{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "79bc2499",
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy import stats\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f1356c7",
   "metadata": {},
   "source": [
    "Parameter Estimation\n",
    "======================\n",
    "\n",
    "Assume we are given a samples from a Gaussian:\n",
    "\n",
    "$$p(x) = \\sigma^{-1} (2\\pi)^{-1/2} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$\n",
    "\n",
    "You already know the formulas for estimating means and variances of Gaussians:\n",
    "\n",
    "$$\\mu \\approx \\hat{\\mu} = \\bar{x} = {1 \\over n} \\sum_{k=1}^n x_k $$\n",
    "\n",
    "$$\\sigma^2 \\approx \\hat{\\sigma}^2 = \\bar{(x-\\bar{x})^2} = {1 \\over n} \\sum_{k=1}^n (x_k-\\bar{x})$$\n",
    "\n",
    "We're asking the following questions here:\n",
    "\n",
    "- Where do these formulas come from?\n",
    "- Are they \"correct\"?\n",
    "- Are they the whole story?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad4587c7",
   "metadata": {},
   "source": [
    "The $x_k$ above are an i.i.d. sample from the original sample density:\n",
    "\n",
    "$$D=\\{x_1,\\ldots,x_n\\}$$\n",
    "\n",
    "We want to estimate the parameters of the density $p(x)$.  \n",
    "Let's call the parameters generically $\\theta$.\n",
    "For the univariate normal density, $\\theta = (\\mu,\\sigma)$.\n",
    "\n",
    "$$p(x_i) = p(x,\\theta) = p(x|\\theta)$$\n",
    "\n",
    "Now we can write a density for the entire dataset:\n",
    "\n",
    "$$ p(D|\\theta) = p(x_1,\\ldots,x_n|\\theta) = p(x_1|\\theta)\\cdot\\ldots\\cdot p(x_n|\\theta) = \\prod_{i=1}^n p(x_i|\\theta)$$\n",
    "\n",
    "We call $\\ell(\\theta) = p(D|\\theta)$ the _likelihood_ of the data.\n",
    "Note that likelihoods are parameterized densities viewed as functions of the parameters.\n",
    "\n",
    "Maximum Likelihood Estimate\n",
    "============================\n",
    "\n",
    "The _maximum likelihood estimate_ is given by:\n",
    "\n",
    "$$\\hat{\\theta} = \\arg\\max_\\theta ~ p(D|\\theta)$$\n",
    "\n",
    "This seems like a reasonable thing to do: choose the parameter that was\n",
    "most likely to produce the data set.\n",
    "\n",
    "However, in general, there is little reason why the maximum of the likelihood function\n",
    "should mean anything.\n",
    "For example, we can easily modify the likelihood function to put a tiny spike in it\n",
    "that moves the maximum somewhere arbitrary without actually changing the problem much at all.\n",
    "\n",
    "Maximum A Posteriori Estimate\n",
    "=============================\n",
    "\n",
    "By analogy to Bayesian methods, we can also ``multiply in'' a prior:\n",
    "\n",
    "$$\\hat{\\theta} = \\arg\\max_\\theta ~ p(D|\\theta) p(\\theta)$$\n",
    "\n",
    "This is called the _maximum a-posterior estimate_ (MAP) for the parameter $\\theta$.\n",
    "\n",
    "MAP vs Bayesian Methods\n",
    "========================\n",
    "\n",
    "Although MAP looks like the derivation may have involved _Bayes rule_, it is\n",
    "not a Bayesian method at all.\n",
    "Bayesian methods are *not* methods that involve Bayes rule somewhere.\n",
    "\n",
    "**Bayesian methods are methods that result in decisions that minimize expected loss.**\n",
    "\n",
    "In order to minimize expected loss, you actually need a loss function.\n",
    "Maximum likelihood or MAP estimation does not attempt to minimize a loss function,\n",
    "hence it is not a Bayesian method.\n",
    "For individual distributions, ML or MAP estimation may implicitly optimize some loss\n",
    "function, but that loss function may depend on the distribution, sample size, and parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0d39ee6",
   "metadata": {},
   "source": [
    "Maximum Likelihood Estimation of the Mean of a Normal Density\n",
    "=========================================\n",
    "\n",
    "Let's derive a maximum likelihood estimator, for the mean of a Gaussian\n",
    "we want to maximize the likelihood function:\n",
    "\n",
    "$p(x_1,...,x_n|\\mu)=\\prod_i p(x_i|\\mu)$\n",
    "\n",
    "Let's take logarithms\n",
    "\n",
    "$$l(\\mu) = \\sum_i p(x_i|\\mu) = \\sum -{1\\over 2} \\ln((2\\pi)^d |\\Sigma|) - {1\\over 2} (x_i-\\mu)\\Sigma^{-1}(x_i-\\mu)$$\n",
    "\n",
    "If $l$ is sufficiently well behaved (it is), then a necessary condition for a local maximum is that the gradient in the parameter to be estimated is 0\n",
    "\n",
    "$$\\nabla_\\mu l(\\mu) = 0$$\n",
    "\n",
    "The gradient is\n",
    "\n",
    "$$\\nabla_\\mu l(\\mu) = \\sum_i \\Sigma^{-1} (x_i-\\mu)$$\n",
    "\n",
    "So,\n",
    "\n",
    "$$\\nabla_\\mu l(\\hat{\\mu}) = \\sum_i \\Sigma^{-1} (x_i-\\hat{\\mu}) = 0$$\n",
    "\n",
    "Multiply by $\\Sigma$\n",
    "\n",
    "$$\\sum_i (x_i-\\hat{\\mu}) = 0$$\n",
    "\n",
    "$$\\sum_i x_i = n\\cdot\\hat{\\mu}$$\n",
    "\n",
    "$$\\hat{\\mu} = {1 \\over n} \\sum_{i=1}^n x_i$$\n",
    "\n",
    "This is nice because it says that the maximum likelihood estimate of the mean of a Gaussian is just the arithmetic mean of the samples (the empirical mean or sample mean).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a657f6f3",
