Probability.ipynb

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     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Probability"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Preface - "
     ]
    },
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     "metadata": {},
     "source": [
      "These are notes I wrote for myself while studying for PhD qualifying exams at Stanford. \n",
      "\n",
      "I wanted to published them so that others could benefit.\n",
      "\n",
      "They draw from sources:\n",
      "\n",
      "* CS229, an excellent machine learning class at Stanford taught by Andrew Ng.\n",
      "* An Introduction to Statistical Learning (ISL), an excellent text by G. James, D. Witten,  T. Hastie and R. Tibshirani (Springer, 2013).\n",
      "* A pretty good, free online textbook: http://www.statsoft.com/Textbook/Experimental-Design\n",
      "* Another textbook (actually on Deep Learning, but with some probability): http://www.iro.umontreal.ca/~bengioy/dlbook/prob.html\n",
      "* http://nbviewer.ipython.org/gist/fonnesbeck/5850483\n",
      "\n",
      "Some of the figures are taken from ISL with permission from the authors."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "The idea behind probability."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Probability theory is a mathematical framework for representing uncertain statements. \n",
      "\n",
      "Sources of uncertainty:\n",
      "\n",
      "* Incomplete observability: When we cannot observe something, we are uncertain about its true nature.\n",
      "* Simplifications: Uncertainty can also arise from the simpli\ufb01cations we make in order to model real-world processes.\n",
      "\n",
      "Types of probability:\n",
      "\n",
      "* Frequentist: Related directly to the rates at which events occur (e.g., drawing a card).\n",
      "* Bayes: Related to qualitative levels of certainty (e.g., change of getting cancer).\n",
      "\n",
      "A random variable:\n",
      "\n",
      "* A variable that can take on di\ufb00erent values randomly.\n",
      "\n",
      "A probability distribution:\n",
      "\n",
      "* A description of how likely a random variable is to take on each of its possible states. \n",
      "* Probability mass function (PMF): probability distribution over discrete variables.\n",
      "* Probability density function (PDF): distribution over continuous random variables.\n",
      "\n",
      "Note that we can integrate the PDF to \ufb01nd the actual probability mass over a range of points.\n",
      "\n",
      "Conditional probability:\n",
      "\n",
      "The probability of some event, given that some other event has happened, $ P (y \\mid x) $.\n",
      "\n",
      "Joint probability distribution:\n",
      "\n",
      "Can be decomposed into conditional distributions over only one variable\n",
      "\n",
      "$P(A,B,C) = P(A \\mid B,C) P(B \\mid C) P(C) $\n",
      "\n",
      "Independence:\n",
      "\n",
      "Two random variables $X$ and $Y$ are independent if their probability distribution can be expressed as a product of two factors.\n",
      " "
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Key terms."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Basics - "
     ]
    },
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     "metadata": {},
     "source": [
      "Expectation (expected value):\n",
      "\n",
      "* The average value that a function $f(x)$ takes when x is drawn from $P(x)$.\n",
      "* For a discrete random variable: $ E(x) = \\sum\\limits_x F(x) P(x) $\n",
      "\n",
      "Variance:\n",
      "\n",
      "* The variance gives a measure of how much the di\ufb00erent values of a function are spread apart.\n",
      "* This is simply the sum of the squared deviations from the mean $ TSS = \\sum\\limits (y^i-y_{mean})^2 $.\n",
      "* For a set of $n$ equally likely values $ v = \\frac{TSS}{n-1} $\n",
      "* Of course, the standard deviation is simply $ v = \\sigma^2 $.\n",
      "\n",
      "Covariance:\n",
      "\n",
      "* The variance is a special case of covariance, which is a measure of how two variables are linearly related.\n",
      "* $ Cov(a,b) = \\frac{1}{n-1} \\sum\\limits_{i=1}^{n} (a_i - \\mu_a) (b_i - \\mu_b) $\n",
      "* If the sign of the covariance is positive, then the values tend to change in the same direction.\n",
      "\n",
      "Correlation (Pearson correlation):\n",
      "\n",
      "* When the covariance is normalized by standard deviation, one obtains the Pearson correlation coefficient:\n",
