emcee_example.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import emcee"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Choose the \"true\" parameters.\n",
    "m_true = -0.9594\n",
    "b_true = 4.294\n",
    "f_true = 0.534\n",
    "\n",
    "# Generate some synthetic data from the model.\n",
    "N = 50\n",
    "x = np.sort(10*np.random.rand(N))\n",
    "yerr = 0.1+0.5*np.random.rand(N)\n",
    "y = m_true*x+b_true\n",
    "y += np.abs(f_true*y) * np.random.randn(N)\n",
    "y += yerr * np.random.randn(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "throw_dim=2# throw away first 2 points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def lnlike(theta, x, y, yerr):#throw away points\n",
    "    m, b, lnf = theta\n",
    "    model = m * x[throw_dim:] + b\n",
    "    inv_sigma2 = 1.0/(yerr[throw_dim:]**2 + model**2*np.exp(2*lnf))\n",
    "    return -0.5*(np.sum((y[throw_dim:]-model)**2*inv_sigma2 - np.log(inv_sigma2)))\n",
    "\n",
    "def lnlike2(theta, x, y, yerr):#marginalize over points\n",
    "    m, b, lnf = theta[:3]\n",
    "    ythrow= theta[3:]\n",
    "    model = m * x[throw_dim:] + b\n",
    "    model=np.append(ythrow,model)\n",
    "    inv_sigma2 = 1.0/(yerr**2 + model**2*np.exp(2*lnf))\n",
    "    return -0.5*(np.sum((y-model)**2*inv_sigma2 - np.log(inv_sigma2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ndim, nwalkers = 3, 100\n",
    "pos0 = [[m_true,b_true,f_true] + 1e-4*np.random.randn(ndim) for i in range(nwalkers)]\n",
    "\n",
    "ndim2= 3+throw_dim\n",
    "pos2 = [np.append([m_true,b_true,f_true],x[:throw_dim]) + 1e-4*np.random.randn(ndim2) for i in range(nwalkers)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, lnlike, args=(x, y, yerr))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "burn=50\n",
    "nstep=500\n",
    "thin=2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pos, prob, state =sampler.run_mcmc(pos0,burn,storechain=0)\n",
    "sampler.reset()\n",
    "pos, prob, state =sampler.run_mcmc(pos,nstep,thin=thin)\n",
    "chain=sampler.flatchain#.reshape((-1, ndim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sampler2 = emcee.EnsembleSampler(nwalkers, ndim2, lnlike2, args=(x, y, yerr))\n",
    "pos2, prob2, state2 =sampler2.run_mcmc(pos2,burn,storechain=0)\n",
    "sampler2.reset()\n",
    "pos2, prob2, state2 =sampler2.run_mcmc(pos2,nstep,thin=thin)\n",
