{ "cells": [ { "cell_type": "markdown", "id": "71f30505-b755-403a-9b5c-9c88bade649f", "metadata": {}, "source": [ "(awkward_binning)=\n", "(binning)=\n", "# Estimating parameters of a distribution from awkwardly binned data\n", ":::{post} Oct 23, 2021\n", ":tags: binned data, case study, parameter estimation\n", ":category: intermediate\n", ":author: Eric Ma, Benjamin T. Vincent\n", ":::" ] }, { "cell_type": "markdown", "id": "cc8f267d", "metadata": {}, "source": [ "## The problem\n", "Let us say that we are interested in inferring the properties of a population. This could be anything from the distribution of age, or income, or body mass index, or a whole range of different possible measures. In completing this task, we might often come across the situation where we have multiple datasets, each of which can inform our beliefs about the overall population.\n", "\n", "Very often this data can be in a form that we, as data scientists, would not consider ideal. For example, this data may have been binned into categories. One reason why this is not ideal is that this binning process actually discards information - we lose any knowledge about where in a certain bin an individual datum lies. A second reason why this is not ideal is that different studies may use alternative binning methods - for example one study might record age in terms of decades (e.g. is someone in their 20's, 30's, 40's and so on) but another study may record age (indirectly) by assigning generational labels (Gen Z, Millennial, Gen X, Boomer II, Boomer I, Post War) for example.\n", "\n", "So we are faced with a problem: we have datasets with counts of our measure of interest (whether that be age, income, BMI, or whatever), but they are binned, and they have been binned _differently_. This notebook presents a solution to this problem that [PyMC Labs](https://www.pymc-labs.io) worked on, supported by the [Gates Foundation](https://www.gatesfoundation.org/). We _can_ make inferences about the parameters of a population level distribution.\n", "\n", "![](gates_labs_logos.png)" ] }, { "cell_type": "markdown", "id": "4f1f4b0b", "metadata": {}, "source": [ "## The solution\n", "\n", "More formally, we describe the problem as: if we have the bin edges (aka cut points) used for data binning, and bin counts, how can we estimate the parameters of the underlying distribution? We will present a solution and various illustrative examples of this solution, which makes the following assumptions:\n", "\n", "1. that the bins are order-able (e.g. underweight, normal, overweight, obese),\n", "2. the underlying distribution is specified in a parametric form, and\n", "3. the cut points that delineate the bins are known and can be pinpointed on the support of the distribution (also known as the valid values that the probability distribution can return).