   "metadata": {},
   "source": [
    "Example\n",
    "=======\n",
    "\n",
    "Let's try this out in 1D.\n",
    "\n",
    "We assume that $\\sigma=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b54745f2",
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "D = randn(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "977ca0bd",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAD9CAYAAACoXlzKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mus = linspace(-5,5,10000)\n",
    "l_of_mu = [-(x-mus)**2 for x in D]\n",
    "for l in l_of_mu: plot(mus,l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "46f40547",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.398539853985"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAD9CAYAAABQvqc9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nboRFn4jIjbDoExG5kf8Dro5HxweiOHUAAAAASUVORK5CYII=\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "l = sum(array(l_of_mu),0)\n",
    "plot(mus,l)\n",
    "print mus[argmax(l)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "064f3c74",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.39899698828239921"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a941c492",
   "metadata": {},
   "source": [
    "ML estimator of Mean and Variance\n",
    "=================================\n",
    "\n",
    "Consider the 1D Gaussian, but assume that we want to estimate both $\\mu$ and $\\sigma^2$\n",
    "\n",
    "Let's use the variance $v=\\sigma^2$\n",
    "\n",
    "The log likelihood is\n",
    "$$\\def\\pd#1{\\frac{\\partial}{\\partial#1}}$$\n",
    "\n",
    "$$l(\\mu,\\sigma^2) = \\sum_i^n \\ln p(x_i|\\mu,v) = -{1\\over2} \\ln 2\\pi v - {1 \\over {2v}} (x_i - \\mu)^2$$\n",
    "\n",
    "As before, to find a potential maximum, \n",
    "we calculate the derivatives of the log likelihood function and set them equal to zero\n",
    "\n",
    "$$\\pd{\\mu} l =  \\sum_i^n {1 \\over {\\hat{v}}} (x_i - \\hat{\\mu}) = 0$$\n",
    "\n",
    "$$\\pd{v} l =  \\sum_i^n - {1\\over {2\\hat{v}}} + \\frac{(x_i-\\hat{\\mu})^2}{2\\hat{v}^2} = 0$$\n",
    "\n",
    "So, from the first, we obtain again that\n",
    "\n",
    "$$\\hat{\\mu} = {1 \\over n} \\sum_i^n x_i$$\n",
    "\n",
    "For the second, we get\n",
    "\n",
    "$$\\pd{\\hat{v}} l =  \\sum_i^n - {1\\over {2\\hat{v}}} + \\frac{(x_i-\\hat{\\mu})^2}{2\\hat{v}^2} = 0$$\n",
    "\n",
    "Multiply by $v^2$, we get\n",
    "\n",
    "$$ n v  = \\sum_i^n (x_i-\\hat{\\mu})^2 $$\n",
    "\n",
    "Or\n",
    "\n",
    "$$ \\sigma^2 = v = \\frac{1}{n} \\sum_i^n (x_i-\\hat{\\mu})^2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ad48f03",
   "metadata": {},
   "source": [
    "Bias\n",
    "====\n",
    "\n",
    "Above, we calculated the maximum likelihood estimate of the variance.\n",
    "\n",
    "Wouldn't it be nice if the variance were actually, on average, the same a the true variance?\n",
    "\n",
    "Let's see whether that is true.\n",
    "\n",
    "The expected value of the variance is\n",
    "\n",
    "$$E[\\frac{1}{n}\\sum_i^n (x_i-\\bar{x})]$$\n",
    "\n",
    "Consider the case $n=1$.\n",
    "\n",
    "Then $\\bar{x} = \\hat{\\mu} = x_1$, so\n",
    "\n",
    "$$E[\\frac{1}{1}\\sum_{i=1}^1 (x_1-x_1)] = 0$$\n",
    "\n",
    "Well, $0 \\neq \\sigma^2$\n",
    "\n",
    "Tt turns out that an unbiased estimator for variance is\n",
    "\n",
    "$$E[\\frac{1}{n-1}\\sum_i^n (x_i-\\bar{x})]$$\n",
    "\n",
    "Note that this is undefined for $n=1$, so the above problem doesn't exist.\n",
    "\n",
    "Some terminology:\n",
    "\n",
    "- If an estimator is not completely unbiased but becomes approximately unbiased \n",
    "for large $n$, we call it {\\em asymptotically unbiased}\n",
    "- If an estimator is unbiased for all distributions, it is called {\\em absolutely unbiased}.\n",
    "\n",
    "So, we now have two different estimators for the variance:\n",
    "\n",
    "- the first one maximizes the likelihood\n",
    "- the second one has an expected value that equals the true value\n",
    "\n",
    "Which one should we choose?\n",
    "\n",
    "There is no correct answer; the question itself is wrong.\n",
    "\n",
    "The reason is that our knowledge about the true variance is uncertain.\n",
    "Any single value we estimate for the variance is incomplete \n",
    "without a statement of the uncertainty associated with that value\n",
    "\n",
    "The point here is not that you should pick unbiased estimators (you may still choose\n",
    "to do so if you have nothing better to do), rather it is to illustrate that\n",
    "there are many choices in statistics that sound good on the surface but really \n",
    "don't accomplish anything useful."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4aaa3f97",
   "metadata": {},
   "source": [
    "Bayesian Parameter Estimation\n",
    "=============================\n",
    "\n",
    "In Bayesian parameter estimation, we need to decide what purpose we want \n",
    "to use the estimate for.\n",
    "\n",