    "chain2=sampler2.flatchain#.reshape((-1, ndim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((25000, 3), (25000, 5))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chain.shape,chain2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cov=np.cov(chain,rowvar=0)\n",
    "cov2=np.cov(chain2[:,:3],rowvar=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eig_v,eig_vec=np.linalg.eig(cov)\n",
    "eig_v2,eig_vec2=np.linalg.eig(cov2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.16667044,  0.00028052,  0.01791589]),\n",
       " array([ 0.18748079,  0.00141624,  0.03932537]))"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eig_v,eig_v2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ea44a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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Dq1cOFCP1QEIBiY770AwK+8dScXau/kyZPq1auXCgoKTI/p3r27\nli9f7sOqAFxFv/ScP4bFVuvWq1fPKzmBYFebflndCbPy8vJaDQFqqi71hpKSEvXt21fbtm0zPSYi\nIkIjR47UqFGjlJSUZHrcmTNntHHjRr3yyiv6+uuvDY8pLCxUv379tHr1avXr18/j+gFYo2d6rqY1\n3HHHHfrjH//oUa5+/fpp6NChWrFihWF8165d+uabb5SYmOhRHgDG6Jn2KC8v16hRowzvdP7DH/6g\n22+/3Q9VAQgG3FkMABbOnTunBx54QEePHjU9plOnTlq3bl1AXYUIALXRoEED09ilS5e8krOkpMQ0\nVr9+fa/kBFBzVn1BojdI/95VxmpQ/OCDD+r777/X/PnzLQfFkhQTE6OhQ4dq7969Wrp0qa677jrD\n48rKyjR48GDl5uZ6VDsAe9EzPath0qRJCgnx/BTllClTLAdWH3/8scc5AHiOnmluzpw52r9/f5XX\n4+Pj9cILL/ihIgDBgmExAJi4cOGCevfubfncucTERGVlZSkiIsKHlQGAvaye8XbhwgWv5LRat0mT\nJl7JCaDmrPqCRG948803tWHDBtN4RkaGNm7cqKZNm9Z67eHDh2vXrl2Kjo42jJeWlmrEiBFeO5EK\noPbomcYiIyOrPSYuLk4DBw60JV/btm3Vo0cP0/imTZtsyQPAM/RMY4cPH9aLL75Y5XWHw6E33nhD\n4eHhfqgKQLBgWAwABi5duqSHH35Yu3fvNj2mbdu22rRpU7UfcgEg0MXExJjGioqKvJLz559/No1Z\n1QPAN6x+D10uV1D3hqKiIk2aNMk0/utf/1qvvfaaRzkSEhL00Ucfme5cc+TIEc2ZM8ejHADsQ890\nv4aePXvaut1snz59TGP79u2Ty+WyLRcA99Azq3K5XBo9erSuXLlSJTZo0CDL3gYAdmBYjGve1S2I\nHA5HnfqC/1y5ckX9+/e33FawVatWysnJ0fXXX+/DygDvol8Gr9jYWNPYqVOnvJLT6jnw/v5DHahO\nMPTL8PBwy7sXvNEbrly5YnlyMFB6w+LFi03vTAkPD9dbb71lS5777rtP6enppvEFCxYYnlAEAg09\nM3h7Zk12V0hOTrY1p9V6JSUlOnbsmK35ALvRM4OzZy5atMjwPGRkZKTmz5/vh4oABJswfxcAeNu9\n996rixcv+ruMWguUZ2UEm7KyMqWmplpuT9WiRQvl5OSoefPmPqwM8D76ZfC66aabTGNWQ11PnDx5\n0jTWsmVLr+QE7BIs/fLmm2/WoUOHDGPe6A1WfcHhcARMb1iyZIlpLD09XXFxcbblmjJlihYvXqyy\nsrIqsVOnTumzzz5TSkqKbfkAb6BnBm/PbNq0qerVq6fS0lLTYxITE23NabWey+VSfn6+WrVqZWtO\nwE70zODsmc8++6zh6y+99JKaNWvm1dzsuABAYliMIBASEqLrrrvO32WgDqioqNCQIUP06aefmh7T\nrFkz5eTkKD4+3neFAT5CvwxerVu3No398MMPcrlctb5avDp5eXmmMU