      "* $ r = Cor(a,b) = \\frac{Cov(a,b)}{\\sigma_a \\sigma_b} $\n",
      "\n",
      "Coefficient of determination:\n",
      "\n",
      "* An alternative measure of linear dependence is $R^2$.\n",
      "* This is the proportion of variability in Y that can be explained using X.\n",
      "* $ R^2 = 1 - \\frac{RSS}{TSS} $\n",
      "* Where RSS measures the amount of variability that is left unexplained after performing the regression.\n",
      "* This is $ \\sum\\limits (y^i-\\theta^T x^i)^2 $.\n",
      "* $r2=0.9$, then 90% of the variance in Y can be explained by variation our model, a funtion of $x$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Distributions for discrete random variables -"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Bernoulli"
     ]
    },
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     "source": [
      "Single binary random variable that takes value 1 with success probability $\\phi$:\n",
      "\n",
      "$P(x=1) = \\phi $\n",
      "\n",
      "$P(x=0) = 1 - \\phi $"
     ]
    },
    {
     "cell_type": "code",
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     "input": [
      "import numpy as np\n",
      "from scipy import stats\n",
      "from scipy.stats import bernoulli, poisson, binom\n",
      "import matplotlib\n",
      "import matplotlib.pyplot as plt \n",
      "%matplotlib inline\n",
      "\n",
      "plt.subplot(2,2,1)\n",
      "a = np.arange(2)\n",
      "p=0.2\n",
      "plt.bar(a,bernoulli.pmf(a, p),label=p,alpha=0.5)\n",
      "plt.xticks(a)\n",
      "plt.title('P=%s'%p)"
     ],
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     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Binomial"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simply, this tells us the number of heads in $n$ independent flips of a coin with equal heads probability $p$.\n",
      "\n",
      "Compute the probability of observing $n$ successes for $N$ trials given a probability $p$ for each without regard for order. \n",
      "\n",
      "If $N=5$, $p=0.5$, $n=2$, then the first success can occour in any of the 5 trials. \n",
      "\n",
      "The second success can occour in any of the remaining 4 trials, giving 5 $\\times$ 4 permutations.\n",
      "\n",
      "In short, if the first success was on trial 1 then the second can be any of the 4 remaining; for each of the 5 trials for success 1.\n",
      "\n",
      "So, let's say success 1 happened on trial 1 and success 2 happened on trial 2.\n",
      "\n",
      "We don't reall care if the inverse happened (2 on trial 1).\n",
      "\n",
      "So, we correct for order: $\\frac{5 \\times 4}{n!} = 10$ combinations of 2 successes. \n",
      "\n",
      "The number of combinations that satisfy the condition, $n$, is generally given by the binomial coefficient.\n",
      "\n",
      "This states: \n",
      "\n",
      "* (1) there are $N$ choose $n$ combinations with \n",
      "* (2) the probability of each combination $p^n (1-p)^{N-n}$ (n successes, $N-n$ failures)\n",
      "\n",
      "$$ P(n) = {N \\choose n} p^n (1-p)^{N-n} $$\n",
      "\n",
      "$$ {N \\choose n} = \\frac{N!}{n!(N-n)!} $$\n",
      "\n",
      "In short, the use is evaluating a series of equally likely events, such as dice rolls. \n",
      "\n",
      "Also, the expected value is simply $Np$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Poisson"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's say the expected value of a Binomial random variable is $\\lambda = Np$. \n",
      "\n",
      "Let's assume $\\lambda$ is the expected number of cars observed per hour. \n",
      "\n",
      "If we let N=60 (one trial per minute over the course of the hour), then we can compute the probability of success per trail:\n",
      "\n",
      "$$p=\\frac{\\lambda}{60}$$\n",
      "\n",
      "OK, so let's say we want to compute the probability of $n$ cars:\n",
      "\n",
      "$$ P(n=k) = {60 \\choose k} \\frac{\\lambda}{60}^k (1-\\frac{\\lambda}{60})^{60-k} $$\n",
      "\n",
      "The problem is that each trial (minute) is binary: there can only be one or zeo cars!\n",
      "\n",
      "In practice, we could have > 1 car per minute. \n",
      "\n",
      "So, we can get more granular. \n",
      "\n",
      "So, let's assume the number of trials is large, far less than a minute for each one!  \n",
      "\n",
      "And the probability of success at each is correspondingly smaller. \n",
      "\n",
      "As $N$ approaches infinity, Binomal becomes the Poisson distribution."
