bayesian-statistics/1_model-comparison-with-kombine.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "executive-version",
   "metadata": {},
   "source": [
    "# Model Comparison using Kombine\n",
    "We perform a more advanced task in this notebook and use an established bayesian modeling tool: [Kombine](https://github.com/bfarr/kombine)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "grateful-difference",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy.optimize import curve_fit\n",
    "from scipy import stats\n",
    "from corner import corner\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import kombine"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sunrise-horizon",
   "metadata": {},
   "source": [
    "## Step 1: Gather data\n",
    "We are going to use data from [Dutta and Sherrill (2003)](https://aip.scitation.org/doi/10.1063/1.1531658) who computed the bond energy of the HF molecule as a function of distance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "union-consent",
   "metadata": {},
   "outputs": [],
   "source": [
    "data = pd.read_csv('datasets/hf-bond-energy.csv')\n",
    "data = data.sample(12).sort_values('R')  # Reduce the amount of data as an experiment later"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "decimal-oakland",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'E (Ha)')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(data['R'], data['E'], '--o')\n",
    "\n",
    "ax.set_xlabel('R ($\\\\AA$)')\n",
    "ax.set_ylabel('E (Ha)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "straight-radius",
   "metadata": {},
   "source": [
    "We see a bond that has a minimum around $0.9 \\unicode{xC5}$ that plateaus out to be around -100, which is the energy of the broken state.\n",
    "\n",
    "For simplicity, we are going to compute an energy where broken is 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "exposed-brazil",
   "metadata": {},
   "outputs": [],
   "source": [
    "data['dE'] = data['E'] - data['E'].max()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "rapid-redhead",
   "metadata": {},
   "source": [
    "## Step 2: Introducing the Models\n",
    "There are many models that can explain bond energy that look like this, we are going to use 2 of them:\n",
    "\n",
    "- [Lennard-Jones](https://en.wikipedia.org/wiki/Lennard-Jones_potential): $E = 4\\epsilon [(\\sigma/r)^{12} - (\\sigma/r)^6]$\n",
    "- [Morse](https://en.wikipedia.org/wiki/Morse_potential): E = $D_e(1 - e^{-\\alpha(r-r_e)})^2 - D_e$\n",
    "- [Buckingham](https://en.wikipedia.org/wiki/Buckingham_potential): $E = A\\exp(-Br) - \\frac{C}{r^6}$\n",
    "\n",
    "Before getting started using Bayesian methods, we must establish reasponable priors for these model.\n",
    "We will do that by finding the optimal parameters by fitting the parameters to the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "portable-regard",
   "metadata": {},
   "outputs": [],
   "source": [
    "def lj(r, e, s):\n",
    "    \"\"\"Compute the energy using Lennard-Jones potential\n",
    "    \n",
    "    Args:\n",
    "        r: Bond distances\n",
    "        e: Bond energy term (epsilon)\n",
    "        s: Bond length term (sigma)\n",
    "    Return:\n",
    "        Energies at each distance\n",
    "    \"\"\"\n",
    "    rn = np.divide(s, r)\n",
    "    return 4 * e * (np.power(rn, 12) - np.power(rn, 6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "sustainable-landing",
   "metadata": {},
   "outputs": [],
   "source": [
    "def morse(r, d, a, r_e):\n",
    "    \"\"\"Compute the energy using Lennard-Jones potential\n",
    "    \n",
    "    Args:\n",
    "        r: Bond distances\n",
    "        d: Bond energy term (D_e)\n",
    "        a: Bond stretching term (alpha)\n",
    "        r_e: Bond length term (r_e)\n",
    "    Return:\n",
    "        Energies at each distance\n",
    "    \"\"\"\n",
    "    return d * np.power(1 - np.exp(-a * np.subtract(r, r_e)), 2) - d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "third-cliff",
   "metadata": {},
   "outputs": [],
   "source": [
    "def buckingham(r, A, B, C):\n",
    "    \"\"\"Compute the energy using a Buckingham potential\n",
    "    \n",
    "    Args:\n",
    "        r: Bond distances\n",
    "        A: Repulsion strength\n",
    "        B: Repulsion distance\n",
    "        C: Attraction strength\n",
