A3: logaddexp

A3-logaddexp.ipynb

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    "`np.logaddexp`:"
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    "$$\n",
    "\\log \\mathcal{L}(\\theta \\mid x, y) = \\log \\prod_{i=1}^{n}{\\Big[g_i P(y_i \\mid a, b, x, e) + (1-g_i) P(y_i \\mid a, b, x, \\sigma)\\Big]}=\\\\\n",
    "=\\sum_{i=1}^{n}{\\log{\\Big[g_i P(y_i \\mid a, b, x, e) + (1-g_i) P(y_i \\mid a, b, x, \\sigma)\\Big]}}\\\\\n",
    "=\\sum_{i=1}^{n}{\\log{\\Big[L1 + L2\\Big]}}\n",
    "$$\n",
    "$L1,L2$ can be very small, and can exceed the range of normal floating point numbers. If we want precise calculation here, we do the trick:\n",
    "\n",
    "$$\n",
    "\\sum_{i=1}^{n}{\\log{\\Big[L1 + L2\\Big]}}=\\sum_{i=1}^{n}{\\log{\\Big[\\exp(\\log(L1)) + exp(\\log(L2))\\Big]}}\\\\\\\\\n",
    "=np.logaddexp(logL1, logL2).sum()\n",
    "$$\n"
   ]
  }
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