     ]
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     "source": [
      "So, the Binomal was brittle: the interval (e.g., one minute) was specified and assigned an associated probability.\n",
      "\n",
      "For Poisson, probability of an event in a time interval is proportional to the size of the interval!\n",
      "\n",
      "* $P = \\lambda dt$ for interval $dt$, where $\\lambda$ indicates the number of events per unit time.\n",
      "* We want to compute $P(t)$, the probability that an event happens at time $t$ starting at $t=0$.\n",
      "* For this to occour, the event should not happen until time $t$.\n",
      "* We define this as $P_o(t)$, starting from $t=0$.\n",
      "* We break $t$ into a set of small intervals and the probability of no event in one interval is given by $1-\\lambda dt$.\n",
      "* $P_o = (1-\\lambda dt)^N$ is the probability of no events in the $N$ intervals such that $dtN=t$. \n",
      "* This can we re-written as $P_o = (1-\\lambda dt)^{t dt}$ and is $P_o = e^{-\\lambda t}$ in the limit of $dt \\to 0$.\n",
      "* So, to compute $P(t)$ we determine the probabiity of not happening until time $t$, $e^{-\\lambda t}$.\n",
      "* Also include the  probability that it happens in time $dt$, $\\lambda dt$.\n",
      "* The product is $P(t)=\\lambda e^{-\\lambda t}$.\n",
      "\n",
      "Poisson process for many events:\n",
      "\n",
      "* Either $n$ events in the time leading up to the final interval or it is the final event occouring in the last interval, $dt$.\n",
      "* $P_n(t+dt) = P_n(t)(1-\\lambda dt) + P_{n-1}(t) \\lambda dt$.\n",
      "* This can be solved to yield:\n",
      "\n",
      "$$ P_n(t) = \\frac{(\\lambda t)^n}{n!} e^{- \\lambda t} $$"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "An intuitive application is sequence:\n",
      "    \n",
      "There are a large number of trial (reads).\n",
      "\n",
      "The probability of a read hitting a given base is low for each trial.\n",
      "\n",
      "The expected number of reads per base is given by the read count, read length, and genome size.\n",
      "\n",
      "So, we can work out the likelihood of observing a given number of reads per base."
     ]
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      "n=np.arange(0,50) # Range of coverage values.\n",
      "l=100 # Read length.\n",
      "N=3*10**9 # Size of human genome.\n",
      "reads=[10**8,10**8.5,10**9] # Sequencing reads.\n",
      "coverage=[float(read_number)*l/N  for read_number in reads] # Lambda: expected reads per base.\n",
      "colors = matplotlib.rcParams['axes.color_cycle']\n",
      "\n",
      "plt.subplot(2,1,1)\n",
      "for u,c in zip(coverage,colors):\n",
      "    y=stats.poisson.pmf(n,u)\n",
      "    plt.plot(n,y,'-',linewidth=2,color=c,alpha=1,label='$\\lambda$ =%.3f'%u)\n",
      "    plt.fill_between(n,y,color=c,alpha = .25)\n",
      "    plt.xticks(fontsize=8)\n",
      "    plt.yticks(fontsize=8)\n",
      "    plt.title('Probability mass funtion',fontsize=8)\n",
      "    plt.xlabel('n (base)',fontsize=8)\n",
      "    plt.ylabel('P(n reads/base)',fontsize=8)\n",
      "    plt.legend(loc='upper right', scatterpoints = 1,prop={'size':8})\n",
      "    plt.grid()\n",
      "\n",
      "plt.subplot(2,1,2)\n",
      "for u,c in zip(coverage,colors):\n",
      "    y=stats.poisson.cdf(n,u)\n",
      "    plt.plot(n,y,'-',linewidth=2,color=c,alpha=1,label='$\\lambda$ =%.3f'%u)\n",
      "    plt.fill_between(n,y,color=c,alpha = .25)\n",
      "    plt.xticks(fontsize=8)\n",
      "    plt.yticks(fontsize=8)\n",
      "    plt.title('Cumulative distribution funtion',fontsize=8)\n",
      "    plt.xlabel('n (base)',fontsize=8)\n",
      "    plt.ylabel('P(< or = n reads/base)',fontsize=8)\n",
      "    plt.legend(loc='upper right', scatterpoints = 1,prop={'size':8})\n",
      "    plt.grid()\n",
      "\n",
      "plt.tight_layout()\t"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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W+S1D0VpBVkEWgV8Hcuz6MZ21r2/Uk7qCihF60h6hK91T6XDfhQsXaNGiBX36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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a247650>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As coverage increaces, the PMF shows that the probability if getter a higher number of reads per base increaces. \n",