    "    Return:\n",
    "        Energies at each distance\n",
    "    \"\"\"\n",
    "    return A * np.exp(np.multiply(-B, r)) - np.divide(C, np.power(r, 12))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "joined-percentage",
   "metadata": {},
   "source": [
    "Fit them using scipy's [`curve_fit` function](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "accredited-spectrum",
   "metadata": {},
   "outputs": [],
   "source": [
    "lj_opt, _ = curve_fit(lj, data['R'], data['dE'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "careful-chemical",
   "metadata": {},
   "outputs": [],
   "source": [
    "morse_opt, _ = curve_fit(morse, data['R'], data['dE'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "unauthorized-flood",
   "metadata": {},
   "outputs": [],
   "source": [
    "buck_opt, _ = curve_fit(buckingham, data['R'], data['dE'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "historical-chrome",
   "metadata": {},
   "source": [
    "Plot them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dental-begin",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa8bc84c850>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(data['R'], data['dE'], '--o', label='Data')\n",
    "\n",
    "r = np.linspace(0.7, 4, 128)\n",
    "ax.plot(r, lj(r, *lj_opt), label='LJ')\n",
    "ax.plot(r, morse(r, *morse_opt), label='Morse')\n",
    "ax.plot(r, buckingham(r, *buck_opt), label='Buckingham')\n",
    "\n",
    "ax.set_ylim(-0.25, 0.01)\n",
    "\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "stuck-baseline",
   "metadata": {},
   "source": [
    "Morse is clearly the best fit to the data, but let's be precise about it using Bayesian methods"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "integrated-bridge",
   "metadata": {},
   "source": [
    "## Defining the Model Fitting as a Bayesian Problem\n",
    "As shown in the previous notebook, we define our problem by establishing a prior distribution and a \"likelihood\" function.\n",
    "\n",
    "But first, we add an error term to the model that describes our expectation of the model's reliability. Let's choose a Guassian form for that error model, which makes the Morse potential a 4-parameter model:\n",
    "\n",
    "$E(r|D_e,\\alpha,r_e,\\epsilon) = D_e(1 - e^{-\\alpha(r-r_e)})^2 - D_e + N(0, \\epsilon)$\n",
    "\n",
    "where $N(0, \\epsilon)$ is a normal distribution with a mean of 0 and a standard deviation of $\\epsilon$. This will be important when defining our logarithmic function.\n",
    "\n",
    "### Defining a Prior Distribution\n",
    "The prior distribution defines our initial expectation for the values of the parameters. We'll need to establish one for each of the parameters and the error term in the above model, and we'll use the data from the initial fit to build those. \n",
    "\n",
    "Specifically, we'll assume a normal distribution for the model parameters with a mean of those we fit previously and standard deviation of 10% of the mean.\n",
    "\n",
    "We will let the error term be equal to the root mean square error of the model from our previous fitting and assume an expotential distribution.\n",
    "\n",
    "To implement these, we define each distribution using [SciPy](https://docs.scipy.org/doc/scipy/reference/stats.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "chief-edwards",
   "metadata": {},
   "outputs": [],
   "source": [
    "priors = [\n",
    "    stats.norm(p, 0.1 * p) for p in morse_opt\n",
    "]\n",
    "\n",
    "# For the error distribution\n",
    "rmse = np.sqrt(np.power(morse(data['R'], *morse_opt) - data['dE'], 2).sum())\n",
    "priors.append(stats.expon(scale=rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "removable-casting",
   "metadata": {},
   "source": [
    "These objects give us useful utilities such as being about to plot the distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "foreign-circuit",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 468x180 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1, 4, figsize=(6.5, 2.5))\n",
    "\n",
    "# Make the distributiosn for each parameter\n",
    "for ax, prior, m, l in zip(axs, priors, morse_opt, ['$D_e$', '$\\\\alpha$', '$r_e$']):\n",
    "    r = np.linspace(m - 0.4 * m, m + 0.4 * m, 128)\n",
    "    ax.plot(r, prior.pdf(r))\n",
    "    \n",
    "    ax.set_title(f'Prior for {l}')\n",
    "    ax.set_xlabel(l)\n",
