{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "P23m4BV5gVoq"
   },
   "source": [
    "## Bayes on Grids\n",
    "\n",
    "Concepts taught here:\n",
    "\n",
    "* Probabilistic generative models (PGM)\n",
    "* How posterior distributions and evidence answer interesting science questions\n",
    "* How these can be computed numerically\n",
    "* Prior predictive checks\n",
    "* Bayesian model comparison\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "9sZIm-IFfDRS"
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from numpy import log10, exp, log, pi\n",
    "\n",
    "plt.rcParams[\"figure.figsize\"] = (8,6)\n",
    "plt.rcParams['figure.dpi'] = 100\n",
    "\n",
    "# if you cannot see plots, you may need to include: %matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "hCOxySkINcfU"
   },
   "source": [
    "# Introduction\n",
    "\n",
    "Lets say you are given a data set of two observables, period and luminosity, and are told: **\"Fit a line to these data\"**.\n",
    "\n",
    "To have a concrete example, lets generate some data:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ukxrTFG0fRWN"
   },
   "source": [
    "## Generative process (unknown to observers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bOp-aLiolinM"
   },
   "source": [
    "We assume there is a physical, perhaps stochastic process which generates observations. This includes making the objects of interest in the Universe, our observing equipment. \n",
    "\n",
    "For simulating, we choose this process with all its details. (Later, as observers, we will not know it and will try to infer it.)\n",
    "\n",
    "A **probabilistic generative model (PGM)** is also a story. Here is the story of this simulated Universe:\n",
    "\n",
    "![PGM](img/PGM.png)\n",
    "\n",
    "First, cepheids are created from Gaussian distributions of periods and metallicities. \n",
    "\n",
    "For those interested: short videos on [Cepheids](https://www.youtube.com/watch?v=7ohkKiZTJOg), a type of pulsating, massive, yellow stars and [metallicities](https://www.youtube.com/watch?v=CLrVwVeTN78), a measure of heavier-than-helium element content.\n",
    "\n",
    "Each cepheid follows the luminosity-period-metallicity law, which deterministically determines its luminosity as follows:\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "8gWKZBBDSLhn"
   },
   "source": [
    "$\\log L=0.2 \\times (\\alpha + \\beta \\times (\\log(P / {10 \\mathrm{days}}) + \\gamma \\times Z))$\n",
    "\n",
    "with $\\alpha=-4.4$, $\\beta=-2.8$ $\\gamma=-0.3$. In code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "id": "vEeTw8aYmfXe"
   },
   "outputs": [],
   "source": [
    "def luminosity_model(period, metallicity, alpha, beta, gamma, P0=10.0):\n",
    "    return 10**(-0.2 * (alpha + beta * (log10(period/P0) + gamma * metallicity)))\n",
    "\n",
    "true_alpha = -4.4\n",
    "true_beta  = -2.8\n",
    "true_gamma = -0.3\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "JkAOcsPame6y"
   },
   "source": [
    "Here is a plot of our law:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 299
    },
    "id": "52vz7gqASL-t",
    "outputId": "19b496bf-b97d-4076-e1fa-43c2d827d95f"
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid_P = np.linspace(10, 1000, 4000)\n",
    "plt.plot(grid_P, luminosity_model(grid_P, 0, true_alpha, true_beta, true_gamma))\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')\n",
    "plt.xlabel(\"Period [days]\")\n",
    "plt.ylabel(\"Luminosity\")\n",
    "plt.title(\"True Law (Z=0)\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "g-_UoT-WTJF-"
   },
   "source": [
    "Lets now create our cepheids. Here are their true properties:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 299
    },
    "id": "F8OpgGrwNbZD",
    "outputId": "b7c91353-0f53-469b-c40d-76b08926b84d"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# set the random number generator seed so that we get a consistent data sample:\n",
    "np.random.seed(42)\n",
    "\n",
    "N_obs = 20\n",
    "\n",
    "true_metallicity = np.random.normal(0, 0.1, size=N_obs)\n",
    "true_period = 10**np.random.normal(1.8, 0.25, size=N_obs)\n",
    "true_luminosity = luminosity_model(true_period, true_metallicity, true_alpha, true_beta, true_gamma)\n",
    "\n",
    "plt.plot(grid_P, luminosity_model(grid_P, 0, true_alpha, true_beta, true_gamma))\n",
    "plt.plot(true_period, true_luminosity, 'x ')\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')\n",