      "\n",
      "Of course, the probability of a specific position having *exactly* 30 reads is low < 0.1. \n",
      "\n",
      "But the CDF shows that the probability of getting 30 or more reads spanning the base when $\\lambda=30$ is reasonable at $0.524$ ($1-y[29]$)."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Distributions for continuous random variables -"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Exponential"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Exponential distribution models the waiting time between events in a Poisson process:\n",
      "\n",
      "* The probability of seeing no hits at time $t$ for a Poisson process is $P_o(t) = e^{- \\lambda t}$.\n",
      "* The probability of observing an event in time $T < t$ is $1-P_o(t) = 1-e^{- \\lambda t}$.\n",
      "* Thus, the CDF for the exponential distribution is $1-e^{- \\lambda t}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x=np.arange(0,4,0.1)\n",
      "colors = matplotlib.rcParams['axes.color_cycle']\n",
      "lambda_ = [0.5,1,2]\n",
      "\n",
      "plt.subplot(2,1,1)\n",
      "u_vals=[2,1]\n",
      "for u,c in zip(coverage,colors):\n",
      "    y=u*np.exp(-u*x)\n",
      "    plt.plot(x,y,'-',linewidth=2,alpha=1,color=c,label='$\\lambda$ =%.3f'%u)\n",
      "    plt.fill_between(x,y,color=c,alpha = .25)\n",
      "plt.legend()\n",
      "plt.xticks(fontsize=5)\n",
      "plt.yticks(fontsize=5)\n",
      "plt.title('Probability density funtion',fontsize=8)\n",
      "plt.xlabel('x (distance)',fontsize=8)\n",
      "plt.ylabel('P(x)',fontsize=8)\n",
      "plt.legend(loc='upper right', scatterpoints = 1,prop={'size':8})\n",
      "plt.xlim([0,1])\n",
      "plt.grid()\n",
      "\n",
      "plt.subplot(2,1,2)\n",
      "for u,c in zip(coverage,colors):\n",
      "    y=1-np.exp(-u*x)\n",
      "    plt.plot(x,y,'-',linewidth=2,alpha=1,color=c,label='$\\lambda$ =%.3f'%u)\n",
      "    plt.fill_between(x,y,color=c,alpha = .25)\n",
      "plt.legend()\n",
      "plt.xticks(fontsize=5)\n",
      "plt.yticks(fontsize=5)\n",
      "plt.title('Cumulative distribution funtion',fontsize=8)\n",
      "plt.xlabel('x (distance)',fontsize=8)\n",
      "plt.ylabel('P(event <= x)',fontsize=8)\n",
      "plt.legend(loc='upper right', scatterpoints = 1,prop={'size':8})\n",
      "plt.grid()\n",
      "\n",
      "plt.tight_layout()\t"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ghBDOzKlu91mSpO6EXg85eS6m5JWe5WZKYMb1vscXMuLcfDZ5dePxWsvIyXIh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\nTUpERBzr/38ABRsxE2sYAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10aa624d0>"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Normal"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$ P(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} exp(-\\frac{(x-\\mu)^{2}}{2\\sigma^2}) $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that the standard z-score is the number of standard deviations away from the mean:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$ Z=\\frac{(x-\\mu)}{\\sigma} $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Specify parameters for normal distribution\n",
      "colors = matplotlib.rcParams['axes.color_cycle']\n",
      "stdVal=3\n",
      "\n",
      "plt.subplot(2,1,1)\n",
      "mean = [5,10,20]\n",
      "for meanVal,c in zip(mean,colors):\n",
      "    #  Generates a normal continuous random variable with specified mean (loc), std (scale) \n",
      "    rv = stats.norm(loc=meanVal,scale=stdVal)   \n",
      "    x = np.arange(.1*meanVal,meanVal*10,1)\n",
      "    plt.plot(x,rv.pdf(x),lw=2,linewidth=2,alpha=1,color=c,label='$\\mu$ =%.3f'%meanVal)\n",
      "    plt.fill_between(x,rv.pdf(x),color=c,alpha = .25)\n",
      "plt.legend()\n",
      "plt.xticks(fontsize=5)\n",
      "plt.yticks(fontsize=5)\n",
      "plt.xlim([0,np.max(mean)*1.5])\n",
      "plt.title('Probability density funtion',fontsize=7.5)\n",
      "plt.xlabel('Random variable (x)',fontsize=5)\n",
      "plt.ylabel('P(x)',fontsize=5)\n",
      "plt.legend(loc='upper right', scatterpoints = 1,prop={'size':5})\n",
      "plt.grid()\n",
      "\n",
      "plt.subplot(2,1,2)\n",
      "for meanVal,c in zip(mean,colors):\n",
