main.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "University of Helsinki, Master's Programme in Data Science  \n",
    "DATA20047 Probabilistic Cognitive Modelling - Spring 2023  \n",
    "Luigi Acerbi  \n",
    "Adam El Kholy\n",
    "\n",
    "# Calculating Response Distribution and Model Fitting\n",
    "\n",
    "## References\n",
    "\n",
    "- \\[**MKG22**\\] Ma WJ, Körding K, and Goldreich D. \"Bayesian Models of Perception and Action: An Introduction\". MIT Press, 2022.\n",
    "- \\[**AWV12**\\] Acerbi L, Wolpert DM, Vijayakumar S. \"Internal Representations of Temporal Statistics and Feedback Calibrate Motor-Sensory Interval Timing\". *PLoS Computational Biology*, 2012. [Link](https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002771)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# set-up -- do not change\n",
    "import numpy as np\n",
    "import numpy.random as npr\n",
    "import scipy as sp\n",
    "import scipy.stats as sps\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "npr.seed(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 2.1 (7 pts)\n",
    "\n",
    "> This question is about computing the total (root mean squared) error (RMSE) for a Bayesian observer, as seen in Week 3 of the course. The take-home message here is that the Bayesian observer whose prior matches the true empirical distribution of stimuli will perform best at the task (lower RMSE), compared to a Bayesian observer with an incorrect (aka *mismatched*) belief about the distribution of stimuli (i.e., whose prior does not match the true stimulus distribution). See Chapter 4.5 of \\[**MKG22**\\] and the lecture notes for Week 3.\n",
    "\n",
    "A Bayesian observer is estimating a stimulus with empirical distribution $p(s) = \\text{Uniform}(s; -5, 5)$.\n",
    "The measurement distribution and likelihood are Gaussian $p(x|s) = \\mathcal{N}\\left(x; s, \\sigma^2 \\right)$ with $\\sigma = 2$.\n",
    "We assume that the observer uses the posterior mean estimator $\\hat{s}_{PM}$ and we ignore response noise. However, we consider the observer uses as prior a distribution $q(s)$ which might differ from the true prior (mismatched prior).\n",
    "\n",
    "- a) Compute the total RMSE assuming $q(s) = p(s)$, i.e. the observer uses the true stimulus distribution as prior.\n",
    "- b) Compute the total RMSE assuming the observer uses an approximate Gaussian prior, $q(s) = \\mathcal{N}\\left(s; \\mu_s, \\sigma_s^2 \\right)$ with mean $\\mu_s$ and variance $\\sigma^2_s$ equal to the mean and variance of the true stimulus distribution. *Hint*: You can find the variance of a continuous uniform distribution [here](https://en.wikipedia.org/wiki/Continuous_uniform_distribution).\n",
    "- c) Compute the total RMSE assuming the observer uses as prior a mismatched, wider Uniform distribution, $q(s) = \\text{Uniform}\\left(s; -8, 8 \\right)$.\n",
    "\n",
    "Report your results in Moodle. The accepted tolerance is $\\pm 0.01$ from the true value.\n",
    "\n",
    "*Hints*: \n",
    "- Remember that the (total) RMSE of an estimator $\\hat{s}$ is computed as\n",
    "$$\n",
    "\\text{RMSE}[\\hat{s}] = \\sqrt{\\int \\text{MSE}\\left[\\hat{s}|s\\right] p(s) ds}\n",
    "$$\n",
    "  where $p(s)$ is the true empirical distribution and $\\text{MSE}\\left[\\hat{s}|s\\right]$ is the mean squared error at each stimulus, defined as\n",
    "$$\n",
    "\\text{MSE}\\left[\\hat{s}|s\\right] = \\mathbb{E}_{\\hat{s}|s}\\left[\\left(\\hat{s}-s\\right)^2|s \\right] = \\text{Bias}\\left[\\hat{s}|s\\right]^2 + \\text{Var}\\left[\\hat{s}|s\\right],\n",
    "$$\n",
    "  where the definitions for bias and variance can be found in the textbook or lecture notes.\n",
    "- Note that changing the prior $q(s)$ will change $\\hat{s}(x)$, but nothing else! So once you manage to compute (a), you should be able to compute (b) and (c) with a small change to the code (only where $\\hat{s}(x)$ is computed).\n",
    "- You may want to check out Exercise 3.3 of the workshops."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_posterior_mean_1d(s_grid, prior_pdf, likelihood):\n",
    "    \"\"\"Compute s_hat_PM (posterior mean) for an arbitrary prior and likelihood in 1d.\"\"\"\n",
    "    ds = s_grid.flatten()[1] - s_grid.flatten()[0] # grid spacing\n",
    "    protoposterior = prior_pdf * likelihood    \n",
    "    normalization_constant = sp.integrate.romb(protoposterior, dx=ds, axis=0)\n",
    "    posterior_pdf = protoposterior / normalization_constant\n",
    "    posterior_mean = sp.integrate.romb(s_grid * posterior_pdf, dx=ds, axis=0)\n",
    "    return posterior_mean\n",
    "\n",
    "def compute_and_plot_metrics(r_col, s_hat, stimulus_pdf, label):\n",
    "    \"\"\"Compute bias, variance, conditional and overall MSE.\"\"\"\n",
    "    bias = sp.integrate.romb(s_hat * sps.norm.pdf(x_row,r_col,sigma),dx=dx,axis=1) - r_col.flatten()\n",
    "    std = np.sqrt(sp.integrate.romb(s_hat**2 * sps.norm.pdf(x_row,r_col,sigma),dx=dx,axis=1) \n",
    "        - sp.integrate.romb(s_hat * sps.norm.pdf(x_row,r_col,sigma),dx=dx,axis=1)**2)\n",
    "    rmse = np.sqrt(bias**2 + std**2)\n",
    "    \n",