\n", "\n", "The approach used is heavily based upon the logic behind [ordinal regression](https://en.wikipedia.org/wiki/Ordinal_regression). This approach proposes that observed bin counts $Y = {1, 2, \\ldots, K}$ are generated from a set of bin edges (aka cutpoints) $\\theta_1, \\ldots, \\theta _{K-1}$ operating upon a latent probability distribution which we could call $y^*$. We can describe the probability of observing data in bin 1 as:\n", "\n", "$$P(Y=1) = \\Phi(\\theta_1) - \\Phi(-\\infty) = \\Phi(\\theta_1) - 0$$\n", "\n", "bin 2 as:\n", "\n", "$$P(Y=2) = \\Phi(\\theta_2) - \\Phi(\\theta_1)$$\n", "\n", "bin 3 as:\n", "\n", "$$P(Y=3) = \\Phi(\\theta_3) - \\Phi(\\theta_2)$$\n", "\n", "and bin 4 as:\n", "\n", "$$P(Y=4) = \\Phi(\\infty) - \\Phi(\\theta_3) = 1 - \\Phi(\\theta_3)$$\n", "\n", "where $\\Phi$ is the standard cumulative normal.\n", "\n", "![](ordinal.png)\n", "\n", "In ordinal regression, the cutpoints are treated as latent variables and the parameters of the normal distribution may be treated as observed (or derived from other predictor variables). This problem differs in that:\n", "\n", "- the parameters of the Gaussian are _unknown_, \n", "- we do not want to be confined to the Gaussian distribution,\n", "- we have observed an array of cutpoints,\n", "- we have observed bin counts, \n", "\n", "We are now in a position to sketch out a generative PyMC model:\n", "\n", "python\n", "import pytensor.tensor as pt\n", "\n", "with pm.Model() as model:\n", " # priors\n", " mu = pm.Normal(\"mu\")\n", " sigma = pm.HalfNormal(\"sigma\")\n", " # generative process\n", " probs = pm.math.exp(pm.logcdf(pm.Normal.dist(mu=mu, sigma=sigma), cutpoints))\n", " probs = pm.math.concatenate([[0], probs, [1]])\n", " probs = pt.extra_ops.diff(probs)\n", " # likelihood\n", " pm.Multinomial(\"counts\", p=probs, n=sum(counts), observed=counts)\n", "\n", "\n", "The exact way we implement the models below differs only very slightly from this, but let's decompose how this works.\n", "Firstly we define priors over the mu and sigma parameters of the latent distribution. Then we have 3 lines which calculate the probability that any observed datum falls in a given bin. The first line of this\n", "python\n", "probs = pm.math.exp(pm.logcdf(pm.Normal.dist(mu=mu, sigma=sigma), cutpoints))\n", "\n", "calculates the cumulative density at each of the cutpoints. The second line \n", "python\n", "probs = pm.math.concatenate([[0], probs, [1]])\n", "\n", "simply concatenates the cumulative density at $-\\infty$ (which is zero) and at $\\infty$ (which is 1).\n", "The third line\n", "python\n", "probs = pt.extra_ops.diff(probs)\n", "\n", "calculates the difference between consecutive cumulative densities to give the actual probability of a datum falling in any given bin.\n", "\n", "Finally, we end with the Multinomial likelihood which tells us the likelihood of observing the counts given the set of bin probs and the total number of observations sum(counts).