    "plt.xlabel(\"Period [days]\")\n",
    "plt.ylabel(\"Luminosity\")\n",
    "plt.title(\"True sample properties\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "_U-vA05dTaB_"
   },
   "source": [
    "Notice there is some scatter, because of the metallicity effect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rzP90ARTTh0o"
   },
   "source": [
    "**Important**: All of the above are properties we do not know. \n",
    "\n",
    "The final step in the generating process is to take observations:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "OKntzRhgNWOm"
   },
   "source": [
    "# Generate some observed data\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "_ru9G61cftlV"
   },
   "source": [
    "Lets simulate what the observer can get their hands on, which are noisy measurements:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 299
    },
    "id": "V0m0qRODMIEn",
    "outputId": "6ab2ac19-9e6d-4580-da6b-effb5a2e35ed"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "experimental_noise_period = np.random.uniform(0, 0.1, size=N_obs)\n",
    "experimental_noise_luminosity = np.random.uniform(1, 4, size=N_obs)\n",
    "\n",
    "obs_period = np.random.normal(true_period, experimental_noise_period)\n",
    "obs_luminosity = np.random.normal(true_luminosity, experimental_noise_luminosity)\n",
    "\n",
    "plt.errorbar(x=obs_period, xerr=experimental_noise_period, \n",
    "             y=obs_luminosity, yerr=experimental_noise_luminosity, \n",
    "             marker='x', ls=' ', capsize=2, elinewidth=1)\n",
    "plt.xlabel(\"Period [days]\")\n",
    "plt.ylabel(\"Luminosity\")\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.title(\"Observed sample properties\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "dJBXrWmqu6iN"
   },
   "source": [
    "Here is the full generative process as a graphical model:\n",
    "\n",
    "![PGM](img/PGM-full.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "BXwkyQJAUhx1"
   },
   "source": [
    "# Inference Problem\n",
    "\n",
    "## Motivation\n",
    "\n",
    "If we knew the period-luminosity relationship, we could infer the intrinsic luminosity by just measuring the period. --> This **predictiveness** is useful for cosmology\n",
    "\n",
    "The period-luminosity relationship can also tell us something about the interior of stars --> Comparing the data to models gives insights about **physical processes** in the Universe.\n",
    "\n",
    "## Inference\n",
    "\n",
    "We lack information, which introduces uncertainty. \n",
    "\n",
    "1. One source of uncertainty is that we do not know the true process. For example, we may not be aware that metallicity (which we did not measure) is part of the generative model. This is **systematic uncertainty** (aka epistemic uncertainty).\n",
    "2. Another source of uncertainty is our limited data: The instrument does not allow us to measure periods and luminosities perfectly and we only have a finite data set. This is **statistical uncertainty** (alearic uncertainty).\n",
    "\n",
    "We address (2) by quantifying the uncertainty assuming a probabilistic generative model (**parameter estimation**). We address (1) by either criticising them in isolation (**model criticism**), for example with visualisation or exploring several models and comparing them (**model comparison**).\n",
    "\n",
    "## Assumptions\n",
    "\n",
    "So we want to learn the period-luminosity relationship. We assume it has the shape:\n",
    "\n",
    "$\\log L=0.2 \\times (\\alpha + \\beta \\times (\\log(P / {10 \\mathrm{days}}))$\n",
    "\n",
    "As you see, our model is a simplification and approximation of the true process.\n",
    "\n",
    "From our sample, we have uncertain Gaussian measurements of L and uncertain measurements of P. \n",
    "\n",
    "## Goals\n",
    "\n",
    "We want to know the two parameters:\n",
    "\n",
    " * $\\alpha$: related to the luminosity for 10 days-periodic cepheids and \n",
    " * $\\beta$: related to how strongly the luminosity and period are related.\n",
    "\n",
    "But what does that really mean to \"know\" or \"constrain\" parameters?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Lepn_9VDgR9d"
   },
   "source": [
    "# Why Bayesian inference\n",
    "\n",
    "* There is a true value out there in the Universe, but we will never be able to measure it perfectly with infinite precision.\n",