    "    # Plot only where the support of the Uniform stimulus distribution is nonzero\n",
    "    s_range = r_col.copy()\n",
    "    s_range[np.logical_or(s_range < a, s_range > b)] = np.nan\n",
    "\n",
    "    plt.plot(s_range.flatten(),bias.flatten(),label='bias')\n",
    "    plt.plot(s_range.flatten(),std.flatten(),label='standard deviation')\n",
    "    plt.plot(s_range.flatten(),rmse.flatten(),label='RMSE')\n",
    "    plt.plot((a,b),(0,0),':k')\n",
    "    plt.xlabel(r'$s$')\n",
    "    plt.ylabel(r'Metrics')\n",
    "    plt.title('Bias, standard deviation and RMSE (' + label + ')')\n",
    "    plt.legend()\n",
    "    plt.show()\n",
    "\n",
    "    total_rmse = np.sqrt(sp.integrate.romb(rmse**2 * stimulus_pdf,dx=ds))\n",
    "    print('Total RMSE (' + label + '): {}'.format(total_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a)\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total RMSE (posterior mean estimate): 1.6048573201415854\n"
     ]
    }
   ],
   "source": [
    "a = -5.\n",
    "b = 5.\n",
    "sigma = 2.\n",
    "\n",
    "print('a)')\n",
    "Nx = 2**9+1\n",
    "Ns = 2**9+1\n",
    "\n",
    "# define the grid\n",
    "lb = a - sigma*5.\n",
    "ub = b + sigma*5.\n",
    "x_row = np.linspace(lb, ub, Nx).reshape((1,Nx)) # make x a row vector\n",
    "r_col = np.linspace(lb, ub, Ns).reshape((Ns,1)) # make s a column vector\n",
    "dx = x_row.flatten()[1] - x_row.flatten()[0]\n",
    "ds = r_col.flatten()[1] - r_col.flatten()[0]\n",
    "\n",
    "prior_pdf = sps.uniform.pdf(r_col, a, b-a)\n",
    "\n",
    "likelihood = sps.norm.pdf(x_row, r_col, sigma) \n",
    "s_hat_row = compute_posterior_mean_1d(r_col, prior_pdf, likelihood).reshape((1,Nx)) # keep as a row vector\n",
    "\n",
    "stimulus_pdf = prior_pdf.copy().flatten()# As in most cases we consider, these two pdfs are the same\n",
    "compute_and_plot_metrics(r_col, s_hat_row, stimulus_pdf, 'posterior mean estimate')    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "b)\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total RMSE (posterior mean estimate): 1.6446507755833726\n"
     ]
    }
   ],
   "source": [
    "a = -5.\n",
    "b = 5.\n",
    "sigma = 2.\n",
    "\n",
    "print('b)')\n",
    "Nx = 2**9+1\n",
    "Ns = 2**9+1\n",
    "\n",
    "# define the grid\n",
    "lb = a - sigma*5.\n",
    "ub = b + sigma*5.\n",
    "x_row = np.linspace(lb, ub, Nx).reshape((1,Nx)) # make x a row vector\n",
    "r_col = np.linspace(lb, ub, Ns).reshape((Ns,1)) # make s a column vector\n",
    "dx = x_row.flatten()[1] - x_row.flatten()[0]\n",
    "ds = r_col.flatten()[1] - r_col.flatten()[0]\n",
    "\n",
    "variance = ((b - a)**2)/12\n",
    "mean = (b + a)/2\n",
    "sigma = np.sqrt(variance)\n",
    "prior_pdf = sps.norm.pdf(r_col, mean, sigma)\n",
    "\n",
    "#New sigma for likelihood\n",
    "sigma=2\n",
    "likelihood = sps.norm.pdf(x_row, r_col, sigma) \n",
    "s_hat_row = compute_posterior_mean_1d(r_col, prior_pdf, likelihood).reshape((1,Nx)) # keep as a row vector\n",
    "\n",
    "stimulus_pdf = prior_pdf.copy().flatten()# As in most cases we consider, these two pdfs are the same\n",
    "compute_and_plot_metrics(r_col, s_hat_row, stimulus_pdf, 'posterior mean estimate')    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c)\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total RMSE (posterior mean estimate): 1.6842547306881315\n"
     ]
    }
   ],
   "source": [
    "a = -8.\n",
    "b = 8.\n",
    "sigma =1.9\n",
    "\n",
    "print('c)')\n",
    "Nx = 2**9+1\n",
    "Ns = 2**9+1\n",
    "\n",
    "\n",
    "# define the grid\n",
    "lb = a - sigma*8.\n",
    "ub = b + sigma*8.\n",
    "x_row = np.linspace(lb, ub, Nx).reshape((1,Nx)) # make x a row vector\n",
    "r_col = np.linspace(lb, ub, Ns).reshape((Ns,1)) # make s a column vector\n",
    "dx = x_row.flatten()[1] - x_row.flatten()[0]\n",
    "ds = r_col.flatten()[1] - r_col.flatten()[0]\n",
    "\n",
    "\n",
    "prior_pdf = sps.uniform.pdf(r_col, a, b-a)\n",
    "\n",
    "likelihood = sps.norm.pdf(x_row, r_col, sigma) \n",
    "s_hat_row = compute_posterior_mean_1d(r_col, prior_pdf, likelihood).reshape((1,Nx)) # keep as a row vector\n",
    "\n",
    "stimulus_pdf = prior_pdf.copy().flatten()# As in most cases we consider, these two pdfs are the same\n",
    "compute_and_plot_metrics(r_col, s_hat_row, stimulus_pdf, 'posterior mean estimate')    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 2.2 (6 pts)\n",
    "\n",
    "\n",
    "> In this question, we compute the response distribution for \\[**JS10**\\] under different assumptions about the Bayesian observer.\n",
    "\n",
    "Consider the time perception experiment from \\[**JS10**\\] which we analyzed in Exercise 3.4.\n",
    "We recall the setup below. Note that there are differences from Exercise 3.4 (marked as **NEW**):\n",
    "- In this experiment, an observer is asked to judge the time interval $s$ between two flashes, measured in milliseconds (ms). In each trial, the duration is drawn from an interval distribution $p(s)$. \n",
    "- The experiment consist of three separate blocks of sessions run over multiple days. Each experimental block is identical except for the distribution of intervals $p(s)$. The distribution of time intervals in the three blocks are: \n",