\n", "\n", "Hypothetically we could have used base python, or numpy, to describe the generative process. The problem with this however is that gradient information is lost, and so completing these operations using numerical libraries which retain gradient information allows this to be used by the MCMC sampling algorithms.\n", "\n", "The approach was illustrated with a Gaussian distribution, and below we show a number of worked examples using Gaussian distributions. However, the approach is general, and at the end of the notebook we provide a demonstration that the approach does indeed extend to non-Gaussian distributions." ] }, { "cell_type": "code", "execution_count": 1, "id": "e1299722", "metadata": { "tags": [] }, "outputs": [], "source": [ "import warnings\n", "\n", "import arviz as az\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pandas as pd\n", "import pymc as pm\n", "import pytensor.tensor as pt\n", "import seaborn as sns\n", "\n", "warnings.filterwarnings(action=\"ignore\", category=UserWarning)" ] }, { "cell_type": "code", "execution_count": 2, "id": "a8fa52a9", "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "%config InlineBackend.figure_format = 'retina'\n", "plt.rcParams.update({\"font.size\": 14})\n", "az.style.use(\"arviz-darkgrid\")\n", "rng = np.random.default_rng(1234)" ] }, { "cell_type": "markdown", "id": "87989a59", "metadata": {}, "source": [ "## Simulated data with a Gaussian distribution\n", "\n", "The first few examples will be based on 2 hypothetical studies which measure a Gaussian distributed variable. Each study will have it's own sample size, and our task is to learn the parameters of the population level Gaussian distribution. Frustration 1 is that the data have been binned. Frustration 2 is that each study has used different categories, that is, different cutpoints in this data binning process.\n", "\n", "In this simulation approach, we will define the true population level parameters as:\n", "- true_mu: -2.0\n", "- true_sigma: 2.0\n", "\n", "Our goal will be to recover the mu and sigma values given only the bin counts and cutpoints." ] }, { "cell_type": "code", "execution_count": 3, "id": "b8e01e91", "metadata": {}, "outputs": [], "source": [ "# Generate two different sets of random samples from the same Gaussian.\n", "true_mu, true_sigma = -2, 2\n", "x1 = rng.normal(loc=true_mu, scale=true_sigma, size=1500)\n", "x2 = rng.normal(loc=true_mu, scale=true_sigma, size=2000)" ] }, { "cell_type": "markdown", "id": "58fa85c2", "metadata": {}, "source": [ "The studies used the following, different, cutpoints for their data binning process." ] }, { "cell_type": "code", "execution_count": 4, "id": "2a12cff1", "metadata": {}, "outputs": [], "source": [ "# First discretization (cutpoints)\n", "d1 = np.array([-2.0, -1.0, 0.0, 1.0, 2.0])\n", "# Second discretization (cutpoints)\n", "d2 = np.array([-5.0, -3.5, -2.0, -0.5, 1.0, 2.5])" ] }, { "cell_type": "code", "execution_count": 