MCMC_sample.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "fb73b119",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import emcee\n",
    "import scipy.stats as ss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "38891483",
   "metadata": {},
   "outputs": [],
   "source": [
    "def total_model(param1,param2,param3,param4):\n",
    "    #insert some functions of param1,param2,param3,param4...\n",
    "    some_value=2*param1+param2+3*param3+param4\n",
    "    return(some_value)\n",
    "def lprior(p):\n",
    "    param1,param2,param3,param4 = p[0],p[1],p[2],p[3]\n",
    "    prior = 0.0\n",
    "    prior += ss.norm.logpdf(param1,loc=5.,scale=3)\n",
    "    prior += ss.norm.logpdf(param2,loc=3.,scale=1.)+ss.uniform.logpdf(param2,loc=0.,scale=10)\n",
    "    prior += ss.norm.logpdf(param3,loc=2.,scale=1.)\n",
    "    prior += ss.norm.logpdf(param4,loc=3.,scale=2.)+ss.uniform.logpdf(param4,loc=0.,scale=10)\n",
    "    if np.isnan(prior):\n",
    "        return(-np.inf)\n",
    "    return(prior)\n",
    "def log_probability(p):\n",
    "    param1,param2,param3,param4 = p[0],p[1],p[2],p[3]\n",
    "    model= total_model(param1,param2,param3,param4)\n",
    "    lp = lprior(p) + np.nansum(np.log(model))\n",
    "    if np.isnan(lp):\n",
    "        return(-np.inf)\n",
    "    return lp\n",
    "def confidenceInterval(y):\n",
    "    median=np.median(y)\n",
    "    pct15=np.percentile(y,15)\n",
    "    pct85=np.percentile(y,85)\n",
    "    list1=np.array([median,pct15,pct85])\n",
    "    return list1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "482024cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "nSamp=200\n",
    "nBurn=100\n",
    "ndim=4\n",
    "nwalkers=10*ndim\n",
    "\n",
    "guess=[5,3,2,3]\n",
    "p0 = np.zeros((nwalkers,ndim))\n",
    "for idx,thisguess in enumerate(guess):\n",
    "    p0[:,idx] = (np.random.randn(nwalkers)*0.001+1.0)*thisguess\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "9de2f17b",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/f4/vlzp24jx6r9962ljc6g1mcn00000gn/T/ipykernel_89771/579553167.py:18: RuntimeWarning: invalid value encountered in log\n",
      "  lp = lprior(p) + np.nansum(np.log(model))\n"
     ]
    }
   ],
   "source": [
    "#sample\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, log_probability)\n",