    "  - $p_\\text{short}(s) = \\text{Uniform}\\left(s; 494, 847\\right)$\n",
    "  - $p_\\text{medium}(s) = \\text{Uniform}\\left(s; 671, 1023\\right)$\n",
    "  - $p_\\text{long}(s) = \\text{Uniform}\\left(s; 847,1200\\right)$\n",
    "- The observer's measurement distribution follows *Weber's law* (known in time perception as the \"scalar property\" of temporal judgment). According to this empirical law, the measurement noise is roughly linearly proportional to the magnitude of the stimulus. In formulas, $$p(x|s) = \\mathcal{N}\\left(x|s,\\sigma^2(s)\\right) \\qquad \\text{with} \\quad \\sigma(s) = w_s \\cdot s$$\n",
    "  where $w_s$ is known as *Weber's fraction*. Typical values of $w_s$ in timing are around 0.05-0.2, here we assume $w_s = 0.1$.\n",
    "- It is assumed that, after some practice, the observer develops a prior $p(s)$ which matches the stimulus distribution used in that block of sessions (and that the likelihood also matches the measurement distribution).\n",
    "- **NEW**: The observer responds with a deterministic estimate $\\hat{s}_\\text{MAP}$ which we assume is the mode of the posterior (also known as *maximum-a-posteriori* or MAP estimate). \n",
    "- **NEW**: The response is corrupted by motor noise which is proportional to the estimate:\n",
    "$$p(r|\\hat{s}) = \\mathcal{N}\\left(r; \\hat{s}, \\sigma_\\text{m}^2(\\hat{s})\\right) \\qquad \\text{with} \\quad \\sigma_\\text{m}(\\hat{s}) = w_\\text{m} \\cdot \\hat{s}$$ \n",
    "  where $w_\\text{m}$ represents the Weber's fraction for the motor noise. Here we assume $w_\\text{m} = 0.05$.\n",
    "  \n",
    "-------------------------------\n",
    "\n",
    "In this exercise, we look at the *distribution of responses* $p(r|s)$ that the experimenter would observe for a given stimulus in the three different experimental blocks (short, medium, or long). We consider the stimulus $s^\\star = 847$ ms which appears in all three experimental blocks.\n",
    "\n",
    "- a) Compute $p(r|s = s^\\star)$ for the \"short\" block. Compute the mean and standard deviation of $p(r|s = s^\\star)$ and report them on Moodle.\n",
    "- b) Compute $p(r|s = s^\\star)$ for the \"medium\" block. Compute the mean and standard deviation of $p(r|s = s^\\star)$ and report them on Moodle.\n",
    "- c) Compute $p(r|s = s^\\star)$ for the \"long\" block. Compute the mean and standard deviation of $p(r|s = s^\\star)$ and report them on Moodle.\n",
    "\n",
    "The accepted tolerance for the solutions is $\\pm 0.2$ ms for (a) and (b), and $\\pm 0.5$ ms for (c).\n",
    "\n",
    "*Hints*: \n",
    "- Be careful that the likelihood, $p(x|s)$ as a function of $s$, is *not* Gaussian, because $\\sigma(s)$ is not constant in $s$. As a consequence, the posterior will *not* be Gaussian. This affects the MAP estimate, $\\hat{s}_\\text{MAP}$, which you will need to compute numerically.\n",
    "- To compute the response distribution, remember the definition:\n",
    "$$\n",
    "p(r|s) = \\int p(r|\\hat{s}(x)) p(x|s) dx,\n",
    "$$\n",
    "  which you can solve via numerical integration.\n",
    "- It is recommended that you first compute $\\hat{s}_\\text{MAP}(x)$ for a grid of $x$, and then compute the response distribution numerically via the integral above.\n",
    "- The MAP estimate $\\hat{s}_\\text{MAP}$ is the value of $s$ that maximizes the posterior $p(s|x)$. Note that this value does not depend on the normalization constant, so you can compute $p(s|x) \\propto p(s) p(x|s)$ for a (fine) grid of values `s_grid` and take the argument $s$ that maximizes this quantity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sanity_check(x_row,r_col,integrand, s_hat_row, response_integral_pdf):\n",
    "    if x_row[0][0] == 345.79999999999995:\n",
    "        idx = (np.abs(x_row[0] - 759)).argmin()\n",
    "        print(\"Sanity check: x=759, sMap=751.50\")\n",
    "        print(f\"x={x_row[0][idx]}, sMAP={s_hat_row[0][idx]}\")\n",
    "        \n",
    "        print('x_row.shape: {}'.format(x_row.shape))\n",
    "        print('r_col.shape: {}'.format(r_col.shape))\n",
    "        print('s_hat_row.shape: {}'.format(s_hat_row.shape))\n",
    "        print('integrand.shape: {}'.format(integrand.shape))\n",
    "        print('response_integral_pdf.shape: {}'.format(response_integral_pdf.shape))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_distribution(x, y, x_label, y_label, title):\n",
    "    plt.bar(x, y , color='red', width=0.5)\n",
    "    plt.xlabel(x_label)\n",
    "    plt.ylabel(y_label)\n",
    "    plt.title(title)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------short------------\n",
      "mean= 809.0869476150397\n",
      "std= 65.67980760174596\n",
      "1 ~= 0.9999997133477272\n",
      "------------medium------------\n",
      "mean= 839.0043753778564\n",
      "std= 91.38405383325505\n",
      "1 ~= 0.9999999992244515\n",
      "------------long------------\n",
      "mean= 876.7796740577605\n",
      "std= 63.521941134265894\n",
      "1 ~= 0.9999997133484282\n"
     ]
    },
    {
     "data": {