    "* We can instead talk about it being within some interval, or not.\n",
    "* With probability theory, we can talk about the probability of the parameter to be within a interval.\n",
    "  * We use parameter probability distributions to describe our current state of information.\n",
    "  * These do not imply the value really has a distribution. (It also does not mean that if you created many Universes, they would follow that distribution)\n",
    "\n",
    "So if we have some data $D$, we would like a method to produce a probability distribution for us, $P(\\alpha)$.\n",
    "\n",
    "Since the probability is conditional on what data we feed it, we can write $P(\\alpha|D)$, which is read as \"The probability of $\\alpha$ given D\".\n",
    "\n",
    "The probability distribution is normalised ($\\int_{-\\infty}^\\infty P(\\alpha|D) d\\alpha=1$).\n",
    "\n",
    "Unfortunately, there is no method which can produce $P(\\alpha|D)$ just from D. \n",
    "However, if we knew $\\alpha$, we can produce data sets! This generating is what we did above.\n",
    "\n",
    "So we can talk about how frequently some dataset D is produced given $\\alpha$: $P(D|\\alpha)$. This probability distribution is normalised, but over all possible data outcomes: ($\\int_{-\\infty}^\\infty P(D|\\alpha) dD=1$).\n",
    "\n",
    "## Enter Bayes theorem\n",
    "\n",
    "![Conditional probability](img/conditional-probability.png)\n",
    "\n",
    "A simple Venn diagram (above) can tell you that\n",
    "\n",
    "$P(A \\cap B) P(A) = P(B \\cap A) P(B)$\n",
    "\n",
    "The law of conditional probabilities is analogous:\n",
    "\n",
    "$P(A | B) P(A) = P(B | A) P(B)$\n",
    "\n",
    "If we translate this to our problem, and move one term over, we get Bayes' theorem:\n",
    "\n",
    "$P(\\alpha|D) = \\frac{P(D|\\alpha) \\times P(\\alpha)}{P(D)}$\n",
    "\n",
    "where the terms are:\n",
    "\n",
    "* $P(\\alpha|D)$: Posterior probability distribution\n",
    "  * This is the information we want: A probability over $\\alpha$ given the data.\n",
    "\n",
    "* $P(D|\\alpha)$: Sampling distribution\n",
    "  * This is information we have: It encodes how data are probabilistically generated, assuming our model and given the model parameter $\\alpha$.\n",
    "  * It is a function of $\\alpha$, not a probability density over $\\alpha$. It helps us update a auxiliary probability density over $\\alpha$, namely:\n",
    "\n",
    "* $P(\\alpha)$: Prior probability distribution (of $\\alpha$)\n",
    "  * This probability distribution is a starting point we have to use.\n",
    "  * It is our state of information before we take the data.\n",
    "  * For example:\n",
    "    * A maximally ignorant state (very wide flat distribution) (Jeffrey's priors maximize the entropy)\n",
    "    * A informed state from previous experiments\n",
    "    * A informed state from prior theoretical knowledge (part of the model)\n",
    "\n",
    "* The Bayesian evidence or marginal likelihood  \n",
    "\n",
    "  * We can also compute this because we know that the posterior needs to be normalised: $P(D)=\\int_{-\\infty}^\\infty P(\\alpha|D) d\\alpha=\\int_{-\\infty}^\\infty P(D|\\alpha) \\times P(\\alpha) d\\alpha$.\n",
    " \n",
    "\n",
    "## Bayes theorem in practice (with grids)\n",
    "\n",
    "### Prior\n",
    "\n",
    "Lets use uninformative priors. \"Let the data speak for themselves\" they say. Lets see what that means.\n",
    "\n",
    "Lets try a uniform prior probability on $\\alpha$ between -10 and 0:\n",
    "\n",
    "  $P(\\alpha) = 1 / 10$ for $\\alpha\\in[-10,0]$ and zero elsewhere.\n",
    "\n",
    "Another way to write it is like this:\n",
    "\n",
    "  $\\alpha$ ~ Uniform(-10, 0)\n",
    "\n",
    "Read \"~\" as \"is distributed as a\".\n",
    "\n",
    "Lets generate a grid to represent this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "DoEwJEJCUTmt",
    "outputId": "367d5d98-29de-4386-a72f-0df02cfa2d26"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-9.9875, -9.7375, -9.4875, -9.2375, -8.9875, -8.7375, -8.4875,\n",
       "       -8.2375, -7.9875, -7.7375, -7.4875, -7.2375, -6.9875, -6.7375,\n",
       "       -6.4875, -6.2375, -5.9875, -5.7375, -5.4875, -5.2375, -4.9875,\n",
       "       -4.7375, -4.4875, -4.2375, -3.9875, -3.7375, -3.4875, -3.2375,\n",
       "       -2.9875, -2.7375, -2.4875, -2.2375, -1.9875, -1.7375, -1.4875,\n",