    "pos,prob,state=sampler.run_mcmc(p0, nBurn)\n",
    "sampler.reset()\n",
    "pos,prob,state=sampler.run_mcmc(pos, nSamp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "ce4e530e",
   "metadata": {},
   "outputs": [],
   "source": [
    "#print out distribution\n",
    "samples = total_model(sampler.flatchain[:,0],sampler.flatchain[:,1],sampler.flatchain[:,2],sampler.flatchain[:,3])\n",
    "valu=confidenceInterval(samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "18dc9585",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best estimation of thing :  25.838741276701388\n",
      "Upper error :  6.616662953869447\n",
      "Lower error :  6.894353849918538\n"
     ]
    },
    {
     "data": {
      "image/png": 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fAXu2JChjjDEmlwS1DPghbL1b6omzHHvSHjjLpxsTWg899BAPPfRQ0GF4du2UKVwbkTuTqNWtCa9cEtQ44AIReRF4EfgaSJ1f/3vApz7EZkzepIy/iIQotUFFrW5NeOXSBnUbTjvUT4ENwPmquh5ARHYDBgP3+BRf6NhA3XiYNGlS0CHEltWt8cr3gbqqugn4lftKV4XT/lST7XmjwgbqGmOMP5obqNui9aBEpK2I7CUixQCqukVVN6hqbpOXGdNK7rvvPu67776gw4glq1vjl5wSlIgcLiKv4dwxfQEc427fQ0SmisgAH2M0xndTp05l6tSpQYcRS1a3xi9ZP+ITkd44A3VXA/8ALkzuU9WVItIeuIDtu54bEyoTJ04MOoTYsro1fsmlk8TNwFc4vfXaARel7Z8KRGdUoTFmB/UNDVRVVeV8fFFREe3atfMxIlOIcklQPwZuU9VqEck0g+YXwDdbFpYx+XXXXXcBcPXVVwccSfjU1dfzQUUFDw4bltPigzPnz6eoY0eemzTJkpRpkVwSVDuc7uWN2TXHWIxpNbNmzQo6hKx06dCh1T4r0dBAm9paLuncmS4lJVkfP3vGDJasWkUikbAEZVoklwT1KXBEE/uPA/6bWzjGtI7nnnsu6BCyEsRcfCVFRTktMzLujDMYtWJFHiIyhSaXXnz/BM5L66mnACJyFc5s50/6EFsoJQfq2kh5Y4xpmfLycsrKysDHFXXvAk4AXgY+wklO94hIN6AUeBWI7URcNlA3Hm6//XYAhg+PxsowyXn4bhsQ/hEcd8+axVtVVVwVdCAm9HxfUVdVN4vICcDvcJbVqAO+g7MEx93Afaq6pQUxG5N3FRUVQYeQlTW1tUGH4Nm8lSupjFC8JrxyWvJdVetx5tuL7Zx7Jt7GjRsXdAhZGR2heR/HDBlibVDGFy2a6igoIjJNRLSR12S3TI8myuyedr52InKniKwQkVoRmSUixwZyccYYYwAPd1Aicn4uJ1bVf+RynEe/Ycfu7EfhPGJMH8Z+W4Zt6SMQ/wacAgzDWXDxt8DLInKUqlb4EbAJRl1dHYnEjlND3nHHHQBcc801jR5bVVVFQ0NDo/tbU5nbKScKd1J3zJjBDGuDMj7w8ohvDE5HCMnivIozDVJeqOoO3dhF5BJgM856Vak+U9W3GzuXiBwG/BK4SFWfcLdNBxbgzJox2K+4Teuqq6vj/hEjqKms3GHfxDffBGDnJtZY2pRIsHjePOpLSyGH7tZ+WrxmTaCfn42P166NVJuZCS8vCap/3qNoIXf+v58D5aq6NsvDBwMJYHxyg6rWi8g4YLiItHWXGDERk0gkqKms5NIuXSgpLt5u31UexhVVVlczrLY2NHdRUfHYoEHWBmV80WyCUtXpzZUJgdOAjsDfM+y7TUT+F9iIszT99ar6Ycr+g4Alqpq+htUCoBjYz/23iaiS4uKcBpxWbbK/S4wJUiQ7SWRwPrASSF3KcxPwCHApzl3g1cAhwEwROSClXGdgXYZzrk3Zv9WqVavo06fP1peNiYqmP73+On96/fWgw4ilkW+8wevvvx90GCbkRo8evfV7FOiaqUxO3czBWawQ6Af0dDd9BkxX1bpcz5ljHN8EBuCMv6pPblfVFcCvU4q+6fbwWwBcD5ybPAXuTBjpp870ed26dWPOnDl+hG4CtOzrr4MOIba+rKri65rYLqptfFJWVpacRQIRWZ2pTE4Jyu3ZdzfQiW1f5AqsF5GrVHVMLufN0bk4d4KZHu9tR1WXichbwJEpm9cC+2Qo3illv4mZJ4YMCTqE2Hr4lFOsDcr4IpcFC8/C6dn3Bc60R//FSVIH4tyx/E1EalV1fKMn8df5wAeq+oHH8ul3TAuAn4lIh7R2qANxegV+4k+YxhhjspFLG9R1OHPwHaqqt6vqRFWdoKq3AYfiTHl0vZ9BNkZE+uB0cmj27sktvw/wI+CdlM0TgSKcXoDJcjsDZwGvWA++eLp2ypSt89sZf42YNo0p770XdBgmBnJ5xNcL+B9V3eEhvqpuEJEngBEtDcyj84F6nBnWtyMio3AS8CxgFU7c1wJbgFuT5VS1QkTGA/eKSBGwBLgM2BdnrkETQ1Ebp/OdLl2CDsGztbW11FoPSOODXBJUJU0P2t0C/F9u4XjnJpNfAJNVNdPnLcBJNENxuqCvBl4DblLVRWllLwRuAUYCuwMfAANVdW5egjeBi8KMDKmiFO/9J51kbVDGF7kkqDHAUBF5SFWrU3eIyK7ARcATPsTWJFVNAN2a2P848LjHc9UCf3BfxhhjQiCXBPUmcCrwoYg8xLY1oQ7EuWNZjdOle7vJVlX1jRbGaoxvrn7lFQDuOvHEgCPxJkpz8V3/2mvMqa62ufhMi+WSoF5N+fcdbOsRl3zs9620Mslec21y+KzQSa6om1xoy0RTbYYJZMOsS/v2QYfgWW19PQmbHsp4UF5enlyd3LcVdS9sUUQRZyvqxsODp5wSdAhZicJKukl3n3iitUEZT/Kxoq6nLt3GGGNMS8RlLj5jsnLl5MlcOXly0GF4dvrTT3P6008HHYYn10yZwuR33w06DBMDOSUoEfmliMwQkZUi0pDhVd/8WYwxXq2pqWGNzW9nCkwuUx3dANyEM9ZpJplnAjcm1O4dODDoEGLrjgEDrA3K+CKXThK/AabhDGSNVlcoY4wxkZFLgtoVeNqSk4my3/7nP0D0evNFwR9eeYWKjRttHJRpsVwS1PtAd78DMaY1tS8qCjqE2Gq/884UtYnFsEcTsFwS1A3AcyLyfCHOVWcDdeMhKjNIRNEtxx1nbVDGE98H6qrqdBH5FfC2iMwClgLpw8ZVVX+V7bmjwAbqGmOMP3wfqCsiP8CZMHZn4MfuK50CsUxQJh6iNLdd1FwxaRLzamqsDcq0WC6P+O4DEsAQ4E1VXe9rRMa0gijNbQdw1N57Bx2CZ53bt6e9zcVnfJBLgjoUGKGq5X4HY0xridLcdhCteEf062dtUMYXucwksRLY7HcgxhhjTKpcEtTjwLkiksvdlzGhcOGECVw4YULQYXgWpbn4LvvPf5jw1ltBh2FiIJck8xbOgoVvuwsWLmHHXny2QKEJte677hp0CFmJUhvUXh07skyk+YLGNCOXBDUl5d+PsW3BwqRYLVBo4unm/v2DDiErVx99dNAheHbDscdaG5TxhS1YmCUbqGuMMf7Ix0Ddgl6w0AbqxsO5zz8PwNjTTgs4Em/6jRkDwLShQwONw4uLy8tZWFtr46BMs3wfqGtMHPTq0iXoEGJr/86dWVlVFXQYJgZyTlAi0gf4AdCJHXsDqqr+uSWBGZNP/9O3b9AhxNY1P/qRtUEZX+Qy1VF74HngRLZ1iEh22dGUbZagjDHG5CyXcVB/wklOtwD9cRLSBcBJwJvAbOBAvwI0Jh/OfvZZzn722aDDiKWhEybw7PTpQYdhYiCXBHUG8Iyq/gmY7277UlVfBgYAxcBQf8IzJj96l5bSu7Q06DBi6dA99qC0c+egwzAxkEsbVHfgbvffyQG6xQCqWi8i/wIuA65teXjG5MfwY44JOoTY+sNRR5FYvpyqHDtKFBUV0a5dO5+jMlGUS4KqSjmuCtgCfDNl/wYgtn+a2jgoY5pWV1/PBxUVPDhsGEU5rFzcobSUK0aMsCRVAHwfBwV8CnwHQFUbRGQBzmO/x0VEgNOAZbmFG342DioekvPaPXfmmQFHEj9DX3iB91es4K/9+tGlpCSrY6s3b+aRykoSiYQlqAKQj3FQU4CLRORKVW0AHgEeEJFPcXrv7Qtcl3PExrSCKM1tB3Dqd74TdAieHbHnnny+bh0lRUV0bNs26HBMhOWSoG4HnsTtWq6qD4lIO+BcnDapR4G/+BahMXkQpbntIFrx/rpPH+Z99VXQYZgYyLoXn6pWq+oiVa1P2Xa3qh6uqkeq6h2qmj6BrK9EpJ+IaIbX+rRynUTkMRFZLSIbRWSKiByS4XztROROEVkhIrUiMktEjs3nNRhjjGla1Kc6ugJn3FXS1qTptodNxHnk+DtgHU7PwtdFpLeqLk857m/AKcAw4DPgt8DLInKUqlbk9QpMIAb/618ATPzFLwKOxJsozcU3dMIEPqysDDoMEwMtTlDuwoXfB/YC/quqC1oclXcLVfXtRvYNBo4BjlPV1wFEZBbO+lV/xEluiMhhwC+Bi1T1CXfbdGABcLN7HhMzx++7b9AhZGVo795Bh+DZMd27s2JDxjZvY7LiKUGJSD+c3nm3qmplyvZ9gReAg1O2/V1VL/I1ytwMBr5KJicAVd0gIuXAENwE5ZZLAONTytWLyDhguIi0VdVNrRi3aQW//+EPgw4hK1FKUBcffjhzly9vvqAxzfDaBjUUGJKanFx/Bw4BZgL3AP8FLhCRC3yLsGlPiUiDiKwRkX+KyD4p+w5i20wXqRYA+4hISUq5Japak6FcMbCf71Ebk6XVNTWsrkn/FTUm3rwmqCOB8tQNIvJdnEdob6jqj1X1apxHfR8D5/sa5Y42AKOAi4HjcCamHQDMEpE93DKdcdqd0q113zt5LLfdnC2rVq2iT58+W182JiqaTnrqKU566qmgw/DsjKef5gx37FbYnfvvfzPLevGZZowePXrr9yjQNVMZr21QewKL07b1wxn39Fhyg6rWisg/cTol5I2qvg+8n7Jpuoi8AbyL8+juBrbNqp5OMvzspRwA3bp1Y86cOVnHbMJlUITGFUXNgJ49WVNdHXQYJuTKysooKysDQERWZyrjNUG1BWrTth3pvqdPW7yMRqatyCdVnSsii9kW11rS7n5cyTundSnl9mmi3NoM+0zE/ebII5svZHIy9LDDePfzz4MOw8SA10d8X+C01aQ6BlipqunTGnUA1rcwrlyl3g0tYMeYwVkK5AtVrU4pt6+IdMhQbjPwST4CNcYY0zSvCepN4PzkIFcR+RmwPzApQ9lDgC/9Cc87d4Xf7wDvuJsmAnuJSN+UMrsCg9x9pJQrAn6eUm5n4CzgFevBF08D/vEPBvzjH0GHEUtnPfssM6wNyvjA6yO+24BzgAoRWQN0wbm7GJVaSETa4HTbfs7PINOJyFM445nm4tytfQ9nEO6XwF/dYhOBWcBYERnGtoG6QspUTKpaISLjgXtFpMg972U4A3zPyed1mOCcdVCmm2vjh8G9elFVVxd0GCYGPCUoVV3i3onciNPt+l1gZIZBuf2BNcAEX6Pc0XzgFzidMToAlTjL0N+oqqvdmLeIyKnAXcBDQDuchNU/w2PJC3FWCB4J7A58AAxU1bl5vg4TkEuOOCLoEGLrnEMOYcaSJUGHYWLA80wSqjoH5/FYU2Wm4DziyytVvQ3nrq65cmuBi9xXU+VqgT+4L2OMMSEQ9bn4jMlJlOa2i5oznnmGxasz9ho2JiuWoLJkK+rGQ5SmDoJoxfvzAw/kyfffb76gKXj5WFG3oNmKuvEQpS98iFa8Zx10ENM//TToMEwENLeibtbrQRkTB4mGBhINDUGH4VmU5uJLNDSwJb9LwpkCYXdQpiCd8OSTQHTaoJLz8EUh3l88/7y1QRlfWIIyBeniww8POoSsXHXUUUGH4NkvDj6YMXNthIZpuawTlDsbwwvAVe6krcbkVV1dHYlEIuvjqqqqaGjkMd65hx7a0rBa1aBevYIOwbPTDziAqR9/HHQYJgZyuYMqwpnJvBOAiOyCM3vDX1T1I/9CM8ZJTvePGEFNDkuIb0okWDxvHvWlpdC27Xb7atyE16GoyJc4822R+8isV9eMqxKESm0iQf2WLUGHYWLA64q6zwIzcBYmTJ+FoR1wATAWsARlfJVIJKiprOTSLl0oKS7O6tjK6mqG1dZmvIs62V0LKgptOgCXvvgiEI14z3vhBWuDMr7wege1C/AnnL7qCZwZw88SkY04c9dlXDvJGL+UFBfTMe0uqDlVmxqf5/cyZ5E0kwfnHXooj7/3XtBhmBjw1M1cVU/CWVupN3A9TkL6Jc7cdp/gJKxTReR7IhLrZJUcqOsOLjMRddbBB3PWwQcHHUYsDenVi71KSoIOw0RAeXl5ctHCjAN1PY+DUsc84Al30xDgMOAOnIR1OTAHWCsiL7Yk6DBLDtS1WSSibUNdHRtsxu28+HrTpkiNMTPBGTRoUHLig4wDdb22QU3CaYN6C/jM3ayq+qGIrAD+DJyCs6RFX+DHLYzbmLwaMm4cEI02nai5aOJEa4MyvvDaBpXAWdriZqAB55HeUPdpXrJjRL074/kc0taJMiZsrvjBD4IOIbYu6t2b0bNnBx2GiQGv60ENBhCR/YETgAdwlt44D6jDSVini0gtMEdV6/MTrjH+OO2AA4IOIbZO3n9/Xly4MOgwTAxkNRefqn4MjHd/PAP4LnATThvUUJxu6OtFZIqPMRrjuyjNbRc1a2tr2WRtUMYHLZrqSFUXi8hjOIsHDgZW4AziPbbloRmTP1Ga2y5qyl58sUVtUPUNDVRVVeV8fFFREe3atcv5eBMeuSSoOuDvwFfpO1R1IbAQeLiFcRmTV1Ga2w6iFW/Z4Yfzv+++m9OxdfX1fFBRwYPDhlGU4ywfHUpLuWLECEtSMZB1glLVjcCFKZsaTVjGhFWU5raDaMV74re/zQsLFuR0bKKhgTa1tVzSuTNdchhLVb15M49UVpJIJCxBxUCLZzPPkLBizVbUjYfK6moASiMyoDRKc/Gt3LiRuvqW9ZMqKSrKeuYQEz22oq7PbEXdeDj72WeB6LRBRWkuvt+89JKNgzKeNLeiriUoU5CGH3NM0CFk5dbjjw86BM9+e+SRPPD220GHYWLAEpQpSAP32y/oELJydPfuQYfgWf8ePXhm3rygwzAxkNU4KGPiYtmGDSzbkPGpQijNXLaMmcvSV7oJpy+rqqhtYRuUMWB3UKZAnffvfwPRaNMBuG7qVCAa8f5+8mRrgzK+sARlCtINx9pY8ny54vvf5/5Zs4IOw8SAPeIzBWlAz54M6Nkz6DBi6dhvfYs9OnQIOgwTA3YHZfKurq6ORCKR07FVVVUZl2xvqc/WrQOgZ6dOvp+70H2+fj0bc/zvbUwqS1BZsoG62amrq+P+ESOoqazM6fhNiQSL582jvrQUfBy4edGECUA02nSi5qpXX7U2KOOJDdT1mQ3UzU4ikaCmspJLu3ShpLg46+Mrq6sZVlvr+13UTf36+Xo+s81VRx3FPTNmBB2GiQAbqGtCoaS4OKepa6o2bcpDNNC3R4+8nNfAUXvvzZPt2wcdhomByHWSEJEzROQ5EflcRGpFZJGI3CYiHVPK9BARbeS1e9r52onInSKywj3fLBGxLl4xt2j16q3z2xl/fbJ2LVWbNwcdhomBKN5BXQ18AVwHLAe+B4wA+ovI0aq6JaXsbcDEtOPTF5r5G3AKMAz4DPgt8LKIHKWqFb5Hb0IhSnPbRc3wqVOtDcr4IooJapCqrkr5ebqIrMVZ8qMf8FrKvs9UtdFJwUTkMOCXwEWq+oS7bTqwALgZZxFGE0NRmtsOohXvNT/6EaPeeivoMEwMRC5BpSWnpNnu+15Znm4wkGDbMvaoar2IjAOGi0hbVc1PI4gJVJTmtoNoxXvkN79JZ1uLyfggcm1Qjejrvi9M236biNSLyAYRmSgih6TtPwhYoqo1adsXAMVAtGYUNZ7NX7mS+StXBh2GZ1Gai++j1av5Ok+dW0xhidwdVDoR2QvncdwUVZ3jbt4EPAK8AqwCvovTZjVTRL7vLk0P0BlYl+G0a1P2mxi6/KWXgOi0QUVpLr4bXn/d2qCMLyKdoESkBJgA1JOyqq+qrgB+nVL0TRGZjHNndD1wbvIUgGY6dWOfuWrVKvr06bP157KyMsrKynK9BBOQO084IegQsvLIqacGHYJnN/z4x9zxxhtBh2FCbvTo0aljSjMuFR3ZBCUi7XB66PUE+qrq8qbKq+oyEXkLODJl81pgnwzFO6Xs3063bt2YM2dO+mYTMUfulW1zZbCisNR7Uu/SUjpZG5RpRuof9yKS8ZY7km1QIlIEPAd8HzhZVT/0eijb3zEtAPYVkfSZLQ8ENgOftDRWE04VlZVU5Dj9UhDKFy2ifNGioMPwZP7KlWywNijjg8glKBHZCXgKOB4Y0lQ38rTj9gF+BLyTsnkiUAT8PKXczsBZwCvWgy++rpw8mSsnTw46DM9GzZrFqIgsYTFi+nQ+tDYo44MoPuJ7ECeh3AJsFJEfpuxbrqrLRWQUTvKdhdNJohdwLbAFuDVZWFUrRGQ8cK97V7YEuAzYFzinNS7GBOPegQODDiG2RvTty23TpwcdhomBKCaok9z3691XqptwZpVYgJNohgIdgdU4A3hvUtX05yQX4iS7kcDuwAfAQFWd63/oJix6l5YGHUJsHbzHHuzm48zzpnBFLkGpag8PZR4HHvd4vlrgD+7LFIjZX34JRK+zRBRUVFayrq4u6DBMDEQuQRnjh2GvvgpEY1xR1Ix8800Wr1kTdBgmBiLXSSJoyQUL3UW2TEQ9cPLJPHDyyUGHEUsj+/fn0Ah1izfBKS8vT3Y1twUL/WALFsbDwXvsEXQIsfXdrl3Z1dqgjAfNLVhod1CmIEVpbruomf3VV6y1NijjA7uDMgUpSnPbRc0dM2YE2gZV39BAVVX6sm/eFBUV0c5mwQgNS1CmIEVpbjuIVry3H388N732WvMF86Cuvp4PKip4cNgwioqKsj6+Q2kpV4wYYUkqJCxBmYIUpbntIFrx7te5Mx2LiwP57ERDA21qa7mkc2e6lJRkdWz15s08UllJIpGwBBUSlqBMQZq+dCkAfXv0CDQOr5Lz8A3q1SvgSJo3a/lyVtfWBhpDSVERHa2jRuRZgjIF6cZp04DotEEl5+GLQoIaNWsWi9fusBCAMVmzBGUK0uNDhgQdQlaePfPMoEPwbNQJJ/AntxOKMS1h3cyzZAN146Fnp0707NSp+YIh0bVDB7p2SF8VJpy+tfvu7JJDBwVTeGygrs9soG48TPnsMwAG9OwZcCTejKmoAGBo796BxuHFG59/zsqamqDDMBHQ3EBdS1DGk7q6OhKJRNbHVVVV0dDQkIeIWmakuyS5JSj/3f/uuyxety7oMEwMWIIyzaqrq+P+ESOoyWEF2k2JBIvnzaO+tBRC1KvqyZ/9LOgQYuu+gQO5wZ2M15iWsARlmpVIJKiprOTSLl0oyXJ8S2V1NcNqa0N3F9V9t4yPvI0P9urYkfY721eLaTn7LTKelRQXZz22pGrTpjxF0zKTP/kEgIH77RdwJPHz+tKl/J+1QRkfWIIyBen2t94CLEHlw4OzZ/OxtUEZH1iCMgVp3BlnBB1CbD108skMf/nloMMwMWAJyhSk0iznaTPe7bHLLrSLaBtUS2ZCB5sN3W/R/C0KUHKgbrL/vommKM1tFzWvfPoplRs3Bh1G1lo6EzrYbOjZKi8vT056YAN1/WADdeMhSnPbRc3ouXP5ZP36oMPIWktmQgebDT0XNlDXmAyiNLcdRCve0aeeyrDJk4MOI2c2E3p4WIIyBSkq89olRSnezu3b07ZNm6DDMDFgk8WagvT8woU8v3Bh0GF4NqaiYut0R2H30scf81V1ddBhmBiwOyhTkO5/5x0ATjvggIAj8SZKc/E9XlHBZxsyNikYkxVLUKYgTTj77KBDyEpUFlYEeHzwYK566aWgwzAxYAnKFKTdrJdV3uzati1F1gZlfGBtUKYgjZ8/n/Hz5wcdhmd3zZzJXTNnBh2GJxMWLeJLa4MyPrA7qCxFdaBurus5QXjXdGqJh+fMAeCsgw8OOBJvXly8GICrjz464Eia9+S8eSyxNijjgQ3U9VmQA3VzTTJ1dXU8+pe/sHnVqpw+N6xrOrXES+ecE3QIsfXkT3/K7//zn6DDMBFgA3Vjwo9FA+897jh2z2E8TVjXdGqJDjlOZWOa176oiJ13stYD03KWoCLCj0UD27dpk9MI+bCu6dQSY+fNA+DcQw8NOJL4eW7hQpa1YMJVY5IsQblEpDtwD3ACIMAU4EpV/SLQwNLEadHAID02dy5gCSof/jV/Pp9//XXQYQSiJbOh20zoO7IEBYhIB+A1YBNwAaDASOB1ETlUVX2Zmtk6KoTHq+edF3QIsfWv007jdy++GHQYra6ls6HbTOg7sgTluAToCfRS1U8ARGQe8DFwKXB3Sz+gJW1I0LodFUa/9x5lRxyR188IWlPjdArh+puztAV3QEVt2rCTiI/RBCPb34OWzIZevXkzD375JWvXrqVjx47Zhgrk5w5s9OjRlJWV+XrObFiCcgwG3k4mJwBVXSIiM4AhNJOgysvLm+1ynt6GNOnjjzlp//09B1hZXc15q1dnfRdVvmgRR+y5Z1bH5PIFXb5oUdZLV7TGMY2tS9TU1EFxuv5cj8klQSU/Z/yCBXzh8fgw10EuvweVGzdmPRt6+aJF/HDvvbO6+1q0bBm9unffbltR166UXXNNo0lq0qRJnHTSSY2eM1OCyyVBefk+9MoSlOMgYEKG7QuAnzd3cC7/QSZ98klWCQqcX/7qRIK2WbQpPb9wId/t2hXA87FbVLdrt9roPpZs6vjnFy6kX48eO2xv6tjGjmnqeC/HpB7bWJ39zW2DOj3DXHzJ6/dy3U1dS3PHN3Ut6cc2bNnS7DGNHZ9NnXmJuzHJzxk3fz5fVFXlVG8t+V1r6vhcftfS/z9oTlO/b01J/j/apraWX5aU0MnD3dcf586lLOWPq7U1NVz78svUrFjRaIIrnzmT/zYxBVVx165ckpbgtmzZ4qlNrbi4mLZuUvYzQYmq+nKiKBORzcDdqjo8bftIYLiq7pyyrYrtZ+BYBSSAbEcm7hbiY7oCq1vhc1rjmEK//lyPsTqwOoD81kFXoJv77y2qusOzTbuD2iZTpt7hQXqmSjTGGOM/G03nWAd0zrC9k7vPGGNMK7ME5ViA0w6V7kDgv60cS16JyN4i8lcRmSUiNSKiItIjQ7lOIvKYiKwWkY0iMkVEDgkgZF+JyBki8pyIfC4itSKySERuE5GOaeVief0AIvITEXlNRCpFZJOILBeRp0XkwLRysa2DTERksvv/w8i07bGsBxHp515v+mt9WrnArt8SlGMi8EMR6Znc4H5p/8jdFyf7AWfi3Bm+mamAiAjOdQ8EfgecDhThjAvbu5XizJergQbgOpzrexi4DHhVRHaC2F8/OE8L3gMuB04ErsX5A+1tEfkWFEQdbEdEfgEclmF7IdTDFcBRKa8ByR2BX7+qFvwL2AX4BPgQp1v5YOAD4DOgJOj4fL7WnVL+fTFO21uPtDJD3O39U7btBqwF7g/6Glp4/d0ybDvfvd7j4n79TdRLL/earyq0OgB2ByqBX7jXPDJlX2zrAejnXtuAJsoEev12BwWoM1PEccBi4EngKWAJzhdWrBa2UdUtHooNBr5S1ddTjtsAlOP8wkaWqmaa0n22+76X+x7b62/CGvc9OdVJIdXBX4AFqvqvDPsKqR4yCfT6LUG5VPULVT1dVXdV1Y6q+lNVXRp0XAE5CMi0mt8CYB8RyW6YfPj1dd8Xuu8Fcf0i0kZEikVkf+ARnLuIce7uQqmDY3DuoH/TSJFCqIenRKRBRNaIyD9FZJ+UfYFevyUok0lnMvdeXOu+d2rFWPJKRPYCbgamqOocd3OhXP87OPNPLgYOxXlisNLdF/s6EJEinMR8l6ouaqRYnOthAzAK51H/ccCfcdqfZonIHm6ZQK/fxkGZTASP48KizP3rbwJQD1yYuosCuH7gPGBXnHkor8bpKHKM++SgEOrgGqA9cEsTZWJbD6r6PvB+yqbpIvIG8C5Ox4kbCPj6LUGZTNbS+LgwiMHYMBFph9M7qSfQV1WXp+yO/fUDqGrykeY7IjIJWAoMB35NzOvAfYx1Pc7dQ1sRSZ08r62I7A5UEfN6SKeqc0VkMXCkuynQ67dHfCaTpsaFfRH1jiPuo53ngO8DJ6vqh2lFYn39majqepyerPu5m+JeBz2BdsBYnC/Z5Aucu8l1wCHEvx4ySb1rCvT6LUGZTCYCe4lIsvMAIrIrMIiIjwtzxzo9BRwPDFHVtzMUi+31N0ZEvgF8F/jU3RT3OqgA+md4gZO0+uMk7LjXw3ZEpA/wHZz2SQj4+m2y2AIkIme4/zwe53HOb3AmvV2lqtPdL/G3gO7AMJy/Jq/FaUg/TFWXtX7U/hCRh3Gu+RYgfVW95aq6PM7XDyAi/wbmAvOAr3G+kP4fUAp8X1UXx70OGiMiCtyiqje4P8e2HkQkOZxmLrAe+B7OtdUAh6vq6sCvP+jBYvZq/RfO7Xum17SUMp2Bx3GeQdcAU91fyMDjb+G1L23i+kfE/frda7sGZyaJ9e61LcLpzdYjrVxs66CJutluoG6c6wEn0czD6c2XAJYBo4E9w3L9dgdljDEmlKwNyhhjTChZgjLGGBNKlqCMMcaEkiUoY4wxoWQJyhhjTChZgjLGGBNKlqCMSSEiY9zBmvn+nORy20Pz/Vm5EpERbow9fD7vNBFZ6rFs6OvJ5I8lKBNJItJBRK4UkTdFZK2IJETk/0TkJREZKiKxnghZRHZ3E0i/gONIJjEvrzFBxmqiJ9b/E5t4EpH9gP/gTNEzBbgNWA3sgbOezRM4k1n+MagYW8HuwI3uv6cFFwbP48xZl+oe9/3/pW3/lOy9gbMkRqK5giZ+LEGZSBGR9jhz6PUETlfV59OK3CEiR7JtuQCTR6o6D2e6nK1EZKS7b6wP598C1LX0PCaa7BGfiZqLgV7AqAzJCQBVna2qDyV/FpETRWS8iHwmIrUisl5EXkmdobk5IlIqIve759gkIitF5FUROSGlzFIRmZbhWE/tKCKyk4hcLyJviEiliGwWkS9E5GER6ZJ6PpxJPgFuTHmEtjTtfGeJyFsiUiUiNSLyTspEwemfe62ILBGROhH5UETO8Vo3uRKRb4rIv0RknYhsFJGXReQ7aWV2qLvUbSJyoYgscP+bfC4iGe+aReQyEVnkXt9iEbncPV6DfkxqGmd3UCZqkl+wo7M4ZijOhJf/AJYDe+Ekuqki0l9V32zqYLeTwAzgG+455gC7AD/EeaT4ahaxNKUYZ8bo53BW+t2Icyf4K+AYETlCVTcDC3Een90D/BvnMRvA1rV53LuY64HJwP8AW4CfAc+IyOWq+mDK594N/B7ncdo9OI9KHwQ+8+m6MtnF/by3geuAfd0YJojIwara4OEcv8b5b/I3nIlvz8W5g16uqv9MFhKRa4DbcWbtvg7ogFPPq3y7GpMfQc+oay97ZfMC1gBfZ3nMLhm2fQOn3eqltO1jnP8tttv2Es4s1z/JcJ6dUv69lJQZ4VO293OPH9rMNgHaZzj+V27ZM1O29SBtBvaUfYe7+27NsO8FnCU2Oro/98JJXlOBNmnn2OKep0eW9b0UWNrE/mnuef+Ytn1Yej03U3dfAbunbO+Ak3RmpWzrDNTiPIZsl7K9FGcWbwX6Bf17ba/ML3vEZ6JmV5wvWM9UdWPy3yJS4j4ua8BZlO0HTR0rIp2BgcBkVX05w7m3ZBNLM3Gqqta6n9vG7anXFXjNLdJkrCnOwfni/buIdE194Swy1xE4yi07BCcx3q0pdy2qOhf/7gwz2QLcn7YteZ37ezzHE+qsBAyAqtbg3JGlHn8Czsq5D6tqXUrZSpyFK02I2SM+EzVf43zBeiYi38ZZoPAnOL3fUjU35mk/nC/w97P5zFyJyJnAVTiLxxWl7e7k8TQH4MT8URNlvuG+93TfM5X9L3Cix8/M1lepCcO1xn3vkl64EZkeQa5JO35f931RhrKZtpkQsQRlomY+cKyI9FTVZttIRKQEp61jF+Be4EOgCucv+GuB45o7hfvuZfBuY2U8/X8mIqcB44F3cdpjluH0YGuD05bk9YmHuLGchHOnmMmClLKQOXbJsM0vTbUxef1cL+1U+bwGk2eWoEzUPAcci9PJ4ToP5Y8HvglcpKpPpO5Idoduxsc4X97f81B2LU6bR7qeGbZlch5OQurvPq4CQES+m6FsUwnzY5zHkl+o6sJmPjM5NukAdrwjOaCZY6Mg2duxF9seIZKyzYSYtUGZqHkM59HM1SIyJFMBETlCRH7j/pj8K1vSypyIhzYdVV0LTAJOEpEBGT4r9byLge+KyF4p+9sCv23uc1JiVVL+v3TPf0OGsskee5kS4pPu+60i0iZDzHuk/DjR/cw/pJYVkcNxeihG3avAJuAyEWmX3CgipThtdSbE7A7KRIqq1ojIqTgzSbwgIq/gfAmtAboB/XHamv7iHvIWUAmMcruLLwd649ytfAgc4uFjLwdmApNE5O/AezizG/wAp8faNW65B4CzgSki8r843cbPA2rST9iIZ4HTgddE5B84bVA/xemdth1VXSMinwBni8inwP8BG1W1XFVni8iNwE1AhYg8g9PjbU/gCOBkNzZU9SMRedC9xtdE5DmcbuaXAx/g7c4xtNx6ugm4FZghImNx6rMM5w+KPnh7fGsCYAnKRI6qfiIi3wMuxflCvx4owXnENge4APinW3a9iCQT1u9wfuffw/mS/hUeEpSqLhGRPjjjiU4GzgfW4XyBj04pN8MdUHodcCfwJfCwG9NUD58zTkQ64oxxusv9jHJgONs6EKQ6B2fc0q04X7qfu+VR1ZtF5D3gCuBKnDa4lThteL9PO8/vcZJ4mRv3xzh3ffsT8QQFoKq3icjXONd5O/AFznUKToKqDTA80wRRtT8ejDGFR0T+inOn+E1VXRF0PGZHlqCMMbEmIu3Su7SLyJ44Xeu/UFUvj3lNAOwRnzEm7vqJyJ04U0Itx5mF4xKcx8LDA4zLNMMSlDEm7j7B6U5/Cc4g3jqcdsHbVHVKkIGZptkjPmOMMaFk46CMMcaEkiUoY4wxoWQJyhhjTChZgjLGGBNKlqCMMcaE0v8Hl2CGY2nqmp8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot histogram\n",
    "fig=plt.figure()\n",
    "ax1=plt.subplot(111)\n",
    "ax1.hist(samples, bins = 20, edgecolor='black', linewidth=1.2, color='red',alpha=0.5)\n",
    "ax1.axvline(x=valu[2], color='k', linestyle=':')\n",
    "ax1.axvline(x=valu[0], color='k', linestyle='-.')\n",
    "ax1.axvline(x=valu[1], color='k', linestyle=':')\n",
    "ax1.set_xlabel('Calculated Thing', fontsize = 18)\n",
    "ax1.set_ylabel('$\\#$ Samples', fontsize =18)\n",
    "plt.tick_params(axis ='both',which='both',direction = 'in', top=True, right=True, labelsize = 16, pad=2)\n",
    "plt.minorticks_on()\n",
    "plt.tight_layout()\n",
    "\n",
    "#print out results\n",
    "print(\"Best estimation of thing : \", valu[0]) \n",
    "print(\"Upper error : \", (valu[2] - valu[0]))\n",
    "print(\"Lower error : \", (valu[0] - valu[1])) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "45aa6a4f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}