5, "id": "ddf0519f", "metadata": {}, "outputs": [], "source": [ "def data_to_bincounts(data, cutpoints):\n", " # categorise each datum into correct bin\n", " bins = np.digitize(data, bins=cutpoints)\n", " # bin counts\n", " counts = pd.DataFrame({\"bins\": bins}).groupby(by=\"bins\")[\"bins\"].agg(\"count\")\n", " return counts\n", "\n", "\n", "c1 = data_to_bincounts(x1, d1)\n", "c2 = data_to_bincounts(x2, d2)" ] }, { "cell_type": "markdown", "id": "e7763acc", "metadata": {}, "source": [ "Let's visualise this in one convenient figure. The left hand column shows the underlying data and the cutpoints for both studies. The right hand column shows the resulting bin counts." ] }, { "cell_type": "code", "execution_count": 6, "id": "3136de42", "metadata": {}, "outputs": [ { "data": { "image/png": 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ERFRprKhap3JtX3HFFR7FXSf6Fiaqvzlz5liFXfHx8Ro+fLjWrFljc1QAaqL58+frgQceUH5+vgzDsG7cm6apunXrqm3btmrYsKHi4uJUq1YthYaGKiQkRLm5ucrJyVFmZqbS0tK0Z88ebdq0SQcPHpT0vxv8GRkZGj16tJ599lldccUVdj5VANXUhg0bNHz4cOsLTMf/rir+ofWWLVt000036eabb9b48eMVEBBgV9hwuAcffFArVqyQVHpBc9E59v7776tjx4668sor9eCDDyojI8Ojf3BwsGrXrq1jx44pOzvbY9/Fixfr5Zdf1n333efjZwdfq4plyEzTLHNcZhms+aZPny7DMLwu7nrkkUc0d+5c673x+L/piyv62YEDB3TDDTdo7ty5atmyZVU8DThEenq6tm/fLqnwnmTv3r1P2P/hhx/Wvn37rHMlICBAPXr0UM+ePRUfH6+YmBjl5uYqLS1Na9eu1eeff67du3db/R955BF16tRJzZo188XT8xsUdwHV0OLFiz2KgE72C7hInz59rOIuqfDDRYq7nKNoKvki4eHhXu97/Ie7ZX1bCNXLJ5984nFOXHfddTZGA6f54IMPtHbtWkn/+7AXKK5p06Ye2wcOHLApEhzv+JlXpcK/z7zRu3dvTZ8+XVLhUo1Lliyh6NcPcV37j71791r/djMMQ5MnT2Y5FwBVYvv27Zo4caJV2GWappo2barBgwerb9++p/Qlkh07dug///mPPvnkE+3YsUOGYSg/P18TJ05U+/btucEPoFyysrJ0zz33KDMz06OgJjY2Vk2aNFFmZqZ27typnJwc6+cFBQVKSkpSSkqKpk6dWq77rPAPixcv1n/+8x+Pcyo+Pl5du3ZVZGSkdu3apcWLF+vIkSMyTVMvv/yy4uLitG7dOuv98pxzztHYsWPVtWtXaynQ7du3a86cOZo9e7bcbrdM09SsWbM0aNAgCidquKpaKIsFuPybt/mfP3++PvroI4+irjp16qhv377q2LGj6tatq+zsbP31119atGiRli1bZv3+K3qf/fjjj0/pi4ioHopmATQMQ2edddYJV/hYu3at/u///s86R5o1a6aXXnpJ7dq1K9G3VatWuvDCC3XbbbfplVde0RtvvCHDMJSXl6dXX31Vzz33XNU8IT/FFQpUQ0Xf3i7SqVMnr/Zr2rSp6tata32Dsvg4sFdsbKwiIiKUlZUlqXDmJm/t2rXLelz8Qz9UT/Pnz7ceR0VF6aKLLrIxGjhJWlqax4e9kyZN4sNelKroG81FOE+cYfXq1dbjevXqef2BafG/91auXElxlx8qfl1z063mevzxx3X06FFJhTP8de7c2eaIANRUU6ZMUXZ2tnXj/o477tAtt9xSofeYpk2b6pZbbtGIESP0+uuv69VXX5VhGMrOztb/t3ff0VXU+/rHnyGFkIQQSihSDgREiqAU6f5AiiCXSLk2RDBICe0gyhUQaYKgIl6poigSUBTliB0pAooFBUQFxCAQEYi0kAIhCWn790dW5mYnEJIwO7vk/VrLtfZ3MuWTcx529p75zHxffvllLV682KryAZQC7777rk6ePGlerA4NDdX06dPVvn17c52UlBRt2rRJK1as0F9//WWu+9133yk8PFyvv/66goODnfdLwOWsWbNG0v89eeu5555T37597da5dOmSJk2apB07digmJsbuAnVYWJhefPHFfE+G+9e//qXJkyerXbt2Gjt2rDIzM5WZmal169bpmWeecfwvBqfKOy3ZjbjalHp5fwZIUmZmpl555RVJ/5eNBx98UJMmTVJAQEC+9cPDw7Vv3z49+eSTOnv2rCTp8OHD2rhxo+69996SKxwl6uTJk+brW265pcB1t2zZIik7T8HBwVqzZo2qVatW4Dbe3t564oknJEmvv/66JGnz5s2aM2eOypYteyOlIxfOBANuKGdKnhxX65S9lsaNG5tTOl68eFHnz59XSEiIpfWheMqUKaOOHTtq69atkrLv1t+3b991L+Ts3bvXbirGu+66y6F1wvESExN1/Phxc3zbbbfJ19fXeQXBpcyaNcu8uH///ferVatWTq4Iripnut4c1/sCBsc7e/as2awhFe2R/