       "       -1.2375, -0.9875, -0.7375, -0.4875, -0.2375])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_grid = 401\n",
    "\n",
    "alphas_grid = np.linspace(-10, 0, N_grid)\n",
    "alphas_middle = (alphas_grid[1:] + alphas_grid[:-1]) / 2\n",
    "alphas_step = (alphas_grid[1:] - alphas_grid[:-1])\n",
    "\n",
    "alphas_middle[::10]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "M1EtVN9KnH-8"
   },
   "source": [
    "Each value is assumed equally probable by our prior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "id": "MRTFxHViMdem"
   },
   "outputs": [],
   "source": [
    "prior_density = 1 / (alphas_grid[-1] - alphas_grid[0])\n",
    "prior = prior_density * alphas_step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "9nMJ_R9H5piQ"
   },
   "source": [
    "But what if we used a flat prior (uniform spacing) on $\\log-\\alpha$ instead?\n",
    "\n",
    "It would be uninformative (flat) in that variable, but the prior on $\\alpha$ would not be flat."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 279
    },
    "id": "rBLS7-u-0dSl",
    "outputId": "ab94c2da-9442-4aad-d278-d0c02dca448b"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(alphas_middle, histtype='step', label=\"Flat prior on $\\\\alpha$\", density=True)\n",
    "plt.hist(-10**np.linspace(-1, 1, N_grid), histtype='step', label=\"Flat prior on $\\log-\\\\alpha$\", density=True)\n",
    "plt.xlabel(\"$\\\\alpha$\")\n",
    "plt.ylabel(\"Probability density\")\n",
    "plt.legend(loc='best');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "UHuMD9sK1Phj"
   },
   "source": [
    "As you can see, there is no objectively uniform prior -- it depends on the parameterization!\n",
    "\n",
    "Conversely, any prior is flat in some reparameterization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "I1rcXFtKnNPY"
   },
   "source": [
    "### Likelihood function\n",
    "\n",
    "Given a $\\alpha$ value and some data D, we need a function that computes the probability for those data to arise from it.\n",
    "\n",
    "So we need to specify a model:\n",
    "\n",
    "![model](img/PGM-modelled.png)\n",
    "\n",
    "We have a deterministic part of the model:\n",
    "\n",
    " $\\log L(P|\\alpha,\\beta)=0.2 \\times \\left(\\alpha + \\beta \\times (\\log(P / {10 \\mathrm{days}})\\right)$\n",
    "\n",
    "For now we assume $\\beta = -3$ to keep the problem 1d.\n",
    "\n",
    "This allows us to predict L given a P. But we only have a measured P and a measured L. For now we ignore the small P uncertainties, but write for the observed L:\n",
    "\n",
    " $ L_i$ ~ Normal($L(P_i|\\alpha,\\beta)$, $\\sigma_i$)\n",
    "\n",
    "We know $L_i$ from our measurements and $\\sigma_i$ from calibration data of our instrument (a sort of model as well).\n",
    "\n",
    "For one data point $d_i=(P_i, L_i,\\sigma_i)$ given $\\alpha$ and $\\beta$, our Gaussian likelihood is therefore\n",
    "\n",
    " $P(d_i|\\alpha) = \\mathrm{GaussianPDF}(L_i | L(P_i|\\alpha,\\beta), \\sigma_i)$\n",
    "\n",
    "or spelling it out:\n",
    "\n",
    " $P(d_i|\\alpha) = (2 \\pi \\sigma_i^2)^{-\\frac{1}{2}} \\exp\\left(-\\frac{1}{2}\\left(\\frac{L_i - L(P_i|\\alpha,\\beta)}{\\sigma_i}\\right)^2\\right)$\n",
    "\n",
    "Since we want to analyse all data points and require this model to hold for all (logical AND), we multiply the probabilities and obtain our likelihood function:\n",
    "\n",
    " $P(D|\\alpha) = \\prod_i P(d_i|\\alpha)$\n",
    "\n",
    "If we drop some constant factors and work in logarithms, we can write:\n",
    "\n",
    " $-2 \\log P(D|\\alpha) = \\sum_i \\left(\\frac{L_i - L(P_i|\\alpha,\\beta)}{\\sigma_i}\\right)^2 $ \n",
    "\n",
    "Which is sometimes referred to as $\\chi^2$ -- it is just the -2 times the log-likelihood of a Gaussian observation model.\n",
    "\n",
    "Lets implement this:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "id": "4EjmGve6nHsU"
   },
   "outputs": [],
   "source": [
    "def loglikelihood(alpha, beta=-3):\n",
    "    expected_luminosities = luminosity_model(obs_period, alpha=alpha, beta=beta, metallicity=0, gamma=0)\n",
    "    chi2 = (((obs_luminosity - expected_luminosities) / experimental_noise_luminosity)**2).sum()\n",
    "    return -chi2 / 2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "id": "qYsfFrwXt9bf"
   },
   "outputs": [],
   "source": [
    "def generate_luminosity_data(alpha, beta=-3):\n",
    "    experimental_noise_period = np.random.uniform(0, 0.1, size=N_obs)  # not used\n",