      "image/png": 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1GFZroOcRsXMvpNl0qzIzGzSS1kREd70y32FuZmalOXmYmVlpTh42cvgEvNkeU/iEuaSDgJcBhwCPkF3l1JMumTUzsw7SMnlIOgZYCBwI/Ap4AHgy8HfAoZKuAC6IiD9UGaiZmbWPIj2P2cA7IuKe2gJJo4HjgVcD3xrk2MzMrE21TB4RcUaTsu3AdwY1IjMza3stT5hLerekia3qmZlZ5yhytdW5wPWS/lvS/5HkB0WZmXW4IsljA9mDCc8FXgTcJumHkk6VtF+l0ZmZWVsqkjwiInZExDUR8TayS3U/D8wkSyxmZtZhilxttdOdVxHxKNkTbldIekolUZmZWVsr0vN4Y6OCiHhkEGMxM7NhomXyiIjf1M6TdGI14ZiZ2XBQ5A7zk2pnAYvTDYJExJVVBGZmZu2ryDmPy4Efkj2WZOD8x77ACWRfyOTkYWbWYYokj5cA55F9NezSiAhJR0fEadWGZmZm7arIOY/VZM+uGgP8WNJ0sh6HmZl1qEKPZE+PXf+spG8CF1YbkpmZtbvC3+cBEBGbgH+oKBazwSH5+8zNKlbkwYjPlrRM0sckPVXSFyTdKumbkia1aDtT0npJvZIW1imXpEWpfK2kaSXavl9SSBpXbFPNzGywFLlJ8MtkJ8sfBlYBtwOzyK7AWtaokaRRwOJUdypwiqSpNdVmAVPSaz6wpEjb9JTfVwNP+I4RMzOrXpHksV9ELImI84D9I+KCiNgYEV8EDmjSbjrQGxEbImIbcBkwp6bOHGB5ZFYBYyUdXKDtZ4AP4BP3w5O/a9xs2CuSPHZIOlzSi4F9JHUDSDoMGNWk3XhgY266L80rUqdh23R3+70RcXOzoCXNl9Qjqae/v79ZVTMzK6nICfMPAN8DdpB9b/mZkl4A7A+8o0m7eh8va3sKjerUnS9pH+As4DWtgo6Ii4GLAbq7u91DMTMbREW+hvZHwBG5WT9LJ6l/FxGPNWnaB+S/gXACsKlgnTEN5h8KTAZuVjb0MQG4UdL0iLiv1baYmdngKDJs9VeSnpPejmuROCA7yT5F0mRJY4C5ZI9yz1sBzEtXXc0AtkTE5kZtI+KWiDgoIiZFxCSy5DPNicPMbM8qdZ8H8HVgWu5nQxGxXdIC4GqycyPLImKdpNNT+VJgJTAb6AW2Aqc1a1syVjMzq0jZ5DGg0OUyEbGSLEHk5y3NvQ/g/xZtW6fOpCJxWAfyjYJmlSo1bGVmZgZOHmZmtgt2NXl4PMDMrIOVTR6q+WlmZh2obPJ4Rc1PMzPrQIWTh6R9yS6nBThE0omS9qomLDMza2dleh4/BZ4saTzwI7J7Mr5cRVA2gvmhiGYjQpnkoYjYCpwEXBQRryd7XLqZmXWYUslD0kuANwFXpXm7epOhWfXcyzGrTJnk8R7gTODb6TEjzwb+q5qwzMysnRXuOUTEdcB1uekNwLurCMrMzNpby+QhaV7BZd0UEWt3Mx4zMxsGivQ8Jhdc1l27EYeZmQ0jRb4M6hxJTwJmRMQv9kBMZmbW5gqdMI+IHcAFFcdiZmbDRJmrra6R9AbJ1z+amXW6Mvdp/AuwL7Bd0p/JHo4YEbF/JZGZmVnbKnOp7n5VBmJmZsNHy2ErSZNalEvShMEKyEYwj3iajRhFznmcL+lbkuZJep6kgyQ9U9Kxks4Ffg48t15DSTMlrZfUK2lhnXJJWpTK10qa1qqtpHNT3ZskXSPpkF3YbjMz2w0tk0dEnAx8BDgCWEz2dN3vAm8H1gPHRsS1te0kjUr1Z5E9QPEUSbUPUpwFTEmv+cCSAm3Pj4jnR8SRwPeBj5bZYDMz231FL9W9DfgY8D3g18CdwGrgioj4c4Nm04HeiNgQEduAy4A5NXXmAMsjswoYK+ngZm0j4g+59vvir8Q1M9vjylyqewnZ8NQi4KL0fnmT+uOBjbnpvjSvSJ2mbSV9XNJGsif81u15SJovqUdST39/f5MwbcTzuRazQVcmeRwREW+PiP9Kr/lkQ1mN1PuPre0lNKrTtG1EnBURE4GvAQvqrTwiLo6I7ojo7urqahKmmZmVVSZ5/ErSjIEJSUeRnSxvpA+YmJueAGwqWKdIW4CvA29oGbmZmQ2qMsnjKOAXku6SdBfwS+CVkm6RVO9puquBKZImSxoDzAVW1NRZAcxLV13NALZExOZmbSVNybU/Ebi9xDaYmdkgKHOH+cwyC46I7ZIWAFcDo4Bl6UukTk/lS4GVwGygF9hK9r3oDdumRZ8n6QhgB3A3cHqZuMzMbPcpYuRfrNTd3R09PT1DHYblT1xHPD6df99qukzd/HQH/J2bDTZJayKiu15ZmWErs+HLV1yZDSonDzMzK83Jw8zMSnPyMDOz0pw8zMysNCcPMzMrzcnDzMxKc/IwM7PSnDzMzKw0Jw8zMyvNycPMzEpz8rDO4seUmA0KJw+rng/YZiOOk4eZmZXm5GFmZqU5eZiZWWlOHtZ5fA7GbLc5eZiZWWlOHmZmVlqlyUPSTEnrJfVKWlinXJIWpfK1kqa1aivpfEm3p/rfljS2ym0wM7Mnqix5SBoFLAZmAVOBUyRNrak2C5iSXvOBJQXaXgv8r4h4PvAb4MyqtsHMzOqrsucxHeiNiA0RsQ24DJhTU2cOsDwyq4Cxkg5u1jYiromI7an9KmBChdtgI5VPmpvtliqTx3hgY266L80rUqdIW4C3Aj+ot3JJ8yX1SOrp7+8vGboNGh+kzUakKpNHvaNGFKzTsq2ks4DtwNfqrTwiLo6I7ojo7urqKhCumZkVNbrCZfcBE3PTE4BNBeuMadZW0qnA8cBxEVGbkMzMrGJV9jxWA1MkTZY0BpgLrKipswKYl666mgFsiYjNzdpKmgl8EDgxIrZWGL+ZmTVQWc8jIrZLWgBcDYwClkXEOkmnp/KlwEpgNtALbAVOa9Y2LfpzwN7AtcrG01dFxOlVbYeZmT2ROmHUp7u7O3p6eoY6jM40cMI8YueT5/npZmW7U7dIWzNrSNKaiOiuV+Y7zM3MrDQnD+tsvpTYbJc4eZiZWWlOHlYdf6o3G7GcPMzMrDQnDzP3kMxKc/IwM7PSnDzMzKw0Jw8z8NCVWUlOHmZmVpqTh5mZlebkYWZmpTl5mJlZaU4eZmZWmpOHWZ6vujIrxMnDzMxKc/IwM7PSnDzMannoyqwlJw8bfD74mo14lSYPSTMlrZfUK2lhnXJJWpTK10qa1qqtpJMlrZO0Q1Ld79Y1M7NqVZY8JI0CFgOzgKnAKZKm1lSbBUxJr/nAkgJtbwVOAn5aVexm7j2ZNVdlz2M60BsRGyJiG3AZMKemzhxgeWRWAWMlHdysbUT8OiLWVxi3mZm1UGXyGA9szE33pXlF6hRpa1Yt9z7MGqoyedT7z4uCdYq0bb5yab6kHkk9/f39ZZqamVkLVSaPPmBibnoCsKlgnSJtm4qIiyOiOyK6u7q6yjS13THSPq2PtO0xGyRVJo/VwBRJkyWNAeYCK2rqrADmpauuZgBbImJzwbZmZjZERle14IjYLmkBcDUwClgWEesknZ7KlwIrgdlAL7AVOK1ZWwBJrwcuArqAqyTdFBGvrWo7zJAgSo2amo14ig74p+ju7o6enp6hDqMzDAzzROw85NNsuqq6u9s2rwP+T8xqSVoTEXXvp/Md5mZmVpqThw0en1w26xhOHmZFOTma/ZWTh5mZlebkYVaGex9mgJOHWXlOIGZOHmZmVp6Th5mZlebkYbarPHxlHczJw3afD6JmHcfJw2x3OHFah3LyMNtdTiDWgZw8bPf4wGnWkZw8zAaLE6l1ECcPs8HkBGIdwsnDrApOIjbCOXnYrvHBsTXvIxvBnDzMquQEYiOUk4eV5wNied5nNsI4eVg5PgjuOsn7z0aMSpOHpJmS1kvqlbSwTrkkLUrlayVNa9VW0oGSrpV0R/p5QJXbYFaJgUTiZGLDVGXJQ9IoYDEwC5gKnCJpak21WcCU9JoPLCnQdiHwo4iYAvwoTVuVfJCrnvexDTNV9jymA70RsSEitgGXAXNq6swBlkdmFTBW0sEt2s4BLknvLwH+rsJt6Ez5A5kPaHteba+k0bTZEBpd4bLHAxtz033AUQXqjG/R9m8iYjNARGyWdFC9lUuaT9abAXhY0vqCcY8DHixYtx1UF2/tAarZdPG6Wby71rZc3d1t2yjedomxWbzDy3CLuZPifVajgiqTR72/7ChYp0jbpiLiYuDiMm0AJPVERHfZdkPF8VbL8VZvuMXseDNVDlv1ARNz0xOATQXrNGt7fxraIv18YBBjNjOzAqpMHquBKZImSxoDzAVW1NRZAcxLV13NALakIalmbVcAp6b3pwLfrXAbzMysjsqGrSJiu6QFwNXAKGBZRKyTdHoqXwqsBGYDvcBW4LRmbdOizwMul/Q24B7g5EEOvfRQ1xBzvNVyvNUbbjE7XkARpU4lmJmZ+Q5zMzMrz8nDzMxK67jkIWmUpF9J+n6abvi4E0lnpsejrJf02iGKd6ykKyTdLunXkl7SzjFLeq+kdZJulXSppCe3W7ySlkl6QNKtuXmlY5T0Ikm3pLJFUjV37jWI9/z0N7FW0rcljW3neHNl75cUksa1e7yS3pViWifpk+0cr6QjJa2SdJOkHknTK483IjrqBfwL8HXg+2n6k8DC9H4h8In0fipwM7A3MBn4LTBqCOK9BHh7ej8GGNuuMZPd3Hkn8JQ0fTnwlnaLF/hbYBpwa25e6RiBG4CXkN2X9ANg1h6M9zXA6PT+E+0eb5o/kewimLuBce0cL3AM8J/A3mn6oDaP95qB9ZFdhPSTquPtqJ6HpAnA64D/yM1u9LiTOcBlEfGXiLiT7Iqw6exBkvYn+0P5IkBEbIuI37dzzGRX8D1F0mhgH7L7c9oq3oj4KfA/NbNLxajsHqP9I+KXkf0nLqeiR+XUizciromI7WlyFdm9UG0bb/IZ4APsfMNvu8b7TuC8iPhLqjNwP1m7xhvA/un903j8vrjK4u2o5AFcSPbHuyM3b6fHnQADjztp9OiUPenZQD/wJWVDbf8haV/aNOaIuBf4FNkl1JvJ7tu5pl3jrVE2xvHpfe38ofBWsk+O0KbxSjoRuDcibq4past4gcOBV0i6XtJ1kl6c5rdrvO8Bzpe0kex/8Mw0v7J4OyZ5SDoeeCAi1hRtUmfenr6ueTRZ93RJRLwQ+BPNnyI8pDGn8wRzyLrHhwD7SnpzsyZ15rXbteOVPUJnMEg