    "For example, we might want to estimate the parameters of a normal\n",
    "density in order to perform classification.\n",
    "\n",
    "In a Bayesian approach, we use all available data:\n",
    "\n",
    "$$p(\\omega_j|x) = p(\\omega_j|x,D)$$\n",
    "\n",
    "In order to perform estimation, we consider the case where...\n",
    "\n",
    "- the form of the density $p(x|\\theta)$ is known (parametric estimation)\n",
    "- we assume that samples for each class can only give us information about that class\n",
    "- assume we have a prior density $p(\\theta)$ for the parameters\n",
    "\n",
    "Then\n",
    "\n",
    "$$p(x|D) = \\int p(x|\\theta) p(\\theta|D) d\\theta$$\n",
    "\n",
    "Now by Bayes formula,\n",
    "\n",
    "$$p(\\theta|D) = \\frac{p(D|\\theta)p(\\theta)}{p(D)}$$\n",
    "\n",
    "and by independence\n",
    "\n",
    "$$p(D|\\theta) = \\prod_{k=1}^n p(x_k|\\theta)$$\n",
    "\n",
    "Note that the maximum likelihood case is equivalent to this if we choose \n",
    "\n",
    "$p(\\theta|D)=\\delta(\\theta,\\hat{\\theta})$\n",
    "\n",
    "$$p(x|\\omega_j,D) = \\int p(x|\\theta) \\delta(\\theta,\\hat{\\theta}) d\\theta$$\n",
    "\n",
    "Whether the maximum likelihood solution works well therefore depends on how\n",
    "closely $p(\\theta|D)$ approximates a $\\delta(\\theta,\\hat{\\theta})$ \n",
    "(note that \"how closely\" depends also on the smoothness of $p(x|\\theta)$)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01147fb1",
   "metadata": {},
   "source": [
    "Parameter Updates\n",
    "=================\n",
    "\n",
    "Above, we have seen how to compute distributions of parameters given\n",
    "some samples.\n",
    "\n",
    "In many cases, it turns out that the distribution of parameters itself\n",
    "has a parameteric form closely related to the parametric form of the\n",
    "distribution we are trying to estimate.\n",
    "\n",
    "For example, the distribution of the estimates of a mean of a Gaussian\n",
    "is itself a Gaussian."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c29c843c",
   "metadata": {},
   "source": [
    "For now, let's look at a particularly simple case.\n",
    "Assume we are given $p(\\theta)$ and a measurement $x$\n",
    "we want to compute $p(\\theta|x)$, the distribution of the parameter values\n",
    "assuming we are given a measurement.\n",
    "\n",
    "So\n",
    "$$p(\\theta|x) = \\frac{p(x|\\theta) p(\\theta)}{p(x)}$$\n",
    "\n",
    "Here,\n",
    "$$p(x) = \\int p(x|\\theta) p(\\theta) dx$$\n",
    "\n",
    "Let's look at this in the case of estimating the probability parameter \n",
    "of a binomial distribution.\n",
    "\n",
    "For a given $q$, we have $p(k) = {n \\choose k} \\theta^k (1-\\theta)^{n-k}$\n",
    "(the parameter $\\theta$ is just the probabilit; I'm avoiding writing $p$ because we have too many $p$'s already)\n",
    "since, initially, we know nothing about $\\theta$, let's assume a uniform prior\n",
    "\n",
    "$p(\\theta) = 1$ \n",
    "\n",
    "this is conveniently normalized, since $\\theta\\in[0,1]$\n",
    "\n",
    "So, what do we get?\n",
    "\n",
    "Well, our measurement is $x=k$, so let's replace that:\n",
    "\n",
    "$$p(\\theta|k) = \\frac{p(k|\\theta) p(\\theta)}{p(k)}$$\n",
    "$$ = \\frac{{n\\choose k} \\theta^k (1-\\theta)^{n-k}}{\\int {n\\choose k} \\theta^k (1-\\theta)^{n-k}}$$\n",
    "$$ = \\frac{ \\theta^k (1-\\theta)^{n-k}}{\\int \\theta^k (1-\\theta)^{n-k}}$$\n",
    "\n",
    "The denominator is called a beta function. It serves as a normalizing factor.\n",
    "\n",
    "We can write\n",
    "$$ p(\\theta|k) = \\frac{ \\theta^k (1-\\theta)^{n-k}}{B(k,n-k)}$$\n",
    "\n",
    "This is the density function for the beta distribution\n",
    "\n",
    "Note that the beta FUNCTION and the beta DISTRIBUTION are different things\n",
    "we will be writing $B(.)$ for the beta function and $\\beta(.)$ for the beta distribution\n",
    "the beta function is the normalizing factor for the beta distribution.\n",
    "\n",
    "The formula for the beta distribution looks like the binomial distribution, with some\n",
    "of the parameters taking on different meanings."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b03cc0c9",
   "metadata": {},
   "source": [
    "Incremental Updates and Conjugate Priors\n",
    "========================================\n",
    "\n",
    "Above, we looked at $p(\\theta|D)$ for a single data set $D$ (observation of $k,n$ in this case)\n",
    "\n",
    "What if we make a second observation $k_2,n_2$?\n",
    "\n",
    "You can observe that having $n_1$ Bernoulli trials with $k_1$, giving\n",
    "parameter estimate $\\theta_1$, followed by $n_2$ Bernoulli trials with $k_2$ successes, \n",
    "giving parameter estimate $\\theta_2$ is the same as if we had just performed a single\n",
    "experiment with $n_1+n_2$ trials and $k_1+k_2$ successes\n",
    "\n",
    "That means that the parameter distribution after the first update should give\n",
    "$p(\\theta|k_1,n_1) = \\beta(\\theta;k_1,n_1)$\n",
    "\n",
    "And after the second update\n",