    "    \n",
    "# Make the distribution for the model error\n",
    "r = np.linspace(1e-6, 0.04, 128)\n",
    "ax = axs[-1]\n",
    "ax.plot(r, priors[-1].pdf(r))\n",
    "ax.set_title(f'Prior for $\\\\epsilon$')\n",
    "ax.set_xlabel('$\\\\epsilon$')\n",
    "\n",
    "\n",
    "    \n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "latin-drill",
   "metadata": {},
   "source": [
    "We can also generate many example model parameters by sampling from these priors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "varied-volunteer",
   "metadata": {},
   "outputs": [],
   "source": [
    "p0 = np.array(list(zip(p.rvs(size=(32,)) for p in priors)))  # Get 32 shapes from each prior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "religious-rainbow",
   "metadata": {},
   "outputs": [],
   "source": [
    "p0 = np.squeeze(p0).T  # Shape them into a (32 x 4) array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "expired-hardware",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'E (Ha)')"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the data\n",
    "ax.plot(data['R'], data['dE'], 'k--o')\n",
    "\n",
    "# Plot the data\n",
    "r = np.linspace(0.6, 4, 128)\n",
    "for p in p0:\n",
    "    e = morse(r, *p[:-1])\n",
    "    ax.plot(r, e, 'b-', lw=0.1, alpha=0.9)\n",
    "    \n",
    "    \n",
    "ax.set_ylim(-0.25, 0.01)\n",
    "\n",
    "ax.set_xlabel('R ($\\\\AA$)')\n",
    "ax.set_ylabel('E (Ha)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "expanded-official",
   "metadata": {},
   "source": [
    "The above shows a big population of models that *could* explain our data. Our next step is to create a function to mark which ones are more likely. \n",
    "\n",
    "### Defining a Likelihood Function\n",
    "\n",
    "As noted in our previous notebook, the likelihood function requires both assessing the likelihood of observing the data given our model and the parameters:\n",
    "\n",
    "$P(\\theta|D,M)=P(D|\\theta,M)P(\\theta|M)$\n",
    "\n",
    "The function has two parts:\n",
    "1. $P(\\theta|M)$: The probability function of the parameters given our prior expectation\n",
    "1. $P(D|\\theta,M)$: The likelihood of observing the data given our parameters\n",
    "\n",
    "Our goal is to create a probability distribution as a function of $\\theta$ for each. Creating these functions often involves multiplying large and small numbers together, which is problematic due to [how computers perform multiplication](https://en.wikipedia.org/wiki/Floating-point_arithmetic#Accuracy_problems). So, we are going to write functions that compute the logarithm of this probability distribution function - \"log-likelihood\"\n",
    "\n",
    "Computing using the prior is relatively easy. We have the prior distribution defined using Scipy, and we can call the \"PDF\" function to get them. As we have multiple parameters, the likelihood of observing a certain collection of parameters is the [product of the probability functions for each](https://en.wikipedia.org/wiki/Chain_rule_(probability))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "later-wisdom",
   "metadata": {},
   "outputs": [],
   "source": [
    "def prior_logpdf(params, priors):\n",
    "    \"\"\"Evalulate the logarithm PDF probability distribution function for the prior distributions\n",
    "    \n",
    "    We compute the logarithm of the PDF to avoid issues around multiplying large numbers\n",
    "    \n",
    "    Args:\n",
    "        params: A certain set of parameters\n",
    "        priors: The prior distributiuon for each parameter\n",
    "    \"\"\"\n",
    "    \n",
    "    # Sum the log-liklihood for the parameters in each prior distribution\n",
    "    logpdf = 0\n",
    "    for prior, th in zip(priors, params):\n",
    "        logpdf += prior.logpdf(th)\n",
    "    return logpdf"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "printable-ultimate",
   "metadata": {},
   "source": [
    "Evaluating the $P(D|\\theta,M)$ term uses the error model we define along with the Morse function (that normal distribution term we defined above).\n",
    "\n",
    "Assume we have an observed energy, $E_{obs}$, at distance $r$ and a model for the energy as $E(r) = \\textrm{Morse}(r, D_e, \\alpha, r_e)+ N(0, \\epsilon)$.\n",
    "\n",
    "We can compute the error between the observed energy and that predicted using the model: $\\Delta E = E_obs - E(r) = N(0, \\epsilon)$\n",