    "    experimental_noise_luminosity = np.random.uniform(1, 3, size=N_obs)\n",
    "\n",
    "    true_luminosity = luminosity_model(obs_period, alpha=alpha, beta=beta, metallicity=0, gamma=0)\n",
    "    obs_luminosity = np.random.normal(true_luminosity, experimental_noise_luminosity)\n",
    "    return obs_luminosity\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XdBdRVDevXSf"
   },
   "source": [
    "## Prior predictive checks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "tXVsG1AOusID"
   },
   "source": [
    "Lets have a look what data each prior grid point would generate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 299
    },
    "id": "GlADlUWZusjs",
    "outputId": "6caf3450-3e65-437b-c079-d95c3f48bffc"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for alpha in alphas_middle[::40][::-1]:\n",
    "  plt.plot(obs_period, generate_luminosity_data(alpha), 'x ', label='$\\\\alpha=%s$' % alpha)\n",
    "\n",
    "plt.plot(obs_period, obs_luminosity, 'o ', color='k', ms=10, label='actual data')\n",
    "\n",
    "plt.legend(loc='upper left', prop=dict(size=8))\n",
    "\n",
    "plt.xlabel(\"Period [days]\")\n",
    "plt.ylabel(\"Luminosity\")\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.title(\"Prior predictive samples\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "QO6bbBqIsE9f"
   },
   "source": [
    "Lets plot our loglikelihood function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 283
    },
    "id": "v3SuC0S2sEjZ",
    "outputId": "5b6a75f0-0d1e-4941-fd8b-d09cd134f20b"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "loglikelihoods = np.array([loglikelihood(alpha) for alpha in alphas_middle])\n",
    "plt.plot(alphas_middle, loglikelihoods, drawstyle='steps-mid')\n",
    "plt.ylim(-1000, 0);\n",
    "plt.ylabel(\"log-likelihood\")\n",
    "plt.xlabel(\"$\\\\alpha$\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "sjAr0BAtwGCw"
   },
   "source": [
    "The model most frequently making data that looks closest to our observed sample  near $\\alpha=-4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "eMojvUKyzXqj"
   },
   "source": [
    "So lets compute the probability distribution:\n",
    "\n",
    "First we compute the normalising constant (**evidence**):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "mthzOpWKzs0s",
    "outputId": "fa3533ce-e9f9-4be7-b3ea-f51049369223"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-10.395592057686736, 0.11442464665918742, -8.227753277270889)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# to avoid overflows and underflows, we subtract a constant\n",
    "posterior_unnormalised = np.exp(loglikelihoods - loglikelihoods.max()) * alphas_step\n",
    "\n",
    "# the same as above, but in log, which is more stable\n",
    "logposterior_unnormalised = loglikelihoods + log(alphas_step)\n",
    "\n",
    "# compute log-evidence, add back the constant\n",
    "logevidence = np.log(posterior_unnormalised.sum()) + loglikelihoods.max()\n",
    "\n",
    "logevidence, posterior_unnormalised.sum(), loglikelihoods.max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Plq9B5b6-FB7"
   },
   "source": [
    "Verify that it is now a **normalised** posterior probability distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "joq1mxEmARG0",
    "outputId": "f0055da0-3052-42eb-ee4f-41b6aeeb7b0b"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000004"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "posterior = np.exp(logposterior_unnormalised - logevidence)\n",
    "\n",
    "posterior.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the **posterior probability distribution**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 279
    },
    "id": "dkYxiZI1zWtr",
    "outputId": "aa25078a-1c5a-4041-fb53-5ac5e7881b89"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(alphas_middle, posterior, drawstyle='steps', label='Posterior')\n",
    "plt.plot(alphas_middle, alphas_step, drawstyle='steps', ls='--', color='gray', label='Prior')\n",
    "plt.ylabel(\"Posterior probability density $P(\\\\alpha|D)$\")\n",
    "plt.xlabel(\"$\\\\alpha$\")\n",
    "plt.legend(loc='best')\n",
    "plt.xlim(-5, -4);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "cvJY8JFQLOQG"
   },
   "source": [
    "The **information gain** (https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) can be quantified (in bits) as:\n",
    "\n",