    "$p(\\theta|k_1,n_1;k_2,n_2) = p(\\theta|k_1+k_2,n_1+n_2) = \\beta(\\theta;k_1+k_2,(n_1+n_2)-(k_1+k_2))$\n",
    "\n",
    "How does this look when we update the parameters directly?\n",
    "\n",
    "We have again\n",
    "$$p(\\theta|x) = \\frac{p(x|\\theta) p(\\theta)}{p(x)}$$\n",
    "\n",
    "Here,\n",
    "$$p(x) = \\int p(x|\\theta) p(\\theta) dx$$\n",
    "\n",
    "and our prior is not uniform anymore, but the Beta distribution that we obtained\n",
    "from the first experiment\n",
    "$p(\\theta) = \\beta(\\theta;k_1,n_1-k_1)$\n",
    "\n",
    "So, writing $b$ for the binomial probability,\n",
    "\n",
    "$$p(k_2,n_2) = \\frac{b(k_2;n_2,\\theta) \\beta(\\theta;k_1,n_1)}{\\int b(k_2;n_2,\\theta) \\beta(\\theta;k_1,n_1) d\\theta}$$\n",
    "\n",
    "Now, recall that the binomial probabilities are\n",
    "$$b(k;n,\\theta) = {n \\choose k} \\theta^k (1-p)^{n-k}$$\n",
    "\n",
    "And the Beta distribution is\n",
    "$$\\beta(\\theta;a,b) = \\frac{1}{B(a,b)} \\theta^{a-1} (1-\\theta)^{b-1}$$\n",
    "\n",
    "So, let $k=a-1$ and $n-k=b-1$, or $a = k+1$ and $b=n-k+1$:\n",
    "$$b(k;n\\theta) = \\beta(\\theta;k+1,n-k+1) B(k+1,n-k+1)$$\n",
    "\n",
    "We can plug that back in:\n",
    "$$p(k_2,n_2) = \\frac{\\beta(\\theta;k_2+1,n_2-k_2+1) \\beta(\\theta;k_1,n_1)} {\\int \\beta(\\theta;k_2+1,n_2-k_2+1) \\beta(\\theta;k_1,n_1) d\\theta}$$\n",
    "\n",
    "What about the products of beta distributions?\n",
    "$$\\beta(\\theta;a,b)\\beta(\\theta;a',b') = \\frac{1}{B(a,b)B(a',b')} \\theta^{a+a'-2} (1-\\theta)^{b+b'-2} = \\frac{B(a+a',b+b')}{B(a,b)B(a',b')} \\beta(a+a'-1,b+b'-1)$$\n",
    "\n",
    "So\n",
    "$$p(k_2,n_2) = \\frac{C\\cdot\\beta(\\theta;k_1+k_2,(n_1+n_2)-(k_1+k_2))} {\\int C\\cdot\\beta(\\theta;k_1+k_2,(n_1+n_2)-(k_1+k_2)) d\\theta}$$\n",
    "\n",
    "Pull out the $C$ from the integral and cancel it and we're left with an integral \n",
    "over a $\\beta$ distribution,\n",
    "which is 1, so\n",
    "$$p(k_2,n_2) = \\beta(\\theta;k_1+k_2,(n_1+n_2)-(k_1+k_2))$$\n",
    "\n",
    "So, what we have in this case is that starting with a uniform density as a prior,\n",
    "every time we take another binomial sample, we get another beta distribution as the\n",
    "density for the parameter we are trying to estimate.\n",
    "\n",
    "We say that the beta distribution is a _conjugate prior_ for the binomial distribution,\n",
    "and we refer to the density as a _reproducing density_"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b17cbee9",
   "metadata": {},
   "source": [
    "Updates to the Normal Density\n",
    "===============================\n",
    "\n",
    "Let's repeat this exercise for the normal density\n",
    "the parameter we are trying to estimate is $\\mu$\n",
    "assume our prior is itself normal \n",
    "\n",
    "$p(\\mu) = N(\\mu;\\mu_0,\\sigma_0)$\n",
    "\n",
    "Furthemore, we assume that\n",
    "\n",
    "$p(x) = N(x;\\mu,\\sigma)$\n",
    "\n",
    "Where $\\mu$ is unknown, but $\\sigma$ is known and given\n",
    "\n",
    "Now, given a dataset $D = \\{x_1,...,x_n\\}$,\n",
    "$$p(\\mu|D) = \\frac{p(D|\\mu)p(\\mu)}{\\int p(D|\\mu)p(\\mu)d\\mu} = C \\prod_i^n N(x_i;\\mu,\\sigma) N(\\mu;\\mu_0,\\sigma_0)$$\n",
    "\n",
    "Dropping the constant factors, we get\n",
    "$$p(\\mu|D) \\propto\\prod_i^n \\exp(\\frac{-1}{2\\sigma^2}(x_i-\\mu)^2) \\exp(\\frac{-1}{2\\sigma_0^2}(\\mu-\\mu_0)^2)$$\n",
    "$$= \\prod_i^n \\exp(\\frac{-1}{2\\sigma^2}(x_i-\\mu)^2 + \\frac{-1}{2\\sigma_0^2}(\\mu-\\mu_0)^2)$$\n",
    "$$= \\exp(\\sum_i (\\frac{-1}{2\\sigma^2}(x_i-\\mu)^2 + \\frac{-1}{2\\sigma_0^2}(\\mu-\\mu_0)^2))$$\n",
    "\n",
    "You can work out that this is equal to a quadratic function for some parameters $a,b,c$:\n",
    "$$= \\exp(\\sum_i a\\mu^2 + b\\mu + c)$$\n",
    "\n",
    "Furthermore, we know that the output must be a density, so the quadratic must be $\\geq 0$\n",
    "what all of that means is that, for some $\\mu_1,\\sigma_1$,\n",
    "$p(\\mu|D) = N(\\mu;\\mu_1,\\sigma_1)$\n",
    "\n",
    "We can compute  $\\mu_1,\\sigma_1$, and eventually get\n",
    "$$\\mu_1 = \\frac{n\\sigma_0^2}{n\\sigma_0^2+\\sigma^2}\\hat{\\mu}_1 + \\frac{\\sigma^2}{n\\sigma_0^2+\\sigma^2}\\mu_0$$\n",
    "$$\\sigma_1^2 = \\frac{\\sigma_0^2\\sigma^2}{n\\sigma_0^2+\\sigma^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ed29b72",
   "metadata": {},
   "source": [
    "Incremental Updates in General\n",
    "===============================\n",
    "\n",
    "Recall\n",
    "$$p(x|D) = \\int p(x|\\theta) p(\\theta|D) d\\theta$$\n",
    "\n",
    "Recall now by Bayes formula,\n",
    "$$p(\\theta|D) = \\frac{p(D|\\theta)p(\\theta)}{p(D)}$$\n",
    "\n",
    "where $p(D) = \\int p(D|\\theta)p(\\theta)d\\theta$\n",
    "and by independence\n",
    "$$p(D|\\theta) = \\prod_{k=1}^n p(x_k|\\theta)$$\n",
    "\n",
    "Now write $D^n$ for the first $n$ samples\n",
    "\n",
    "Then, trivially\n",
    "$$p(D^n|\\theta) = p(x_n|\\theta) p(D^{n-1}|\\theta)$$\n",
    "\n",
    "Substituting that above, we get\n",
    "$$p(\\theta|D^n) = \\frac{p(x_n|\\theta) p(D^{n-1}|\\theta) p(\\theta)}{p(D^n)}$$\n",
    "where $p(D) = \\int {p(x_n|\\theta) p(D^{n-1}|\\theta) p(\\theta)}d\\theta$\n",
    "\n",
    "Applying Bayes formula and canceling the $p(D^{n-1})$ factor, we get\n",
    "\n",
    "$$p(\\theta|D^n) = \\frac{p(x_n|\\theta) p(\\theta|D^{n-1})} {\\int p(x_n|\\theta) p(\\theta|D^{n-1}|\\theta)d\\theta}$$\n",
    "\n",
    "This formula tells us how to update our density for the parameter $\\theta$ when a new sample $x_n$ arrives\n",
    "\n",
    "Note that, up to normaliation:\n",
    "\n",
    "$$p(\\theta|D^n) \\propto p(x_n|\\theta) p(\\theta|D^{n-1})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2250ee8d",
   "metadata": {},
   "source": [
    "Incremental Update Example\n",
    "===========================\n",
    "\n",
    "Assume we know that $x \\sim {\\cal N}(\\mu)$, where $p(\\mu) = {\\cal N}(1,1)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "id": "7b3d0718",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99899944201334256"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXgAAAD9CAYAAAC2l2x5AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = linspace(-4,6,1000)\n",
    "C = (amax(xs)-amin(xs))/xs.size\n",
    "pmu0 = norm.pdf(xs,loc=1)\n",
    "plot(xs,pmu0)\n",
    "C*sum(pmu0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94e09541",
   "metadata": {},
   "source": [
    "Now we obtain a training sample $x=2$ and update our estimate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "id": "086da9c5",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x14e27990>]"
      ]
     },
     "execution_count": 198,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAD9CAYAAAChtfywAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nkcqxyImIVI5FTkSkcixyIiKV+38SQ/ZFQAf2YgAAAABJRU5ErkJggg==\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu1 = norm.pdf(xs,loc=2.0)*pmu0\n",
    "pmu1 /= C*sum(pmu1)\n",
    "plot(xs,pmu1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "299d7c65",
   "metadata": {},
   "source": [
    "We can also compute the map estimate $\\hat{\\mu}$ for the parameter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "id": "bba331d3",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.4954954955"
     ]
    }
   ],
   "source": [
    "mumap = xs[argmax(pmu1)]\n",
    "print mumap"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52c84205",
   "metadata": {},
   "source": [
    "Let's now compute $p(x|D)$ and compare it with $p(x|\\hat{\\mu})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "id": "cf02c73c",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x149c7d10>]"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXgAAAD9CAYAAAC2l2x5AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ndkoSvBBC2ClJ8EIIYackwQshhJ36f7W5GgqEgr9NAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "for i,mu in enumerate(xs):\n",
    "    weight = pmu1[i]\n",
    "    result += weight * norm.pdf(xs,loc=mu)\n",
    "result /= C*sum(result)\n",
    "plot(xs,result)\n",
    "plot(xs,norm.pdf(xs,loc=muml))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbd2beec",
   "metadata": {},
   "source": [
    "Finally, let's compute posteriors using the MAP and Bayesian estimates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "id": "4bb8d132",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x14bfe4d0>]"
      ]
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAD9CAYAAAChtfywAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nqchFRBzc/wPKKg/CrGFxyAAAAABJRU5ErkJggg==\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = norm.pdf(xs,loc=0)\n",
    "p2map = norm.pdf(xs,loc=mumap)\n",
    "p2b = result\n",
    "plot(xs,p2map/(p1+p2map))\n",
    "plot(xs,p2b/(p1+p2b))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7e17c1d",
   "metadata": {},
   "source": [
    "As you can see, the MAP-based estimate underestimates the error rate for smaller values of $x$.\n",
    "\n",
    "In addition, it results in different decision boundaries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "id": "544f5e80",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.754754754755 0.844844844845"
     ]
    }
   ],
   "source": [
    "dbmap = xs[find(p2map/(p1+p2map)>0.5)[0]]\n",
    "dbb = xs[find(p2b/(p1+p2b)>0.5)[0]]\n",
    "print dbmap,dbb"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "637fc3aa",
   "metadata": {},
   "source": [
    "Bayesian vs Maximum Likelihood Example\n",
    "==============================\n",
    "\n",
    "Let's apply this to another, simple example and compare maximum likelihood and bayesian approaches\n",
    "~\n",
    "Assume that the samples $x$ come from a uniform density over the interval $[0,\\theta]$\n",
    "\n",
    "$$p(x|\\theta) = U(x;0,\\theta) = 1/\\theta \\cdot \\lfloor x\\in[0,\\theta]\\rfloor$$\n",
    "\n",
    "We also assume a prior\n",
    "\n",
    "$$p(\\theta) = U(\\theta;0,10)$$\n",
    "\n",
    "That is, $\\theta$ is distributed uniformly over the interval $[0,10]$\n",
    "\n",