    "\n",
    "This relates the error to a distribution, so we can compute the log-likelihood of the value of the error given the error value. We have more than one point, so the probability of seeing all of them is the product of the likelihoods of errors in the model for each points points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "plain-sailing",
   "metadata": {},
   "outputs": [],
   "source": [
    "def data_loglikelihood(params, model, data):\n",
    "    \"\"\"Compute the log-likelihood of observing our data given the model parameters\n",
    "    \n",
    "    Args:\n",
    "        params: Parameters of the model\n",
    "    Return:\n",
    "        Log-likelihood\n",
    "    \"\"\"\n",
    "    \n",
    "    # Compute the values for the Morse function\n",
    "    e_pred = model(data['R'], *params[:-1])  # The first N-1 parameters are of the morse model\n",
    "    \n",
    "    # Compute the error for each point\n",
    "    e_error = data['dE'] - e_pred\n",
    "    \n",
    "    # Sum the log-likelihoods for observing each error\n",
    "    #  based on the width of the error distribution (the last parameter)\n",
    "    return stats.norm(0, scale=params[-1]).logpdf(e_error).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bibliographic-geneva",
   "metadata": {},
   "source": [
    "### Creating a tool to compute the posterior likelihdood.\n",
    "Together, the functions above define the ability to generate the posterior distribution for the parameters.\n",
    "With these in hand, we make a single tool that will compute the log-likelihdoof for any set of parameters using different data sources, models, or prior distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "disciplinary-armenia",
   "metadata": {},
   "outputs": [],
   "source": [
    "class PosteriorPDF(object):\n",
    "    def __init__(self, model, priors, data):\n",
    "        \"\"\"Args:\n",
    "            model: Model used to compute energy\n",
    "            priors: Prior distribution of the parameters\n",
    "            data: Data to use for fitting\n",
    "        \"\"\"\n",
    "        self.model = model\n",
    "        self.priors = priors\n",
    "        self.data = data\n",
    "        \n",
    "    def __call__(self, x):\n",
    "        return prior_logpdf(x, self.priors) + data_loglikelihood(x, self.model, self.data)\n",
    "lnpdf = PosteriorPDF(morse, priors, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "interracial-disposal",
   "metadata": {},
   "source": [
    "We can use this to compute the ratio in posterior probabilities between different sets of parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "civil-morning",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ratio between likelihoods of the first two sampled points: inf\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/wardlt/miniconda3/envs/bayes/lib/python3.7/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp\n",
      "  \n"
     ]
    }
   ],
   "source": [
    "lnpdf_diff = lnpdf(p0[0, :]) - lnpdf(p0[1, :])\n",
    "print(f'Ratio between likelihoods of the first two sampled points: {np.exp(lnpdf_diff):.2e}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "equipped-eleven",
   "metadata": {},
   "source": [
    "We now are ready to MCMC sampling!\n",
    "\n",
    "## Sampling from the Posterior\n",
    "As discussed in our previous notebook, Metropolis-Hastings (MH) sampling is effective but inefficient. The small jumps in parameters lead to correlation between guesses, which slows down the rate at which parameter space is explored (you get stuck around your current guess). The [`kombine` program](https://pages.uoregon.edu/bfarr/kombine/index.html) uses an [ensemble algorithm](https://projecteuclid.org/euclid.camcos/1513731992) that avoids these issues for reasons out-of-scope for this class.\n",
    "\n",
    "Here, we describe how to use it.\n",
    "\n",
    "### Step 1: Seed the sampling algorithm\n",
    "We will generate samples from our prior distribution to produce an initial set of points to begin our sampling.\n",
    "We choose 500 as an experiment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "dental-intake",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created a starting array of shape: (128, 4)\n"
     ]
    }
   ],
   "source": [
    "n_walkers = 128  # Number of samples to start with\n",