    "$\\int P(\\alpha|D)\\log_2\\frac{P(\\alpha|D)}{P(\\alpha)}d\\alpha$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "pjINKdsSLe-U",
    "outputId": "7f029bb3-45a7-4c5b-cdf6-7cf61dd2e821"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Information gain: 5.7 bits\n"
     ]
    }
   ],
   "source": [
    "with np.errstate(divide='ignore', invalid='ignore'):\n",
    "    information_gain_bits = (posterior * np.log2(posterior / prior))[posterior>0].sum()\n",
    "    print('Information gain: %.1f bits' % information_gain_bits)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 296
    },
    "id": "D_Kkq7Fx9FSy",
    "outputId": "71ac976e-6af4-4798-e62e-b94e787f63d6"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(alphas_middle[posterior>0], posterior[posterior>0].cumsum(), drawstyle='steps')\n",
    "plt.ylabel(\"Cumulative posterior probability density $P(<\\\\alpha|D)$\")\n",
    "plt.xlabel(\"$\\\\alpha$\")\n",
    "plt.grid()\n",
    "plt.xlim(-5, -4);\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.581501212469834e-07"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "posterior[alphas_middle < -4.4].sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets see what the probable models look like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 299
    },
    "id": "S07lczViH8rR",
    "outputId": "07e57428-9929-456a-d463-059d8e1bd369"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for alpha, alpha_posterior in zip(alphas_middle, posterior):\n",
    "  plt.plot(\n",
    "      # predict relation with this alpha\n",
    "      grid_P, luminosity_model(grid_P, alpha=alpha, beta=-3, metallicity=0, gamma=0), \n",
    "      'x ', color='r', alpha=alpha_posterior # weigh by their probability\n",
    "  )\n",
    "\n",
    "plt.errorbar(x=obs_period, y=obs_luminosity, xerr=experimental_noise_period, yerr=experimental_noise_luminosity, \n",
    "             ls=' ', marker='o', color='k', ms=4, label='Data')\n",
    "\n",
    "plt.plot(grid_P, luminosity_model(grid_P, 0, true_alpha, true_beta, true_gamma), '--', color='gray', label='truth')\n",
    "plt.plot(true_period, true_luminosity, 'x ', color='gray')\n",
    "\n",
    "plt.legend(loc='upper left', prop=dict(size=8))\n",
    "\n",
    "plt.xlabel(\"Period [days]\")\n",
    "plt.ylabel(\"Luminosity\")\n",
    "plt.yscale('log')\n",
    "plt.xscale('log')\n",
    "plt.ylim(4, 100)\n",
    "plt.title(\"Posterior prediction of the Period-Luminosity relation\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bmGxoMeRKld-"
   },
   "source": [
    "* Note that the posterior models are much narrower than the prior predictions. We learned a lot!\n",
    "\n",
    "* The model is near the data, but there may be some scatter additional to what is expected from the error bars.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6ZzjjRZd1AUj"
   },
   "source": [
    "# Questions to discuss\n",
    "\n",
    "(5 points for contributing something to the discussion!)\n",
    "\n",
    "* What is the most probable value for $\\alpha$? \n",
    "\n",
    " * MAP: maximum a posteriori : $\\alpha_{MAP} = argmax(posterior)$\n",
    "\n",
    "* In which interval is 90% of the posterior probability is contained? Is this credible interval uniquely defined?\n",
    "\n",
    "* What is the probability that $\\alpha<-4.4$?\n",
    "\n",
    "* How did the prior (uniform grid) influence the posterior here?\n",
    "  What would happen if we had parameterized our model with parameter $\\alpha' = \\log(-\\alpha)$ instead and used a flat uniform prior on $\\alpha'$ instead?\n",
    "\n",
    "  * How would the posterior probability density change? In which region is more probability mass placed?\n",
    "\n",
    "* Lets say we manually adjust our prior to be very close to the posterior, for example $\\alpha$~Uniform(-4.5,-4.2).\n",
    "\n",
    "  * Would the prior predictive checks be more diverse or less diverse?\n",
    "\n",
    "  * Would the typical likelihood values become higher or lower?\n",
    "\n",
    "  * How would the posterior probability density normalisation -- the evidence (marginalised/averaged likelihood) -- change? Would it increase or decrease? \n",
    "\n",
    "  * Do the likelihood function values change if we change the prior?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "KAO3XANJDqYo"
   },
   "source": [
    "# Take-aways\n",
    "\n",
    "* Priors are always there when inferring probability distributions. Sometimes they are implicit via the chosen parameterization of the space.\n",