6C9gOfG1gVp1qQxqvpH2As4CP1iuuM68d9u9o4ABgBnAG2b1lahLXUMf7TuC9ETEReC9ptIIK4+2Y5AG8DDhR0l1kT+k9VtJXafy4kyKPV6laH9AXEden6SvIkkm7xvwq4M6I6I+IR4ErgZe2cbx5ZWPs4/Ghovz8PUbSqcDxwJvS0AO0Z7yHkn2guDn9/00AbpT0DNozXtL6r4zMDWSjFeNo33hPJft/A/gmjw//VhZvxySPiDgzIiZExCSyx538OCLeTOPHnawA5kraW9Jksu8cuWEPx3wfsFHSEWnWccBttG/M9wAzJO2TPqUdB/y6jePNKxVjGtr6o6QZaVvnsQcflSNpJvBB4MSI2Jorart4I+KWiDgoIial/78+YFr6+267eJPvAMcCSDqc7GKVB9s43k3AK9P7Y4E70vvq4q3iaoB2fwFH8/jVVk8n+1KpO9LPA3P1ziK7OmE9FV05USDWI4EeYC3ZH/QB7RwzcA5wO3Ar8BWyqzzaKl7gUrJzMo+SHcjetisxAt1pO38LfI70xIY9FG8v2Vj2Tem1tJ3jrSm/i3S1VbvGS5YsvprWfyNwbJvH+3JgDdmVVdcDL6o6Xj+exMzMSuuYYSszMxs8Th5mZlaak4eZmZXm5GFmZqU5eZiZWWlOHmZmVpqTh1mbUMb/kzYs+A/VbAhJmqTse1o+T3Yz2sRWbczagW8SNBtCkiYBG4CXRsSqoY3GrDj3PMyG3t1OHDbcOHmYDb0/DXUAZmU5eZiZWWlOHmZmVppPmJuZWWnueZiZWWlOHmZmVpqTh5mZlebkYWZmpTl5mJlZaU4eZmZWmpOHmZmV9v8BgnYz8V2bI2cAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = np.array((494, 671, 847))\n",
    "b = np.array((847,1023,1200))\n",
    "label = ('short','medium','long')\n",
    "w_s = 0.1\n",
    "w_m = 0.05\n",
    "\n",
    "#res=11, clearance=5 for correct results\n",
    "res = 11\n",
    "Nx = 2**res+1\n",
    "Ns = 2**res+1\n",
    "\n",
    "# calculate sMAP\n",
    "def compute_maximum_a_posteriori_1d(s_grid, prior_pdf, likelihood):\n",
    "    posterior = likelihood * prior_pdf\n",
    "    map_estimate = s_grid[np.argmax(posterior, axis=0)]\n",
    "    return np.transpose(map_estimate) #must be a row vector\n",
    "\n",
    "# calculate response distribution\n",
    "def calculate_response_dist(x_row, r_col, likelihood, prior_pdf, _dx):\n",
    "    s_hat_row = compute_maximum_a_posteriori_1d(r_col, prior_pdf, likelihood)\n",
    "    measurement_dist = sps.norm.pdf(x_row, 847, w_s*847)\n",
    "    integrand = sps.norm.pdf(r_col, s_hat_row, w_m * s_hat_row) * measurement_dist \n",
    "    response_dist = sp.integrate.romb(integrand, dx=_dx, axis=1)\n",
    "    sanity_check(x_row,r_col,integrand, s_hat_row, response_dist)\n",
    "    return response_dist\n",
    "\n",
    "for iter in range(a.size):\n",
    "    # define the grid    \n",
    "    clearance = 5\n",
    "    lb = a[iter] - (w_s*a[iter])*clearance\n",
    "    ub = b[iter] + (w_s*b[iter])*clearance\n",
    "    x_row = np.linspace(lb, ub, Nx).reshape((1,Nx)) # make x a row vector\n",
    "    r_col = np.linspace(lb, ub, Ns).reshape((Ns,1)) # make s a column vector\n",
    "    _dx = x_row.flatten()[1] - x_row.flatten()[0]\n",
    "\n",
    "    #calculate response distribution p(r|s=847)\n",
    "    prior_pdf = sps.uniform.pdf(r_col, loc=a[iter], scale=b[iter]-a[iter])    \n",
    "    likelihood = sps.norm.pdf(x_row, r_col, w_s*r_col)\n",
    "    response_dist = calculate_response_dist(x_row, r_col, likelihood, prior_pdf, _dx)\n",
    "\n",
    "    #mean and SD calculation\n",
    "    flat = r_col.flatten()\n",
    "    mean_r = sp.integrate.romb(flat * response_dist, dx=_dx)\n",
    "    var = sp.integrate.romb((flat - mean_r)**2 * response_dist, dx=_dx)\n",
    "    std = np.sqrt(var)\n",
    "    \n",
    "    #results\n",
    "    print(\"------------\"+ label[iter] + \"------------\")\n",
    "    print(f\"mean= {mean_r}\")\n",
    "    print(f\"std= {std}\")\n",
    "    print(\"1 ~=\", sp.integrate.romb(response_dist, dx=_dx)) #should be apprpx 1\n",
    "\n",
    "plot_distribution(r_col.flatten(), response_dist , \"r\", \"p(r|s=847)\", \n",
    "                  \"Long block response distribution\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 2.3 (6 pts)\n",
    "\n",
    "> The key quantity for model fitting is the log-likelihood for a dataset and some model parameters. In this exercise, we compute the log-likelihood for a Bayesian observer model which also includes the possibility of *lapses*, a common mechanism used in cognitive science to explain away \"random\" responses and subjects' mistakes.\n",
    "\n",
    "In this question, we consider the datasets from Experiment 3 of \\[**AWV12**\\], as seen in Week 4.. The experimental setup which involves time perception and interval reproduction is very similar to \\[**JS10**\\], so we can consider the same type of models.\n",
    "\n",
    "We analyze the data with the `gaussianobserverwithlapse` model, defines as follows:\n",
    "\n",
    "- We assume the observer builds a (mismatched) Gaussian prior $p(s) = \\mathcal{N}\\left(s| \\mu_\\text{prior}, \\sigma_\\text{prior}^2 \\right)$ over the stimuli (time intervals). \n",
    "- We assume that the measurement distribution and likelihood are also Gaussian, $p(x|s) = \\mathcal{N}\\left(x| s, \\sigma^2 \\right)$.\n",
    "- The observer uses the *posterior mean* estimator for the value of the stimulus, $\\hat{s}_\\text{PM}$.\n",
    "- Gaussian motor response noise is added to the estimate, $p(r|\\hat{s}) = \\mathcal{N}\\left(r| \\hat{s}, \\sigma_\\text{motor}^2 \\right)$.\n",
    "- In each trial, the observer lapses with probability $\\lambda$ (the *lapse rate*), in which case the response is drawn from $p_\\text{lapse}(r) = \\text{Uniform}\\left(r; 0, 1500 \\right)$ ms. Otherwise, the observer responds normally (according to $p(r|\\hat{s})$ described above) with probability $1 - \\lambda$. \n",