    "p0 = np.vstack([  # Combines samples from each prior to form one array\n",
    "    p.rvs(size=(n_walkers,)) for p in priors  # Generate random samples from each prior\n",
    "]).T  # Flips it from <parameters>x<walkers> to <walkers>x<parameters>, as expected by kombine\n",
    "print(f'Created a starting array of shape: {p0.shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "single-packet",
   "metadata": {},
   "source": [
    "### Step 2: Make the Sampler\n",
    "The [`Sampler` class](https://pages.uoregon.edu/bfarr/kombine/kombine.html#kombine.sampler.Sampler) is the tool in combine that generates samples from the posterior\n",
    "distribution. We create it by telling how many \"walkers\" to use (we perform sampling from many different random starting points), the number of parameters in the model, and a function that gives their likelihood over time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "arabic-lambda",
   "metadata": {},
   "outputs": [],
   "source": [
    "sampler = kombine.Sampler(n_walkers, 4, lnpdf) \n",
    "# Recall that lnpdf is the log-likelihood given the Morse model and our full dataset we defined earlier"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "greek-translator",
   "metadata": {},
   "source": [
    "### Step 3: Burn In\n",
    "The initial points we provided is not from the posterior distribution. We need to re-run the make-new-guesses-then-choose-whether-to-accept procedure enough times to be able to transform the initial population of points that looks like posterior distribution.\n",
    "\n",
    "Kombine does this by running the sampling until the probability of accepting a new move remains constant over time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "armed-surge",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 2.23 s, sys: 229 ms, total: 2.46 s\n",
      "Wall time: 1min 33s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "output = sampler.burnin(p0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "centered-processing",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Burn-in required 147 steps\n"
     ]
    }
   ],
   "source": [
    "burn_in_length = len(sampler.acceptance_fraction)\n",
    "print(f'Burn-in required {burn_in_length} steps')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "naked-scotland",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'P(Accept)')"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(sampler.acceptance_fraction)\n",
    "\n",
    "ax.set_xlabel('Step')\n",
    "ax.set_ylabel('P(Accept)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "chubby-whale",
   "metadata": {},
   "source": [
    "You can look at the distribution for the $D_e$ parameter to see how it changes over time.\n",
    "\n",
    "We do that by looking at the \"chain,\" which records the state of the sampling at each step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "intended-developer",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The chain is an array of shape: (147, 128, 4). N Steps x N Walkers x N Parameters\n"
     ]
    }
   ],
   "source": [
    "print(f'The chain is an array of shape: {sampler.chain.shape}. N Steps x N Walkers x N Parameters')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "occasional-quality",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$P(D_e)$')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 468x180 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1, 3, figsize=(6.5, 2.5), sharey=True)\n",
    "\n",
    "bins = np.linspace(0.175, 0.225, 128)\n",
    "for i, ax in zip([0, burn_in_length // 4, burn_in_length - 1], axs):\n",
    "    ax.hist(sampler.chain[i, :, 0], bins=bins)\n",
    "    ax.set_xlabel('$D_e$')\n",
    "    ax.set_title(f'Step {i}')\n",
    "    \n",
    "axs[0].set_ylabel('$P(D_e)$')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "discrete-tobacco",
   "metadata": {},
   "source": [
    "Note how the distribution evolves significantly over time\n",
    "\n",
    "## Step 4: Sampling from the Posterior\n",
    "\n",
    "Now that our sampler is ready, we can generate samples from the posterior and use that to evaluate properties of our model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "chief-nursing",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 612 ms, sys: 28.3 ms, total: 640 ms\n",