    "Let's assume we see a sequence of training examples $D = \\\\{4,7,2\\\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "id": "fcd47dbe",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x178d9490>]"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD9CAYAAABdoNd6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nwCQKlVvtYlMAAAAASUVORK5CYII=\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = linspace(-1,11,1000)\n",
    "mus = xs\n",
    "C = (amax(xs)-amin(xs))/len(xs)\n",
    "def pxt(x,mu): return (xs>=0)*(x<=mu)*1.0/maximum(mu,1e-6)\n",
    "plot(xs,pxt(xs,5.5))\n",
    "pmu = (xs>=0)*(xs<=10)*0.1\n",
    "plot(xs,pt)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f016891",
   "metadata": {},
   "source": [
    "Now assume we draw the sample $x_1=4$.  How should we update our estimate?\n",
    "\n",
    "$$p(\\mu|x) = \\frac{p(x|\\mu) p(\\mu)}{p(x)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "id": "a88a213a",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x18290f90>]"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD9CAYAAABdoNd6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ne4myAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu1 = pxt(4,mus)*pmu\n",
    "pmu1 /= C*sum(pmu1)\n",
    "plot(xs,pmu1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f5639af",
   "metadata": {},
   "source": [
    "The maximum likelihood estimate is clearly at $\\mu=4$.  If we now plug this\n",
    "into $p(x|\\mu)$ we get..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "id": "aec6e8ae",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1829cd10>]"
      ]
     },
     "execution_count": 229,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD9CAYAAABDaefJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAEzBJREFUeJzt3H9sXXX9x/HXXe41kqHb+LHC7r1LhXvpvWVbi+msSDDX\njdKlgYtuIymKLKXMpmQBjCYmJmpr4ljlD12sfxTD+KFQBgnaabpr7OQiMtuCm5uBBQrScFu0Scm6\nQSp2uzvfP9D7pWt7dk5/fXY/ez6SG3p3z+55p2FPzt49l4DjOI4AAFZYYnoAAMD8IeoAYBGiDgAW\nIeoAYBGiDgAWIeoAYBHXqN99990qKSnR2rVrZzzmvvvuUzweV0VFhQ4fPjzvAwIAvHONekNDgzKZ\nzIyvd3d3680339TAwIAefvhhNTc3z/uAAADvXKN+4403asWKFTO+vm/fPm3btk2SVF1drbGxMY2M\njMzvhAAAz4Jz+c3Dw8OKRqOF55FIRENDQyopKZl0XCAQmMtpAOCC5fdD/3P+QenZJ5wp4I7jFO3j\nBz/4gfEZLtT5i3l25jf/KPb5Z2NOUQ+Hw8rlcoXnQ0NDCofDc3lLAMAczCnq6XRaTzzxhCSpt7dX\ny5cvn7J6AQAsHted+h133KEXXnhBo6Ojikajam1t1alTpyRJTU1NqqurU3d3t2KxmJYuXapHH310\nUYZebKlUyvQIc1LM8xfz7BLzm1bs889GwJnt4sbPSQKBWe+HAOBCNZt28olSALAIUQcAixB1ALAI\nUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcA\nixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1\nALAIUQcAixB1ALAIUQcAixB1ALAIUQcAixB1ALDIOaOeyWSUSCQUj8fV1tY25fXR0VFt2rRJlZWV\nWrNmjR577LGFmBMA4EHAcRxnphfz+bzKysrU09OjcDis9evXq7OzU8lksnBMS0uL/vOf/+jBBx/U\n6OioysrKNDIyomAw+P8nCQTkchoAwDRm007XK/X+/n7FYjGVlpYqFAqpvr5eXV1dk4658sordfLk\nSUnSyZMndemll04KOgBg8bjWd3h4WNFotPA8Eomor69v0jHbt2/Xhg0btGrVKr3//vt65plnpn2v\nlpaWwtepVEqpVGr2UwOAhbLZrLLZ7JzewzXqgUDgnG+wc+dOVVZWKpvN6q233lJNTY2OHDmiT33q\nU5OO+3jUAQBTnX3B29ra6vs9XNcv4XBYuVyu8DyXyykSiUw65uDBg7r99tslSVdffbU+85nP6PXX\nX/c9CABg7lyjXlVVpYGBAQ0ODmpiYkJ79+5VOp2edEwikVBPT48kaWRkRK+//rquuuqqhZsYADAj\n1/VLMBhUe3u7amtrlc/n1djYqGQyqY6ODklSU1OTvvvd76qhoUEVFRU6c+aMfvzjH+uSSy5ZlOEB\nAJO53tI4byfhlkYA8G3eb2kEABQXog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGAR\nog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4A\nFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHqAGARog4AFiHq\nAGARog4AFjln1DOZjBKJhOLxuNra2qY9JpvN6rrrrtOaNWuUSqXme0YAgEcBx3GcmV7M5/MqKytT\nT0+PwuGw1q9fr87OTiWTycIxY2NjuuGGG/T73/9ekUhEo6OjuuyyyyafJBCQy2kAANOYTTtdr9T7\n+/sVi8VUWlqqUCik+vp6dXV1TTrmqaee0pYtWxSJRCRpStABAIsn6Pbi8PCwotFo4XkkElFfX9+k\nYwYGBnTq1Cl96Utf0vvvv6/7779fX//616e8V0tLS+HrVCrFmgYAzpLNZpXNZuf0Hq5RDwQC53yD\nU6dO6dChQzpw4IDGx8d1/fXX6/Of/7zi8fik4z4edQDAVGdf8La2tvp+D9eoh8Nh5XK5wvNcLldY\ns/xPNBrVZZddposuukgXXXSRvvjFL+rIkSNTog4AWHiuO/WqqioNDAxocHBQExMT2rt3r9Lp9KRj\nbrvtNv35z39WPp/X+Pi4+vr6VF5evqBDAwCm53qlHgwG1d7ertraWuXzeTU2NiqZTKqjo0OS1NTU\npEQioU2bNmndunVasmSJtm/fTtQBwBDXWxrn7STc0ggAvs37LY0AgOJC1AHAIkQdACxC1AHAIkQd\nACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC\n1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHA\nIkQdACxC1AHAIkQdACxC1AHAIkQdACxC1AHAIueMeiaTUSKRUDweV1tb24zHvfzyywoGg3ruuefm\ndUAAgHeuUc/n89qxY4cymYxee+01dXZ26tixY9Me953vfEebNm2S4zgLNiwAwJ1r1Pv7+xWLxVRa\nWqpQKKT6+np1dXVNOe5nP/uZtm7dqssvv3zBBgUAnFvQ7cXh4WFFo9HC80gkor6+vinHdHV16Y9/\n/KNefvllBQKBad+rpaWl8HUqlVIqlZr91ABgoWw2q2w2O6f3cI36TIH+uAceeEC7du1SIBCQ4zgz\nrl8+HnUAwFRnX/C2trb6fg/XqIfDYeVyucLzXC6nSCQy6Zi//vWvqq+vlySNjo5q//79CoVCSqfT\nvocBAMxNwHH5yebp06dVVlamAwcOaNWqVfrc5z6nzs5OJZPJaY9vaGjQrbfeqs2bN08+yX+v4gEA\n3s2mna5X6sFgUO3t7aqtrVU+n1djY6OSyaQ6OjokSU1NTbOfFgAw71yv1OftJFypA4Bvs2knnygF\nAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQ\ndQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCwCFEHAIsQdQCw\nCFEHAIsETQ9QDI4ckX79a9NTXLhiMenOO01PARQHrtQ9ePZZ6YUXTE9xYTp+XPr+901PARQPrtQ9\ncBxpwwbpe98zPcmF5x//kPbtMz0FUDy4UvfAcaQlfKeMWLLko+8/AG9IlQeOIwUCpqe4MAUCRB3w\ng6h7cOYMUTclEPjo+w/AG6LuAesXc1i/AP6QKg9Yv5jD+gXw55xRz2QySiQSisfjamtrm/L6k08+\nqYqKCq1bt0433HCDjh49uiCDmsT6xRzWL4A/rrc05vN57dixQz09PQqHw1q/fr3S6bSSyWThmKuu\nukp/+tOftGzZMmUyGX3jG99Qb2/vgg++mFi/mMP6BfDHNVX9/f2KxWIqLS1VKBRSfX29urq6Jh1z\n/fXXa9myZZKk6upqDQ0NLdy0hrB+MYf1C+CP65X68PCwotFo4XkkElFfX9+Mxz/yyCOqq6ub9rWW\nlpbC16lUSqlUyt+kBrF+MYf1Cy4k2WxW2Wx2Tu/hGvWAj5I9//zz2rNnj1566aVpX/941IsN6xdz\nWL/gQnL2BW9ra6vv93CNejgcVi6XKzzP5XKKRCJTjjt69Ki2b9+uTCajFStW+B7ifMf6xRzWL4A/\nrtefVVVVGhgY0ODgoCYmJrR3716l0+lJx7zzzjvavHmzfvWrXykWiy3osKawfjGH9Qvgj+uVejAY\nVHt7u2pra5XP59XY2KhkMqmOjg5JUlNTk374wx/q+PHjam5uliSFQiH19/cv/OSLiPWLOaxfAH8C\njrPwf2QCgYAW4TQLprlZWrtWuvde05NceE6ckFav/uifwIVmNu3k+tMDdurmsH4B/CHqHpw5w/rF\nFNYvgD+kygOu1M3h7hfAH6LuAVE3h/UL4A9R94D1izmsXwB/SJUHXKmbw/oF8Ieoe0DUzWH9AvhD\n1D1g/WIO6xfAH1LlAVfq5rB+Afwh6h4QdXNYvwD+EHUPWL+Yw39MAX9IlQdcqZvzv+87KxjAG6Lu\nAVE3ixUM4B1R94Com8UPSwHviLoH7NTN4rZGwDtS5QFX6maxfgG8I+oeEHWzWL8A3hF1D1i/mMX6\nBfCOVHnAlbpZrF8A74i6B0TdLNYvgHdE3QPWL2axfgG8I1UecKVuFusXwDui7gFRN4v1C+AdUfeA\n9YtZrF8A70iVB1ypm8X6BfCOqHtA1M1i/QJ4R9Q9cBzWLyaxfgG8I1UenDnDlbpJrF8A74i6B6xf\nzGL9AnhH1D1g/WIW6xfAO1LlAesXs1i/AN4RdQ9Yv5jF+gXwjqh7wPrFLNYvgHekygPWL2axfgG8\nI+oesH4xi/UL4B1R92BsLFvU65dsNmt6hFnLZrNFvX4p5u+9xPzF6JypymQySiQSisfjamtrm/aY\n++67T/F4XBUVFTp8+PC8D2naiRPZor5SL+Z/sbPZbFGvX4r5ey8xfzFyjXo+n9eOHTuUyWT02muv\nqbOzU8eOHZt0THd3t958800NDAzo4YcfVnNz84IObEoxR73YsX4BvHONen9/v2KxmEpLSxUKhVRf\nX6+urq5Jx+zbt0/btm2TJFVXV2tsbEwjIyMLN7EB7NTNKub1C7DoHBfPPvusc8899xSe//KXv3R2\n7Ngx6ZhbbrnFeemllwrPN27c6LzyyiuTjpHEgwcPHjxm8fArKBcBj5enzlmXUWf/vrNfBwAsDNf1\nSzgcVi6XKzzP5XKKRCKuxwwNDSkcDs/zmAAAL1yjXlVVpYGBAQ0ODmpiYkJ79+5VOp2edEw6ndYT\nTzwhSert7dXy5ctVUlKycBMDAGbkun4JBoNqb29XbW2t8vm8GhsblUwm1dHRIUlqampSXV2duru7\nFYvFtHTpUj366KOLMjgAYBq+t/Cz9O1vf9tJJBLOunXrnK985SvO2NjYYp16Tvbv3++UlZU5sVjM\n2bVrl+lxfHnnnXecVCrllJeXO9dee62ze/du0yP5dvr0aaeystK55ZZbTI/i2/Hjx50tW7Y4iUTC\nSSaTzl/+8hfTI/myc+dOp7y83FmzZo1zxx13OB9++KHpkVw1NDQ4K1eudNasWVP4tffee8+56aab\nnHg87tTU1DjHjx83OKG76eafTTcX7XOSN998s1599VUdOXJE11xzjR588MHFOvWseblP/3wWCoX0\nk5/8RK+++qp6e3v185//vKjml6Tdu3ervLzc8w/tzyf333+/6urqdOzYMR09elTJZNL0SJ4NDg7q\nF7/4hQ4dOqS///3vyufzevrpp02P5aqhoUGZTGbSr+3atUs1NTV64403tHHjRu3atcvQdOc23fyz\n6eaiRb2mpkZL/vtZ++rqag0NDS3WqWfNy33657MrrrhClZWVkqSLL75YyWRS7777ruGpvBsaGlJ3\nd7fuueeeoruD6sSJE3rxxRd19913S/polbls2TLDU3n36U9/WqFQSOPj4zp9+rTGx8fP+xsgbrzx\nRq1YsWLSr338czTbtm3Tb37zGxOjeTLd/LPpppH/o8mePXtUV1dn4tS+DA8PKxqNFp5HIhENDw8b\nnGj2BgcHdfjwYVVXV5sexbNvfvObeuihhwr/UheTt99+W5dffrkaGhr02c9+Vtu3b9f4+LjpsTy7\n5JJL9K1vfUurV6/WqlWrtHz5ct10002mx/JtZGSkcONGSUlJUX8w0ms35/VPS01NjdauXTvl8dvf\n/rZwzI9+9CN94hOf0Fe/+tX5PPWCKMa/8k/ngw8+0NatW7V7925dfPHFpsfx5He/+51Wrlyp6667\nruiu0iXp9OnTOnTokO69914dOnRIS5cuPa//6n+2t956Sz/96U81ODiod999Vx988IGefPJJ02PN\nSSAQKNo/03666Xr3i19/+MMfXF9/7LHH1N3drQMHDsznaReMl/v0z3enTp3Sli1bdOedd+rLX/6y\n6XE8O3jwoPbt26fu7m59+OGHOnnypO66667C7bPnu0gkokgkovXr10uStm7dWlRRf+WVV/SFL3xB\nl156qSRp8+bNOnjwoL72ta8ZnsyfkpIS/etf/9IVV1yhf/7zn1q5cqXpkXzz281F+3ttJpPRQw89\npK6uLn3yk59crNPOiZf79M9njuOosbFR5eXleuCBB0yP48vOnTuVy+X09ttv6+mnn9aGDRuKJujS\nRz/PiEajeuONNyRJPT09uvbaaw1P5V0ikVBvb6/+/e9/y3Ec9fT0qLy83PRYvqXTaT3++OOSpMcf\nf7yoLmykWXZzoW7POVssFnNWr17tVFZWOpWVlU5zc/NinXpOuru7nWuuuca5+uqrnZ07d5oex5cX\nX3zRCQQCTkVFReH7vn//ftNj+ZbNZp1bb73V9Bi+/e1vf3OqqqqK7jbe/2lrayvc0njXXXc5ExMT\npkdyVV9f71x55ZVOKBRyIpGIs2fPHue9995zNm7cWBS3NJ49/yOPPDKrbgYcpwgXlgCAaRXfbQUA\ngBkRdQCwCFEHAIsQdQCwCFEHAIsQdQCwyP8BM7zjkDoDL7UAAAAASUVORK5CYII=\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ylim(0,1)\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a18242cf",
   "metadata": {},
   "source": [
    "This is odd because it predicts that only values between 0 and 4 can occur.\n",
    "But the training sample $x_1=4$ only excludes that $\\mu\\lt4$; it doesn't \n",
    "exclude any values greater than $4$.\n",
    "\n",
    "What's the Bayesian estimate?\n",
    "\n",
    "$$p(x|D) = \\int p(x|\\theta) p(\\theta|D) d\\theta \\propto \\int_0^{10} 1/\\theta \\cdot \\lfloor x\\in[0,\\theta]\\rfloor \\cdot 1/\\theta \\cdot \\lfloor \\theta\\in[4,10]\\rfloor d\\theta\n",
    "= \\int_{x_1}^{10} 1/\\theta^2 \\cdot \\lfloor x\\in[0,\\theta]\\rfloor d\\theta$$\n",
    "\n",
    "You can either think about it, or we can simply perform this integral numerically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "id": "20f02fda",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x183d6c10>]"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD9CAYAAABdoNd6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIiILYNgTEVkAw56IyAL+DyYnlQewGq1mAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu1[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "result /= C*sum(result)\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "140ae1ae",
   "metadata": {},
   "source": [
    "Now assume we get another sample, $x_1=7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "id": "e1987183",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19319d50>]"
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD9CAYAAABDaefJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nLHUiIh1hqRMR6cj/A8P5kvs7vvMGAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu2 = pxt(7,mus)*pmu1\n",
    "pmu2 /= C*sum(pmu2)\n",
    "plot(xs,pmu2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "id": "dc1026b3",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x194731d0>]"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD9CAYAAABdoNd6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nN3+xiAhqAQAAAABJRU5ErkJggg==\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu2[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "result /= C*sum(result)\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,7))\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ed89a8e",
