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    "# Solutions for Chapter 5. Bayesian Statistics\n",
    "+ date: 2017-05-08\n",
    "+ tags: mlapp, statistics\n",
    "+ author: Peijun Zhu\n",
    "+ status: draft\n",
    "+ slug: mlapp-ch5-sol"
   ]
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    "## 5.1 Proof that a mixture of conjugate priors is indeed conjugate\n",
    "$$p(\\theta) =\\sum_k p(z = k)p(\\theta|z = k)$$\n",
    "\n",
    "\\begin{align}\n",
    "p(\\theta|D)&=\\frac{p(\\theta, D)}{p(D)}\\\\\n",
    "&=\\frac{\\sum_k p(\\theta, D,z=k)}{p(D)}\\\\\n",
    "&=\\frac{\\sum_k p(D,z=k)p(\\theta|D,z=k)}{p(D)}\\\\\n",
    "&=\\sum_k p(z=k|D)p(\\theta|D,z=k)\\\\\n",
    "p(z=k|D)&=\\frac{p(D,z=k)}{p(D)}\\\\\n",
    "&=\\frac{p(D,z=k)}{\\sum_i p(D,z=i)}\\\\\n",
    "&=\\frac{p(z=k)p(D|z=k)}{\\sum_i p(z=i)p(D|z=i)}\n",
    "\\end{align}"
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