    "\n",
    "* Probabilistic generative models tell a complete story of the physical process of interest and the measurement process.\n",
    "\n",
    "* Posterior probabilities represent our state of information after data were taken.\n",
    "\n",
    "* The Bayesian evidence punishes prediction diversity. Bayes factors and posterior odds ratios are one kind of model comparison.\n",
    "\n",
    "* For problems with several parameters, we need to go beyond grids.\n",
    "\n",
    "--> Next: How to compute posterior probability distributions and the evidence with Monte Carlo methods."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "JQ7UIk12J5pA"
   },
   "source": [
    "# The curse of dimensionality\n",
    "\n",
    "In our model, we made quite a few assumptions, fixing parameters:\n",
    "\n",
    "* we assumed $\\beta=-3$\n",
    "* we assumed $\\sigma_P=0$\n",
    "* we assumed $\\gamma=0$ and no systematic scatter about our law.\n",
    "\n",
    "* If we extend our model to explore other parameters (such as $\\beta$), how does the number of grid points (and thus the number of likelihood evaluations we need to perform) scale with parameters?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ihP-iyUdJ9SS"
   },
   "source": [
    "## 2d grid\n",
    "\n",
    "Lets make a grid in two dimensions (over $\\alpha$ and $\\beta$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "JMt41cvvFm1_",
    "outputId": "d0eebd3d-be90-491b-f3ab-101639b5b164"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(39, 400) (39, 400) (39, 400)\n"
     ]
    }
   ],
   "source": [
    "betas_grid = np.linspace(-5, 2, N_grid // 10)\n",
    "betas_middle = (betas_grid[1:] + betas_grid[:-1]) / 2\n",
    "betas_step = betas_grid[1:] - betas_grid[:-1]\n",
    "\n",
    "grid = np.meshgrid(alphas_middle, betas_middle)\n",
    "grid_cellsize = np.product(np.meshgrid(alphas_step, betas_step), axis=0)\n",
    "\n",
    "print(grid[0].shape, grid[1].shape, grid_cellsize.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "77pn_N6pUGo-"
   },
   "source": [
    "Evaluate the likelihood and get the 2d normalised posterior probability distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "MnmQAjl0UJz2",
    "outputId": "7e14dffe-171d-4534-a0f4-10c1a116d8f7"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999999"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "grid_unnormalised_logposterior = np.vectorize(loglikelihood)(grid[0], grid[1]) + np.log(grid_cellsize)\n",
    "grid_unnormalised_posterior = np.exp(grid_unnormalised_logposterior)\n",
    "grid_logevidence = np.log(grid_unnormalised_posterior.sum())\n",
    "grid_posterior = np.exp(grid_unnormalised_logposterior - grid_logevidence)\n",
    "\n",
    "grid_posterior.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 308
    },
    "id": "N8jqW8VNNosz",
    "outputId": "1eb005fa-7074-459b-e33d-77915135d4ad"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(2, 2, 3)\n",
    "plt.imshow(\n",
    "      grid_posterior[::-1],\n",
    "      extent=(alphas_middle[0], alphas_middle[-1], betas_middle[0], betas_middle[-1]),\n",
    "      aspect='equal', cmap='gray_r')\n",
    "#plt.colorbar(label='log-likelihood')\n",
    "plt.xlabel(\"$\\\\alpha$\")\n",
    "plt.ylabel(\"$\\\\beta$\");\n",
    "plt.subplot(2, 2, 1)\n",
    "plt.plot(alphas_middle, grid_posterior.sum(axis=0))\n",
    "plt.xlabel(\"$\\\\alpha$\")\n",
    "plt.yticks([])\n",
    "plt.subplot(2, 2, 4)\n",
    "plt.plot(grid_posterior.sum(axis=1), betas_middle)\n",
    "plt.ylabel(\"$\\\\beta$\")\n",
    "plt.xticks([])\n",
    "plt.suptitle(\"Marginal and conditional probability distributions\");\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "xvfc5pbDSKXP"
   },
   "source": [
    "## Bayesian model comparison"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "q8FG57__e6E3"
   },
   "source": [
    "We marginalised the likelihoods to model evidences: $Z_1=P(D|M_1)$, $Z_2=P(D|M_2)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "DSx4ryzcSpjA",
    "outputId": "8d764d6b-b2cf-4bb9-a533-36664d6e4bed"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "log10(Z) of 1d model (beta fixed ): -10.40\n",
      "log10(Z) of 2d model (beta varies): -9.61\n"
     ]
    }
   ],
   "source": [
    "print(\"log10(Z) of 1d model (beta fixed ): %.2f\" % (logevidence))\n",