Xk/7x07dsyyuuA+8v67rl69upMqgSNt2bLF/E5QuXJlTZw40ckVAfBUiYmJ2rFjh3nRcNKkSZY+Fdjb21vjxo2Tv7+/eUF8x44dSkxMVIUKFSw7DlzLnj17SuQ4XNguPd59913zfaphw4Zau3ZtvqdNlCtXTv3791efPn20bNkyrVixQlJ2Tn777TcNHjxYK1euVNWqVUu8friepKQkff/992auJk6cmK+xS8q+4XbRokUKCwvTiRMndPDgQUlSjRo1NGfOnAKn/OzcubOGDh2qN954Q5L0xRdf0NxVipQvX77Ap+JcT86N/4ZhqEaNGlaVBQ+1fft2xcTEmO9pjzzyyHXfb1q2bKl3331X//3f/62EhARJ2X9vae7yXBcvXjRfV6pUqcB1c09ZHBERUaTrCmPHjtWHH36o2NhYpaWl6eeff1aHDh2KVzTyobkLcEO5m7t8fX1VuXLlQm9700035dsXzV2uY+TIkdq2bZs5Hd+MGTP09ttvq2LFildd/8KFC5o2bZo5Dg4O1pAhQ0qkVjhOzomCHLfeequk7BNSO3fu1KeffqpDhw7pzJkz5nvAbbfdpi5duqhHjx4FnliAe9u4caN27NghKfsD+P/8z/84uSK4qlOnTmnfvn3mOCgoqEjN4HCMvA36eT+XFaRKlSry8fFRenq6JOmvv/6ytDa4h08//dRu3K5dOydVAke5dOmS5syZY46nTJlCAwQAh9m3b5/S09NlGIbq1Kmj8PBwhxxn6NChWrdunU6cOKGMjAzt27ePG9M82ODBg2/4aSVAjhMnTtg1OcyfP7/AhgkfHx9NmDBBbdu21ZNPPmlesD5y5IgefvhhrVq1qljTzcKzHDx40Dz/XqFCBQ0aNOia6/r6+mrIkCHmZ3TDMPTQQw/Jz8/vuscJDw/XqlWrlJGRofj4eJ08eZL8ebAyZcqYjccZGRkaNGiQhg4dWqxz9U2bNlVmZqak7MYdoCA//PCDpOzrR7Vq1dKUKVMKtd1NN92kyZMnm+sfOHBASUlJCgwMdFitcJ7cN0bknF++ltwPFOnevXuRjuPr66s777xTH330kaTs6xSwDld/ATeU+021evXqRTphkre79syZM5bVhRvXvHlzPfnkk+b4yJEj6tu3r9auXauTJ08qIyND6enp+vvvvxUZGal7773XvLjr5+enRYsWXbfjGq7v6NGjduPatWsrJiZGQ4YM0ciRI/X5558rOjpaycnJSkhI0LFjx7RhwwaNHz9effr0YcpVD5WYmKi5c+eaYy72oiDLli2z+8LWo0cPpm9zATmPOs9RlKcuGYZh9zku777g+Xbv3m3eOSdl3wncqVMnJ1YER5g/f775fa9Dhw7cNQvAoXKfE2rXrp3DGnIMw7CbPo1zUaWDzWZz6H8oHXJugDQMQ7fffrsaNWpUqO3at2+v999/X7Vr1zanRzt16pQGDRqU77wbSp8TJ05Iys7VHXfcIR8fnwLXz/03TFKhn0BSuXJl86ZdSYqKiipipXAn77//vho2bCibzabU1FQtWLBA/fv312+//ebs0uDhcjJmGIYGDRokLy+vQm8bFhZmXlPMysrSgQMHHFIjnC/3taTcfQZXk5SUZL4uzqwBubfJabSHNWjuAtxMamqq2bEvSf7+/kXaPm/HdXJysiV1wTojRozQggULVKVKFUnZF29nz56t7t27q2nTprr11lt199136/nnn1dsbKwkqXXr1lq/fj1Pb/AQiYmJduOMjAw99NBDdhd0r+XYsWMKDw/XF1984ajy4CQvvvii+W++ffv2V31cPCBl362Vc2eMlH3nckREhBMrQo68n7uK+jkuICDAfJ2RkaErV65YUhdcX0pKiqZPn263bOjQoXaZgPvbs2eP1q9fLyn7TseZM2c6uSIAni73dNHBwcEOPVbu/ee+WADPxdO7YIW4uDjzdfPmzYu0bZ06dbRu3To1adLEbPA6d+6cHnnkkXxPzUfpkntqqn/961/XXb9WrVp247p16xb6WA0bNjRf584zPE+zZs20YcMGPfHEE/L19ZXNZtPhw4c1cOBAPfvss3z+gcPkXDOQ8jejXo+Xl5fuuOMOc5zztEx4nnr16pmvf/zxxwLXzT1jWEpKSpGPlZqaar7m3KW1uH0fcDN5LwqWLVu2SNv7+voWuD+4hrCwMHXp0kVz5861u0Cfl2EYevjhhzV+/HiHn4hFycl9gkGSFixYYJ50Dw0N1ahRo9SuXTtVrFhRCQkJ+umnn/Taa6+Zdx6mp6drypQpqlu3rpo2bVri9cN6P/74oz788ENJ2e/js2bNcm5BcFnnzp3TpEmT7O5kj4iIKNTJSjjejX6Oy7v+5cuXi7wPuKdnn31Wx48fN8ehoaEaPny48wqC5dLS0jR9+nTz/XvUqFFFumgEAMVRtWpV87Wjp8vIvf+QkBCHHgvO5ePjo4yMDElSgwYN1KtXL4ccZ9myZQ7ZL1xL7maIgqZjvJZKlSppzZo1GjVqlPbu3SvDMJSQkKDw8HC99tprat26tZXlwk3kvEdJUrly5a67vq+vr7y8vMyb7osyZVnFihXN17mbquGZvLy8FBERoZ49e2r69Onas2ePsrKytG7dOm3dulVTp05V7969nV0mPEzu60l5m1ELo2bNmlfdFzzLbbfdJl9fX6Wlpen8+fP6+uuv1aVLl6uuW7duXfP72/79+4s8c8D+/fvN13lnFMONobkLcDN5n9BwvUcG55W3uSt39yxcx0cffaSFCxded6oCm82mtWvX6qOPPtLIkSMVERFRrDnc4VryXvzP+eLfuXNnLV261O7fcdWqVRUWFqaePXtq/Pjx2rFjh6T/u0C4YcOGkiscDnHlyhXNmDHDHEdERHCxF1d15coVjR8/Xuf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