      "Wall time: 36.3 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "# Get more samples by running with the now-converged sampler\n",
    "outputs = sampler.run_mcmc(64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "distinguished-fourth",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Retrieved 4576 samples from the posterior\n"
     ]
    }
   ],
   "source": [
    "samples = sampler.get_samples()  # Gets only the samples after burn-in\n",
    "print(f'Retrieved {len(samples)} samples from the posterior')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sought-creation",
   "metadata": {},
   "source": [
    "Such as the joint probability distribution between two parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "periodic-toolbox",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$\\\\alpha$')"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 252x252 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(3.5, 3.5))\n",
    "\n",
    "ax.hist2d(samples[:, 0], samples[:, 1], bins=32, cmap='Blues');\n",
    "\n",
    "ax.set_xlabel('$D_e$')\n",
    "ax.set_ylabel('$\\\\alpha$')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "divided-steel",
   "metadata": {},
   "source": [
    "Or confidence intervalues for certain predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "consecutive-source",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Predict the energy for each sampled parameter\n",
    "r = np.linspace(0.75, 1.25, 128)\n",
    "eng_predictions = [morse(r, *s[:-1]) for s in samples]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "double-irish",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'E')"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot between the 2.5th and 97.5th percentiles\n",
    "ax.fill_between(\n",
    "    r, np.percentile(eng_predictions, 2.5, axis=0), np.percentile(eng_predictions, 97.5, axis=0),\n",
    "    alpha=0.5\n",
    ")\n",
    "\n",
    "# Plot the mean\n",
    "ax.plot(r, np.mean(eng_predictions, axis=0))\n",
    "ax.set_xlim(0.75, 1.25)\n",
    "\n",
    "ax.set_xlabel('r')\n",
    "ax.set_ylabel('E')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "after-aaron",
   "metadata": {},
   "source": [
    "## Comparing Models\n",
    "Another key benefit of Bayesian modeling is we can get statistically-robust measures of which models best describe data using the so-called \"marginal likelihood\":\n",
    "\n",
    "$P(D|M) = \\int_\\theta P(D|\\theta,M)P(\\theta|M)d\\theta$\n",
    "\n",
    "As discussed in the lecture, we can compute this integral by averaging the value of $P(D|\\theta, M)$ over many samples of $\\theta$ from our posterior distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "hairy-space",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 8.48 s, sys: 36.6 ms, total: 8.52 s\n",
      "Wall time: 8.47 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "# Using the data_loglikelihood function we defined earlier\n",
    "log_evidence = [\n",
    "    data_loglikelihood(x, morse, data) for x in samples\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "similar-explorer",
   "metadata": {},
   "source": [
    "These are log-likehoods, so we have to use numpy's [logaddexp](https://numpy.org/doc/stable/reference/generated/numpy.logaddexp.html) to avoid numerical problems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "southwest-titanium",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Log model evidence for Morse is: 51.87\n"
     ]
    }
   ],
   "source": [
    "evidence = np.logaddexp.reduce(log_evidence) - np.log(len(samples)) # Subtracting logarithms is equivalent to dividing\n",
    "print(f'Log model evidence for Morse is: {evidence:.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "neither-ottawa",
   "metadata": {},
   "source": [
    "This number is not particularly meaningful on its own, but we can compare that of another model to see which has greater evidence"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "progressive-amsterdam",
   "metadata": {},
   "source": [
    "### Repeating with the Buckingham Model\n",
    "We are going to go through the steps of making priors, defining likelihood functions and sampling the posterior with Kombine.\n",
    "\n",
    "Step 1: Make the prior distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "vulnerable-motorcycle",
   "metadata": {},
   "outputs": [],
   "source": [
    "priors_buck = priors = [\n",