    "- The parameters of this model are $\\mathbf{\\theta} = \\left(\\mu_\\text{prior}, \\sigma_\\text{prior}, \\sigma, \\sigma_\\text{motor}, \\lambda \\right)$.\n",
    "\n",
    "For this question, we consider parameters $\\mathbf{\\theta}_\\star = \\left(\\mu_\\text{prior} = 780, \\sigma_\\text{prior} = 140, \\sigma = 90, \\sigma_\\text{motor} = 60, \\lambda = 0.02 \\right)$. \n",
    "\n",
    "- a) Compute the log-likelihood of model parameter $\\theta_\\star$ for the dataset of subject 2.\n",
    "- b) Compute the log-likelihood of model parameter $\\theta_\\star$ for the dataset of subject 5.\n",
    "\n",
    "Report your results on Moodle with high precision.\n",
    "\n",
    "*Hint*:\n",
    "- If you use code from the lectures, be careful about the model definition, as there may be subtle differences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Subject id</th>\n",
       "      <th>Session id</th>\n",
       "      <th>Run id</th>\n",
       "      <th>Stimulus (ms)</th>\n",
       "      <th>Response (ms)</th>\n",
       "      <th>Stimulus id</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>973.327049</td>\n",
       "      <td>862.947945</td>\n",
       "      <td>6.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>677.519900</td>\n",
       "      <td>574.920276</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>826.253049</td>\n",
       "      <td>870.995615</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>677.854859</td>\n",
       "      <td>695.055098</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>598.501198</td>\n",
       "      <td>632.981845</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Subject id  Session id  Run id  Stimulus (ms)  Response (ms)  Stimulus id\n",
       "0           1           1       1     973.327049     862.947945          6.0\n",
       "1           1           1       1     677.519900     574.920276          2.0\n",
       "2           1           1       1     826.253049     870.995615          4.0\n",
       "3           1           1       1     677.854859     695.055098          2.0\n",
       "4           1           1       1     598.501198     632.981845          1.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Load data of Experiment 3 of [AWV12] from .csv file to a Pandas dataframe\n",
    "df = pd.read_csv('https://www2.helsinki.fi/sites/default/files/atoms/files/awv12_exp3.csv')\n",
    "\n",
    "# Remove unused columns (they deal with performance feedback, which we ignore in this lecture)\n",
    "df.drop(df.columns[[6, 7, 8]], axis=1, inplace=True)\n",
    "\n",
    "# Remove rows with NaNs\n",
    "df.dropna(axis=0, inplace=True)\n",
    "\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "s.shape: (2520,)\n",
      "r.shape: (2520,)\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Example code to extract stimuli and responses for a single subject (here S1)\n",
    "subject = 1\n",
    "s = np.array(df['Stimulus (ms)'][df['Subject id'] == subject])\n",
    "r = np.array(df['Response (ms)'][df['Subject id'] == subject])\n",
    "print('s.shape:', s.shape)\n",
    "print('r.shape:', r.shape)\n",
    "\n",
    "plt.scatter(s, r)\n",
    "plt.xlabel('Stimulus $s$ (ms)')\n",
    "plt.ylabel('Response $r$ (ms)')\n",
    "plt.title('S' + str(subject))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "b)\n",
      "SUBJECT 2: The log-likelihood of theta_1 = (90, 60, 0.02) (dataset S2) is: -8577.318546123512\n",
      "SUBJECT 5: The log-likelihood of theta_1 = (90, 60, 0.02) (dataset S5) is: -9105.655523964238\n"
     ]
    }
   ],
   "source": [
    "def gaussian_response(s,theta):\n",
    "    \"\"\"Compute mean and standard deviation of p(r|s; theta).\"\"\"\n",
    "    # Unpack parameter vector theta\n",
    "    mu_prior = theta[0]\n",
    "    sigma_prior = theta[1]\n",
    "    sigma = theta[2]\n",
    "    sigma_motor = theta[3]\n",
    "\n",
    "    # Compute mean and std of the response\n",
    "    w = sigma_prior**2/(sigma_prior**2 + sigma**2)    \n",
    "    mu_resp = w*s + (1-w)*mu_prior\n",
    "    sigma_resp = np.sqrt(w**2*sigma**2 + sigma_motor**2)\n",
    "    return mu_resp, sigma_resp\n",
    "\n",
    "def idealgaussianobserverwithlapse_loglike(theta,s_vec,r_vec):\n",
    "    \"\"\"Log-likelihood of ideal Gaussian observer with added lapse.\"\"\"\n",
    "    mu_prior = 780\n",
    "    sigma_prior = 140\n",
    "    \n",
    "    sigma = theta[0]\n",
    "    sigma_motor = theta[1]\n",
    "    lapse_rate = theta[2]\n",
    "    lapse_pdf = 1/1500.\n",
    "    mu_resp, sigma_resp = gaussian_response(s_vec,np.array((mu_prior,sigma_prior,sigma,sigma_motor)))\n",
    "\n",
    "    # First, compute log-likelihood without probability of lapse\n",
    "    loglike_vec = sps.norm.logpdf(r_vec,mu_resp,sigma_resp) # Vector of log-likelihood per trials\n",
    "\n",
    "    # Now, add the probability of lapse\n",
    "    if lapse_rate > 0.:\n",
    "        likelihood_vec = np.exp(loglike_vec) # Exponentiate back to the likelihood\n",
    "        likelihood_with_lapse_vec = (1-lapse_rate)*likelihood_vec + lapse_rate*lapse_pdf\n",
    "        loglike_vec = np.log(likelihood_with_lapse_vec)\n",