    "print(\"log10(Z) of 2d model (beta varies): %.2f\" % (grid_logevidence))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rGqF23bBfouD"
   },
   "source": [
    "\n",
    "These can be used to compute relative probabilities of **models**, by applying Bayes' theorem once again:\n",
    "\n",
    "$P(M_1|D) = \\frac{P(D|M_1) * P(M_1)}{\\sum_i P(D|M_i) * P(M_i)}$\n",
    "\n",
    "For this step, we first need to define prior model probabilities $P(M_i)$ for each model.\n",
    "\n",
    "If we assume a 1:100 odds ratio, this is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "ajxVYq_5X1pi",
    "outputId": "e2335881-ea98-4689-9103-9a6020b195eb"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Prior probability P(1dmodel) = 0.009901\n",
      "Prior probability P(2dmodel) = 0.990099\n"
     ]
    }
   ],
   "source": [
    "prior_odds = 0.01\n",
    "model_prior_1d = prior_odds / (1 + prior_odds)\n",
    "model_prior_2d = 1 / (1 + prior_odds)\n",
    "\n",
    "print(\"Prior probability P(1dmodel) = %2f\" % (prior_odds / (1 + prior_odds)))\n",
    "print(\"Prior probability P(2dmodel) = %2f\" % (1 / (1 + prior_odds)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "DtKif-hgYf-q"
   },
   "source": [
    "Posterior model probabilities $P(M_i|D)$ give the relative probability of one model being the right one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "FktklWbvXzX4",
    "outputId": "d37fc29a-16af-4aea-d210-d0359bcff978"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior model probability P(1dmodel|D) = 0.004557\n",
      "Posterior model probability P(2dmodel|D) = 0.995443\n"
     ]
    }
   ],
   "source": [
    "print(\"Posterior model probability P(1dmodel|D) = %2f\" % (np.exp(logevidence) * prior_odds / (np.exp(logevidence) * prior_odds + np.exp(grid_logevidence))))\n",
    "print(\"Posterior model probability P(2dmodel|D) = %2f\" % (np.exp(grid_logevidence) / (np.exp(logevidence) * prior_odds + np.exp(grid_logevidence))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "_zfIcZlbYYdH"
   },
   "source": [
    "A quantification of how much the odds have shifted is the **Bayes factor**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Tq4LBsqIOSJe",
    "outputId": "aef5951b-2b7f-470b-ab59-2c5e969651f0"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Bayes factor: P(D|1dmodel)/P(D|2dmodel) = 0.46\n"
     ]
    }
   ],
   "source": [
    "bayes_factor = np.exp(logevidence - grid_logevidence)\n",
    "\n",
    "print(\"Bayes factor: P(D|1dmodel)/P(D|2dmodel) = %.2f\" % bayes_factor)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "JHBiyvEvgGNu"
   },
   "source": [
    "If the Bayes factor is close to 1, there is \"not much evidence\" in favor of one or the other model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "EXOm7wN1WWOM"
   },
   "source": [
    "Recall that model evidences (marginal likelihoods) are likelihoods averaged over the entire parameter space.\n",
    "\n",
    "Benefits:\n",
    "\n",
    "1. Bayes factors capture the entire model parameter space, not just the \"best fit\". \n",
    "2. Bayes factors punish model diversity (\"built-in Occam's razor\")\n",
    "\n",
    "Limitations:\n",
    "\n",
    "3. Dependence on priors: Because of (1), the results strongly depend on the parameter priors. They also strongly depend on the chosen prior model probabilities, which need to be chosen.\n",
    "\n",
    "   * This means posterior odds can be meaningful and appropriate for competing physical models where these priors can be specified. \n",
    "   * For ad-hoc/empirical models, they are almost never useful --> see model comparison lecture later.\n",
    "\n",
    "4. Bayesian inference assumes that the true hypothesis is within the specified model and parameter set (\"Closed world assumption\", \"M-closed\"). This applies to parameter estimation and model comparison.\n",
    "\n",
    "5. Bayesian inference does not make decisions. It only weighs probabilities.  Posterior odds do not state the probability of making a false decision. For this, a frequentist characterisation of imposing a threshold is needed, based on simulations!\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "7UHt2LsLafNd"
   },
   "source": [
    "# Ticket to leave\n",
    "\n",
    "Fill out the [form below](https://indico.ph.tum.de/event/6875/surveys/5) and then you can leave the class. (+5 points)"
   ]
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   "execution_count": 26,
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