$$\n",
    "\\text{Hellinger}(p, q) = \\sqrt{1 - \\int \\sqrt{p(x)q(x)} \\, dx}\n",
    "$$\n",
    "\n",
    "- Wasserstein Distance (Earth Mover's Distance): This metric measures the minimum amount of \"work\" needed to transform one distribution into another. It considers the underlying structure of the distributions and is often used in optimal transport theory. $\\Gamma(p, q)$ represents the set of all joint distributions with marginals $p$ and $q$. The Wasserstein distance is defined as the minimum \"cost\" required to transform one distribution into another.\n",
    "\n",
    "$$\n",
    "\\text{Wasserstein}_m(p, q) = \\left( \\inf_{\\gamma \\in \\Gamma(p, q)} \\int_{\\mathcal{X} \\times \\mathcal{X}} d(x, y)^m \\, d\\gamma(x, y) \\right)^{\\frac{1}{m}}\n",
    "$$\n",
    "\n",
    "- Maximum Mean Discrepancy (MMD) between two distributions can be defined using the kernel trick. Let $\\phi(x)$ be a feature map $\\phi(x)$, and $\\mathbb{E}_{x \\sim P}[ \\phi(x) ]$ the expected value of the feature map $\\phi(x)$ computed over samples drawn from distribution $p$. In the same way, $\\mathbb{E}_{y \\sim Q}[ \\phi(y) ]$ is the expected value of the feature map $\\phi(y)$ computed over samples drawn from distribution $q$, $\\| \\cdot \\|$ is the Euclidean norm. The feature map $\\phi$ is usually chosen to be a reproducing kernel Hilbert space (RKHS) kernel function, such as the Gaussian kernel $k(x, y) = \\exp \\left( -\\frac{\\| x - y \\|^2}{2\\sigma^2} \\right)$.\n",
    "\n",
    "$$\n",
    "MMD(p, q) = \\left\\| \\mathbb{E}_{x \\sim p}[ \\phi(x) ] - \\mathbb{E}_{y \\sim q}[ \\phi(y) ] \\right\\|^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{margin}\n",
    "For all implementations, use `jax` or `numpyro` functionality. For instance, `import jax.numpy as jnp` instead of `import numpy as np`. And `import numpyro.distributions as dist` for access to distributions.\n",
    "````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 13\n",
    ":class: tip\n",
    "1. Implement numeric evaluation of the KL divergence:\n",
    "\n",
    "```\n",
    "def kl_divergence(p: dist.Distribution, q: dist.Distribution, n: int = 10_000):\n",
    "    \"\"\"\n",
    "    add your code here\n",
    "    \"\"\"  \n",
    "    pass\n",
    "```\n",
    "\n",
    "2. Calculate the following KL divergence. What do we see?\n",
    "    - $\\mathrm{KLD}\\left[\\mathrm{Uniform}(0, 1) \\mid\\mid \\mathrm{Uniform}(0, 1)\\right]$\n",
    "    - $\\mathrm{KLD}\\left[\\mathrm{Beta}(5, 2) \\mid\\mid \\mathrm{Beta}(5, 2)\\right]$\n",
    "  \n",
    "3. Calculate the following KL divergences. What can we say about the relationship between the beta and uniform distributions?\n",
    "    - What is $\\mathrm{KLD}\\left[\\mathrm{Uniform}(0, 1) \\mid\\mid \\mathrm{Beta}(5, 2)\\right]$?\n",
    "    - What is $\\mathrm{KLD}\\left[\\mathrm{Uniform}(0, 1) \\mid\\mid \\mathrm{Beta}(2, 2)\\right]$?\n",
    "    - What is $\\mathrm{KLD}\\left[\\mathrm{Uniform}(0, 1) \\mid\\mid \\mathrm{Beta}(1, 1)\\right]$?\n",
    "  \n",
    "4. What is $\\mathrm{KLD}\\left[ \\mathrm{Beta}(5, 2) \\mid\\mid \\mathrm{Uniform}(0, 1)\\right]$. How does it compare to $D_\\mathrm{KL}\\left[\\mathrm{Uniform}(0, 1) \\mid\\mid \\mathrm{Beta}(5, 2)\\right]$?\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{margin}\n",
    "You can tackle this task during week 2, as we have not looked at kernels yet.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`````{admonition} Task 14 (week 2)\n",
    ":class: tip\n",
    "\n",
    "1. Implement numeric evaluation of MMD taking as inputs two distributions `p` and `q` of type `dist.Distributions`, number of samples `n`. Use RBF gaussian as a default kernel. \n",
    "\n",
    "2. Modify the MMD computation code to accept different kernel functions besides the RBF kernel: linear, polynomial, and exponential kernels.\n",
    "\n",
    "\n",
    "`````"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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