    "\n",
    "        # This code snippet below uses the logsumexp trick, which is numerically more stable\n",
    "        # loglapse = np.log(lapse_rate*lapse_pdf)\n",
    "        # M = np.maximum(loglike, loglapse)\n",
    "        # loglike = np.log((1-lapse_rate)*np.exp(loglike-M) + np.exp(loglapse-M)) + M        \n",
    "    return np.sum(loglike_vec)\n",
    "\n",
    "theta1 = (90,60,0.02)\n",
    "\n",
    "print('b)')\n",
    "\n",
    "subject = 2\n",
    "s = np.array(df['Stimulus (ms)'][df['Subject id'] == subject])\n",
    "r = np.array(df['Response (ms)'][df['Subject id'] == subject])\n",
    "loglike1 = idealgaussianobserverwithlapse_loglike(theta1,s,r)\n",
    "\n",
    "print('SUBJECT 2: The log-likelihood of theta_1 = {} (dataset S{}) is: {}'.format(\n",
    "    theta1, subject, loglike1))\n",
    "\n",
    "\n",
    "###################################################################\n",
    "\n",
    "subject = 5\n",
    "s = np.array(df['Stimulus (ms)'][df['Subject id'] == subject])\n",
    "r = np.array(df['Response (ms)'][df['Subject id'] == subject])\n",
    "loglike1 = idealgaussianobserverwithlapse_loglike(theta1,s,r)\n",
    "\n",
    "print('SUBJECT 5: The log-likelihood of theta_1 = {} (dataset S{}) is: {}'.format(\n",
    "    theta1, subject, loglike1))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 2.4 (6 pts)\n",
    "\n",
    "> When fitting models to data, the experimenter may be interested in how model parameters are represented across the population (here represented by the group of subjects). A simple way to look at this is to look at the distribution of maximum-likelihood estimates for the parameters across subjects, in first instance by looking their mean and variability.\n",
    "\n",
    "We consider here the `idealgaussianobserverwithlapse` model. This model is the same as the `gaussianobserverwithlapse` of Question 2.3, but with $\\mu_\\text{prior} = 787.5$ ms and $\\sigma_\\text{prior} = 128.1$ ms fixed. Thus, the model has three free parameters, $\\theta = \\left(\\sigma, \\sigma_\\text{motor}, \\lambda \\right)$. Fit the model using maximum-likelihood estimation.\n",
    "\n",
    "- a) First, fit the `idealgaussianobserverwithlapse` model to the six subjects' datasets (separately for each subject's data). For each maximum-likelihood estimate (MLE) of parameters $\\sigma, \\sigma_\\text{motor}, \\lambda$, report in Moodle the mean and standard deviation across the six subjects. For the standard deviation, use the correction for degrees of freedom (that is, `np.std(..., ddof=1)`).\n",
    "- b) Now fit the pooled data of all subjects as a single dataset (as if all data were collected from a single uber-subject). Report the maximum-likelihood estimate of $\\sigma, \\sigma_\\text{motor}, \\lambda$ for the pooled data in Moodle.\n",
    "\n",
    "*Hints*: \n",
    "- If you use code for the `idealgaussianobserverwithlapse` model from the lectures, be careful about the model definition.\n",
    "- As a sanity check that you have coded the log-likelihood function correctly, check that the log-likelihood of the dataset of subject 1 for $\\theta_\\star = \\left(\\sigma = 90, \\sigma_\\text{motor} = 80, \\lambda = 0.02\\right)$ is $\\log \\mathcal{L}(\\theta_\\star; \\mathcal{D}_1) \\approx -14709.795\\ldots$\n",
    "\n",
    "*Note*: Fitting individual subjects' data is the best approach to describe invidual behavior in cognitive science, but sometimes you will see studies only looking at pooled/group data. Be careful that pooling might hide what really happens, only giving a snapshot of the average behavior of the group, which might not correspond to what individuals do."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def idealgaussianobserverwithlapse_loglike(theta,s_vec,r_vec):\n",
    "    \"\"\"Log-likelihood of ideal Gaussian observer with added lapse.\"\"\"\n",
    "    mu_prior = 787.5\n",
    "    sigma_prior = 128.1\n",
    "    sigma = theta[0]\n",
    "    sigma_motor = theta[1]    \n",
    "    lapse_rate = theta[2]\n",
    "    lapse_pdf = 1/1500.\n",
    "    mu_resp, sigma_resp = gaussian_response(s_vec,np.array((mu_prior,sigma_prior,sigma,sigma_motor)))\n",
    "\n",
    "    # First, compute log-likelihood without probability of lapse\n",
    "    loglike_vec = sps.norm.logpdf(r_vec,mu_resp,sigma_resp) # Vector of log-likelihood per trials\n",
    "    \n",
    "    # Now, add the probability of lapse\n",
    "    if lapse_rate > 0.:\n",
    "        likelihood_vec = np.exp(loglike_vec) # Exponentiate back to the likelihood\n",
    "        likelihood_with_lapse_vec = (1-lapse_rate)*likelihood_vec + lapse_rate*lapse_pdf\n",
    "        loglike_vec = np.log(likelihood_with_lapse_vec)   \n",
    "    return np.sum(loglike_vec)\n",
    "\n",
    "def multioptimize(target_fun,lower_bounds,upper_bounds,plausible_lower_bounds,plausible_upper_bounds,num_runs=3):\n",
    "    \"\"\"Simple function for multi-start optimization.\"\"\"\n",
    "    # Run num_runs optimization runs from different starting points    \n",
    "    num_params = lower_bounds.shape[0]\n",
    "    theta_res = np.zeros((num_runs,num_params))\n",