   "metadata": {},
   "source": [
    "This is even weirder.  After seeing the first sample, the maximum likelihood\n",
    "estimator predicts only values between 0 and 4 occurring, but after seeing\n",
    "another training sample, it is changing its mind and now predicts that values\n",
    "between 0 and 7 can occur.\n",
    "\n",
    "The Bayesian estimator, in contrast, \"knows\" that the parameter must be greater than 7,\n",
    "so it predicts a uniform distribution for the interval [0...7] and then a tradeoff\n",
    "between the parameter distribution and the uniform distribution of the parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cb3aaad",
   "metadata": {},
   "source": [
    "The last sample illustrates this further.\n",
    "\n",
    "A sample of $x_3=2$ doesn't cause any update to the maximum likelihood estimator,\n",
    "but it does cause an update to posterior distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "id": "4bdc1b8b",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19300650>]"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD9CAYAAABDaefJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu3 = pxt(2,mus)*pmu2\n",
    "pmu3 /= C*sum(pmu3)\n",
    "plot(xs,pmu3)\n",
    "plot(xs,pmu2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "id": "1223dab1",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x194884d0>]"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAD9CAYAAABdoNd6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "total = 0\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu3[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "    total += weight\n",
    "result /= total\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,7))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea37826c",
   "metadata": {},
   "source": [
    "In fact, if we repeat the same process with a lot of samples (in this case\n",
    "the true parameter is 7), we see that the Bayesian parameter estimate\n",
    "becomes an increasingly peaked distribution close to the true value.\n",
    "\n",
    "I.e., if, out of 100 samples, we haven't seen a value greater than 7,\n",
    "then the probability that the mean is significantly greater than 7 must\n",
    "be very small."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "id": "11b36258",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19c3f0d0>]"
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXEAAAD9CAYAAABKgkezAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nA4DBKHEAMBglDgAGo8QBwGCUOAAY7H8FNZkRelCUCQAAAABJRU5ErkJggg==\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = pmu.copy()\n",
    "for i in range(100):\n",
    "    p = pxt(rand()*7,mus)*p\n",
    "    p /= C*sum(p)\n",
    "plot(mus,p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e61ca6ec",
   "metadata": {},
   "source": [
    "Loss Functions for Parameter Estimation\n",
    "========================================\n",
    "\n",
    "Consider $p(\\theta|x)$ from the previous example again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "id": "014aef0b",
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19e5a7d0>]"
      ]
     },
     "execution_count": 241,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD9CAYAAABDaefJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nLHUiIh1hqRMR6cj/A8P5kvs7vvMGAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(mus,pmu2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "116d69e1",
   "metadata": {},
   "source": [
    "Assume now that we are supposed to return a \"best estimate\" of the parameter.\n",
    "\n",
    "By itself, that isn't sufficient.\n",
    "\n",
    "But now assume that we are given a loss function: if our estimate is with in $\\pm 0.5$ of the true value,\n",
    "we don't need to pay a penalty, otherwise, we need to pay a penalty of 1.  What value should we return?\n",
    "\n",
    "The most likely value is 7, but no values less than 7 can occur.\n",
    "\n",
    "Therefore, it is better to return 7.5.  That way, not only do we have the most likely value, but we also\n",
    "get all the probability mass between 7 and 8 as well and our expected loss is about half of what it would be\n",
    "if we had returned 7.\n",
    "\n",
    "Now assume we are penalized if we are outside the range $\\pm 1$ from the true value.\n",
    "By the same reasoning, our parameter estimate should now be 8.\n",
    "\n",
    "As you can see from this simple example, there is not \"best\" answer to the parameter estimation\n",
    "problem; our answer depends on the loss function.\n",
    "\n",
    "But we can see that for any symmetric loss function, the value 7 (the maximum likelihood estimate)\n",
    "is never the optimal answer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c6b52629",
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {},
 "nbformat": 4,
 "nbformat_minor": 5
}