    "    stats.norm(p, 0.1 * abs(p)) for p in buck_opt\n",
    "]\n",
    "\n",
    "# For the error distribution\n",
    "rmse = np.sqrt(np.power(buckingham(data['R'], *buck_opt) - data['dE'], 2).sum())\n",
    "priors_buck.append(stats.expon(scale=rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "overhead-charger",
   "metadata": {},
   "source": [
    "Define the likelihood functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "driving-powell",
   "metadata": {},
   "outputs": [],
   "source": [
    "lnpdf_buck = PosteriorPDF(buckingham, priors_buck, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ranging-montgomery",
   "metadata": {},
   "source": [
    "Define the initial population"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "hawaiian-repeat",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created a starting array of shape: (128, 4)\n"
     ]
    }
   ],
   "source": [
    "p0_buck = np.vstack([p.rvs(size=(n_walkers,)) for p in priors]).T\n",
    "print(f'Created a starting array of shape: {p0_buck.shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "rubber-bumper",
   "metadata": {},
   "source": [
    "Burn in with Kombine sampler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "single-stomach",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 1.8 s, sys: 194 ms, total: 1.99 s\n",
      "Wall time: 50.5 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "sampler_buck = kombine.Sampler(n_walkers, 4, lnpdf_buck) \n",
    "output = sampler_buck.burnin(p0_buck)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "turkish-luxury",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'P(Accept)')"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(sampler_buck.acceptance_fraction)\n",
    "\n",
    "ax.set_xlabel('Step')\n",
    "ax.set_ylabel('P(Accept)')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "collectible-equilibrium",
   "metadata": {},
   "source": [
    "Run MCMC to get more samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "nuclear-investigation",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 1.21 s, sys: 117 ms, total: 1.33 s\n",
      "Wall time: 1min 13s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "output = sampler_buck.run_mcmc(128)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bigger-wilson",
   "metadata": {},
   "source": [
    "Computing the evidence interval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "naked-composer",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Log model evidence for Buckingham is: 34.09\n",
      "CPU times: user 13.9 s, sys: 95.8 ms, total: 14 s\n",
      "Wall time: 14.1 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "samples_buck = sampler_buck.get_samples()\n",
    "# Using the data_loglikelihood function we defined earlier\n",
    "log_evidence_buck = [\n",
    "    data_loglikelihood(x, buckingham, data) for x in samples_buck\n",
    "]\n",
    "evidence_buck = np.logaddexp.reduce(log_evidence_buck) - np.log(len(samples_buck)) # Subtracting logarithms is equivalent to dividing\n",
    "print(f'Log model evidence for Buckingham is: {evidence_buck:.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "muslim-relevance",
   "metadata": {},
   "source": [
    "Taking the difference between the evidences gives you a [\"Bayes Factor.\"](https://en.wikipedia.org/wiki/Bayes_factor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "dress-inspiration",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Bayes Factor for Morse potential compared to Buckingham is 5.28e+07\n"
     ]
    }
   ],
   "source": [
    "bayes_factor = np.exp(evidence - evidence_buck)\n",
    "print(f'The Bayes Factor for Morse potential compared to Buckingham is {bayes_factor:.2e}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cosmetic-postage",
   "metadata": {},
   "source": [
    "Larger values mean greater confidence and a Bayes factor of 100 is labeled as \"decisive.\" So, as we expected earlier, the Morse model is clearly better.\n",
    "\n",
    "Things to try: \n",
    "- Sample fewer points of data in the second cell. How does that change your Bayes factor? \n",
    "- Increase the number of walkers. How does that affect computation time?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "satisfactory-legend",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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