Useful Formulae.ipynb

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    "# Conjugate Normal Distributions (known variance)\n",
    "\n",
    "We are trying to learn about the unknown mean of a normal distribution with known variance.  \n",
    "We choose  a prior distribution is normal with mean $\\mu_{0}$ and variance $\\tau_{0}^2$. \n",
    "We draw $n$ values $y_1,\\ldots, y_n$ from the distribution with known variance $\\sigma^2$.  The posterior distribution\n",
    "$p(\\mu|y_1,\\ldots,y_n)=p(y_1,\\ldots,y_n|\\mu)p(\\mu)$ is again normal.  Let \n",
    "$$\n",
    "\\overline{y}=\\frac{1}{n}\\sum_{i=1}^{n} y_i\n",
    "$$\n",
    "be the sample mean. \n",
    "\n",
    "The posterior variance\n",
    "is\n",
    "$$\\frac{1}{\\tau_1^2}=\\frac{1}{\\tau_0^2}+\\frac{n}{\\sigma^2}$$\n",
    "and the posterior mean is\n",
    "$$\n",
    "\\mu_1=\\frac{\\frac{\\mu_0}{\\tau_0^2}+\\frac{n\\overline{y}}{\\sigma^2}}{\\frac{1}{\\tau_{1}^2}}\n",
    "$$\n",
    "\n",
    "The posterior sampling distribution $\\theta$ is\n",
    "$$\n",
    "p( z |y)=\\int_{\\theta} p(z|\\theta) d\\theta\n",
    "$$\n",
    "is a normal distribution with mean equal to the posterior mean $\\mu_1$ and variance equal to $\\sigma^2+\\tau_1^2$\n",
    "where $\\tau_1$ is the posterior variance.\n",
    "\n",
    "See Pages 39-42 of BDA (Section 2.5) for more information."
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    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "def posterior(prior_mean,prior_variance,sample_mean,pop_variance,n):\n",
    "    post_var=1/((1/prior_variance) + n/pop_variance)\n",
    "    post_mean=(prior_mean/prior_variance+sample_mean*n/pop_variance)/(1/post_var)\n",
    "    return post_mean, post_var\n",
    "\n",
    "def post_sample(y,prior_mean,prior_variance,sample_mean,pop_variance,n):\n",
    "    post_mean,post_var=posterior(prior_mean,prior_variance,sample_mean,pop_variance,n)\n",
    "    return norm.pdf(y,post_mean,np.sqrt(pop_variance+post_var))\n",
    "    "
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