    "    nll_res = np.zeros(num_runs)    \n",
    "    \n",
    "    for index in range(num_runs):\n",
    "        if index == 0:\n",
    "            theta0 = 0.5*(plausible_lower_bounds + plausible_upper_bounds)\n",
    "        else:\n",
    "            theta0 = np.random.uniform(low=plausible_lower_bounds,high=plausible_upper_bounds)    \n",
    "        bounds = sp.optimize.Bounds(lower_bounds,upper_bounds,True) # Set hard bounds\n",
    "        res = sp.optimize.minimize(target_fun, theta0, method='L-BFGS-B', bounds=bounds)\n",
    "        nll_res[index] = res.fun\n",
    "        theta_res[index] = res.x\n",
    "        \n",
    "    # Pick the best solution\n",
    "    idx_best = np.argmin(nll_res)\n",
    "    nll_best = nll_res[idx_best]\n",
    "    theta_best = theta_res[idx_best]        \n",
    "    return nll_best,theta_best"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Thetas (= sigma, sigma_motor, lambda): [array([109.596,  42.538,   0.   ]), array([68.835, 30.883,  0.005]), array([115.919,  71.987,   0.013]), array([139.914,  91.651,   0.   ]), array([63.395, 76.46 ,  0.008]), array([93.898, 73.26 ,  0.007])]\n"
     ]
    }
   ],
   "source": [
    "thetas = []\n",
    "num_runs = 10\n",
    "num_subjects = 6\n",
    "\n",
    "for i in range(num_subjects):\n",
    "    subject = i + 1\n",
    "    s = np.array(df['Stimulus (ms)'][df['Subject id'] == subject])\n",
    "    r = np.array(df['Response (ms)'][df['Subject id'] == subject])\n",
    "    target_fun = lambda theta_: -idealgaussianobserverwithlapse_loglike(np.array(theta_),s,r)\n",
    "\n",
    "    # Define hard parameter bounds\n",
    "    lower_bounds = np.array([1.,1.,0.])\n",
    "    upper_bounds = np.array([2000.,2000.,1.])\n",
    "\n",
    "    # Define plausible range\n",
    "    plausible_lower_bounds = np.array([np.mean(s)*0.05,np.mean(s)*0.05,0.01])\n",
    "    plausible_upper_bounds = np.array([np.mean(s)*0.20,np.mean(s)*0.20,0.05])\n",
    "    \n",
    "    #call multioptimise  for subject\n",
    "    nll2_best,theta2_best = multioptimize(target_fun,lower_bounds,upper_bounds,plausible_lower_bounds,plausible_upper_bounds,num_runs)\n",
    "    thetas.append(theta2_best)\n",
    "\n",
    "print(f\"Thetas (= sigma, sigma_motor, lambda): {thetas}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sigma:\n",
      " mle = 98.59291952413031,\n",
      " std = 29.241153589295255\n",
      "sigma motors:\n",
      " mle = 64.46303211477215,\n",
      " std = 22.91497518125363\n",
      "lambdas:\n",
      " mle = 0.005600439788615318,\n",
      " std = 0.0050677241630049065\n"
     ]
    }
   ],
   "source": [
    "#thetas = [sigma, ]\n",
    "sigmas = [x[0] for x in thetas]\n",
    "sigma_motors = [x[1] for x in thetas]\n",
    "lambdas = [x[2] for x in thetas]\n",
    "\n",
    "#calculate mles and stds\n",
    "sigma_mle = np.mean(sigmas)\n",
    "sigma_motors_mle = np.mean(sigma_motors)\n",
    "lambdas_mle = np.mean(lambdas)\n",
    "sigma_std = np.std(sigmas, ddof=1)\n",
    "sigma_motors_std = np.std(sigma_motors, ddof=1)\n",
    "lambdas_std = np.std(lambdas, ddof=1)\n",
    "\n",
    "print(f\"sigma:\\n mle = {sigma_mle},\\n std = {sigma_std}\")\n",
    "print(f\"sigma motors:\\n mle = {sigma_motors_mle},\\n std = {sigma_motors_std}\")\n",
    "print(f\"lambdas:\\n mle = {lambdas_mle},\\n std = {lambdas_std}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gaussianobserverwithlapse_loglike(theta,s_vec,r_vec):\n",
    "    \"\"\"Log-likelihood of ideal Gaussian observer with added lapse.\"\"\"\n",
    "    mu_prior = 787.5\n",
    "    sigma_prior = 128.1\n",
    "    sigma = theta[0]\n",
    "    sigma_motor = theta[1]\n",
    "    lapse_rate = theta[2]\n",
    "    lapse_pdf = 1/1500\n",
    "    mu_resp, sigma_resp = gaussian_response(s_vec,np.array((mu_prior,sigma_prior,sigma,sigma_motor)))\n",
    "\n",
    "    # First, compute log-likelihood without probability of lapse\n",
    "    loglike_vec = sps.norm.logpdf(r_vec,mu_resp,sigma_resp) # Vector of log-likelihood per trials\n",
    "\n",
    "    # Now, add the probability of lapse\n",
    "    if lapse_rate > 0.:\n",
    "        likelihood_vec = np.exp(loglike_vec) # Exponentiate back to the likelihood\n",
    "        likelihood_with_lapse_vec = (1-lapse_rate)*likelihood_vec + lapse_rate*lapse_pdf\n",
    "        loglike_vec = np.log(likelihood_with_lapse_vec)\n",
    "        \n",
    "        # This code snippet below uses the logsumexp trick, which is numerically more stable\n",
    "        # loglapse = np.log(lapse_rate*lapse_pdf)\n",
    "        # M = np.maximum(loglike, loglapse)\n",
    "        # loglike = np.log((1-lapse_rate)*np.exp(loglike-M) + np.exp(loglapse-M)) + M        \n",
    "    return np.sum(loglike_vec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[98.093 64.117  0.008]\n"
     ]
    }
   ],
   "source": [
    "target_fun = lambda theta_: -gaussianobserverwithlapse_loglike(np.array(theta_),s,r)\n",
    "nll2_best,theta2_best = multioptimize(target_fun,lower_bounds,upper_bounds,plausible_lower_bounds,plausible_upper_bounds,num_runs=50)\n",
    "print(theta2_best[:3])"
   ]
  }
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