7gIdAFS79s1aqV6Um877//\n3vZ8Vn0hIiLC73d8SNLp06f17bffmsbtHtzGxcUpOTlZ27dvN4z//e9/Z1iMgEfPDN6eGRISovj4\neB0+fNj0GLPns7srIiJCTqfT8CIbSTpz5oyt+QC70TODs2cabZEdFhamAwcOaPz48V7L63K5NG3a\nNDVs2NAw3rZtW02YMMFr+QEEDobFACCpsrJSI0aM0Jo1a0yPiYmJ0ebNm/WLX/zCh5UBgPdZDYtL\nS0t1/Phx2y+SOXr0qGnslltusTUXAPdY/S4eOXLE9nx1oS/s3LnTNBYeHq67777b9pw9evQwHRZb\n1QPAt+iZxtq0aWM6LHY4HIqKirI9Z5MmTUy3sS0pKbE9H4Dao2dWr7y8XH/961+9nmfZsmWmsa5d\nuzIsBoIEzywGEPRcLpd++9vfauXKlabHREVFKSsrSwkJCT6sDAB8IzY21nRrfZfLpQMHDtiar6io\nSD/++KNhzOFwKCkpydZ8ANzzq1/9yjRmd1+obs1A6QtW2yLGxcUpLMz+67GttkW02lIRgG/RM40F\nUi2SbN8tB4B76Jl1Az0TCB4MiwEEvfHjxyszM9M03qhRI23cuFF33HGHD6sCAN/q0KGDaWzPnj22\n5rJaLzIykq3+gQBh1Rf++c9/qry83NZ8X331lVu1+NLZs2dNY7GxsV7J2bRpU9MY26kCgYOeaaxj\nx46mMZfLpfPnz9ue89y5c6axYNjeF6gL6JkAEFgYFgMIak899ZTlli4RERFav369Onfu7MOqAMD3\nunTpYhr74osvbM21Y8cO0xj9FggciYmJioiIMIxduXJFu3fvti1Xdesltcv9IQAADUFJREFUJyfb\nlssTVicu/XHnhd0nUgG4j55pzOozpmT/RS/FxcWmzyuWrC/AAeA79EwACCwMiwEEreeff17z5s0z\njYeHh+uTTz7RPffc48OqAMA/evToYRrbsWOHrly5YluunJwc01ivXr1sywPAM06nU/fee69pfPPm\nzbblys3N1eXLlw1jUVFRuuuuu2zL5QmrO9IKCwu9kvP06dOmMbOTrAB8j55pLC4uzvJxTvv377c1\n39dff20aczgcllv7A/AdemZVDofDa1/u5gUQPBgWAwhK06dP16xZs0zj9evX15o1ayyHJwBwLbnz\nzjsVHR1tGLt06ZI2btxoS55Tp06Z3lnscDj0wAMP2JIHgD2sfifXrFljWx6rte6///6AOVl1ww03\nmMZ++ukny7vZ3JWXl+dWPQB8j55prHfv3qaxnTt32prLar0mTZroxhtvtDUfAPfRM/9PZWWlKioq\nvPZlxuFwKC8vz/T7tmzZ4sN3AYA/MSwGEHTmzp2radOmmcadTqfef/99yz9oAeBaExISopSUFNP4\nihUrbMnz3nvvqbKy0jB266236rbbbrMlDwB7PPLII6Yn0Pbv368DBw54nKO0tFQffPCBaTw1NdXj\nHHZp06aNaaykpES5ubm257TajaF169a25wPgPnqmsUcffdQ0tnnzZlu31Le6wJHHnQCBhZ4JAIGD\nYTGAoPKXv/xFzz77rGk8LCxM7777rn7zm9/4sCoACAxDhw41ja1du1bHjx/3aP3Kykq98cYbpvEh\nQ4Z4tD4A+zVv3lzdunUzjVs90qOmVq1apVOnThnGGjVqFFCfyxITE+V0Ok3jH330ka358vPzLe+S\n69Spk635AHiGnmmsY8eOat++vWHs5MmTtvXOQ4cOWV5g06dPH1vyALAHPRMAAgfDYgBBIzMzUxkZ\nGabx0NBQZWZmatCgQT6sCgACx7333mt6Z29FRYWmTp3q0fqZmZk6evSoYax+/fpKT0/3aH0A3jFu\n3DjT2PLly/Wvf/3L7bVLS0v14osvmsZHjhypBg0auL2+3Ro0aKDk5GTT+KJFi1RQUGBbvunTp1tu\nHWh1ghWAf9AzjY0fP940NmvWLMteV1MzZswwjTmdTstddAD4Bz0TAAIDw2IAASM+Pl4hISGGXyNH\njvRo7VWrVlkOIUJCQrRo0SI9/vjjHuUBAF/wZr+cOHGiaWzFihVuP7OooKBAzz33nGl86NChatas\nmVtrA/CugQMH6pZbbjGMlZWVacyYMXK5XG6tPWPGDH3//feGsbCwMD355JNurXuVN/ql1XaFly5d\n0tixY91+P/7Tli1b9Pbbb5vGb7jhBt13330e5wFgL3qmsbS0NMXFxRnG9u3bZznorYlPPvlE7777\nrml8wIABPOcdCED0TAAIDAyLAdQJZs8wqYm1a9dq2LBhph8uHQ6HFixYoCeeeMLtHAAQKDzpl9K/\nt4JOSEgwjLlcLj3++OPKy8ur1ZqXLl1SamqqCgsLDePXXXedx3ctA/Aeh8NheRJ/69atmjx5cq3X\nXbdunWbOnGkaHz16tFq2bFnrdWvK3X45bNgwNWrUyDS+du1aPf300+6WJenfz+kbNGiQ5Z12Y8eO\n9SgHAO+gZxqrV6+eXnjhBdP4zJkz9d5777m19q5du5SWlmYaDwsL07Rp09xaG4B30TMBIDAwLAZw\nTcvOzlZqaqrlibbXXntNo0eP9mFVABC4QkND9frrr5vGT548qe7du2vv3r01Wu/06dN66KGHtGPH\nDtNjJk2apJtuuqnWtQLwncGDB6tr166m8Tlz5uipp56q8Taiy5cvV2pqqunFfDExMR7fZeYtjRs3\nrnYYPG/ePPXt29f0IhkrS5YsUXJyss6fP296TNOmTS0frwLAv+iZxtLT09WlSxfDWHl5uYYNG6a5\nc+eqsrKyxmt+9NFH6tmzp2XPHDdunNq1a1fregH4Bj0TAPwvzN8FAPCOadOm6cyZMzU69sKFC6Yx\nl8uljIyMGm/50rZtW02YMKFGx3pbYWGhBgwYoLKyMtNjWrRooUOHDlk+P8lTXbp0YXtrIIDRL6vq\n1q2bMjIyTIfGJ06cUJcuXZSRkaEJEyYYDnqLioq0bNkyTZ8+3XJYctddd1luTw0gcGRmZqpDhw4q\nKioyjM+bN09btmzR9OnT1adPH4WGhlY5Zvfu3Zo5c6bWrVtnmsfhcGjx4sWKioqyrXa7TZ48We+/\n/76+/fZb02M2btyoVq1aKS0tTWlpaUpMTFRIiPH12oWFhVq/fr1eeeUVHThwwDK3w+HQq6++GtDv\nDwB6ppnly5erY8eOhu9LZWWlnn32WS1btkxTpkxRnz591LBhwyrHlZaWauvWrZozZ45ycnIs8yUk\nJGj27Nm21Q/AO+iZAOBfDpcdD1MCEHDi4+N14sQJn+ft2rWrPv/8c7e+16rmJ554QkuWLKnVeseO\nHVPr1q3dqsVO7tQOwHfol8bKysrUtWtX7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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e0b9250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chain=sampler.flatchain\n",
    "a=hist2d(chain[:,0],chain[:,1],40,norm=LogNorm())\n",
    "show()\n",
    "a=hist2d(chain2[:,0],chain2[:,1],40,norm=LogNorm())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10eac6490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e53a310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chain=sampler.flatchain\n",
    "a=hist2d(chain[:,1],chain[:,2],40,norm=LogNorm())\n",
    "show()\n",
    "a=hist2d(chain[:,1],chain[:,2],40,norm=LogNorm())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e79be50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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uzWmdNrXpLtmt+3b5/j+XwRHh5ujF25csXxfHYN+Ly8PB0+fNjx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\nw4d18uRJ5ebmKi8vTxEREf97HZo1a6aWLVuqS5cu6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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e518d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chain=sampler.flatchain\n",
    "a=hist2d(chain[:,0],chain[:,2],40,norm=LogNorm())\n",
    "show()\n",
    "a=hist2d(chain[:,0],chain[:,2],40,norm=LogNorm())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
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