      "    rv = stats.norm(loc=meanVal,scale=stdVal)   \n",
      "    x = np.arange(.1*meanVal,meanVal*10,1)\n",
      "    plt.plot(x,rv.cdf(x),lw=2,linewidth=2,alpha=1,color=c,label='$\\mu$ =%.3f'%meanVal)\n",
      "    plt.fill_between(x,rv.cdf(x),color=c,alpha = .25)\n",
      "plt.legend()\n",
      "plt.xticks(fontsize=5)\n",
      "plt.yticks(fontsize=5)\n",
      "plt.xlim([0,np.max(mean)*1.5])\n",
      "plt.title('Cumulative distribution funtion',fontsize=7.5)\n",
      "plt.xlabel('Random variable (x)',fontsize=5)\n",
      "plt.ylabel('P(event <= x)',fontsize=5)\n",
      "plt.legend(loc='upper right', scatterpoints = 1,prop={'size':5})\n",
      "plt.grid()\n",
      "\n",
      "plt.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Mvb090dHRODlpZwryZxEZ+VD8+PPPDdNBaXNMKjI9jGU3JwMw4IUJz3RQAFY2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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a537b90>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Conditional distributions."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Bayes rule"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Conditional distributions seek to answer the probability distribution over Y when we know that X must take on a certain value x.\n",
      "\n",
      "Frequentist probability relies on the long-run frequency of events (e.g., probability for each roll of a dice).\n",
      "\n",
      "Bayesian probability assigns a belief, or prior probability, to hypothesis $P(H)$. \n",
      "\n",
      "Note that Frequentist probability is typically assigned based upon a known frequency, whereas our prior can be assinged. \n",
      "\n",
      "We'll use this to assign a the likelihood of our hypothesis given some data, $D$.\n",
      " \n",
      "So, we also must state the probabilty of: \n",
      "\n",
      "* Observing the data, $P(D)$.\n",
      "* Observing data if the hypothesis is true, $P(D|H)$.\n",
      "\n",
      "From this we can estimate the probability that the hypothesis is true given the $P(H|D)$, the posterial probability. \n",
      "\n",
      "$$ P(H|D) = \\frac{P(D|H)P(H)}{P(D)} $$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Information theory."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Shannon entropy - "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The basic intuition behind information theory is that learning that an unlikely event\n",
      "has occurred is more informative than learning that a likely event has occurred.\n",
      "\n",
      "Likely events should have low information content. Unlikely events should have high information content\n",
      "\n",
      "$I(x) = \u2212log P (x)$\n",
      "\n",
      "Shannon entropy of a distribution is the expected amount of information in an event drawn from that distribution.\n",
      "\n",
      "$E(I(x)) = \u2212\\sum p \\times log P (x)$\n",
      "\n",
      "Shannon entropy:\n",
      "\n",
      "* High entropy means $X$ is from a uniform distribution, where every outcome is as likely to occur\n",
      "* Low entropy means $X$ is from varied distribution (with various peaks and valleys)\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Modelling."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Previously (in the statistics notebook), I went through statistical hypothesis tests.\n",
      "\n",
      "These tests seek to esimate whether groups or effects are \"statistically significant.\"\n",
      "\n",
      "Another approach is to be build flexible models with the overarching aim of estimating quantities of interest. \n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Estimation - "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In short, what we want to do is:\n",
      "    \n",
      "* Fit a probability distribution to our data.\n",
      "* Then, extract parameters that correspond to the distribution that best represents our data.\n",
      "\n",
      "We can do this in two ways:\n",
      "\n",
      "* **Method of moments** chooses parameters so sample moments (typically the sample mean and variance) match the theoretical moments of our chosen distribution.\n",
      "* **Maximum likelihood** chooses parameters to maximize the likelihood, which measures how likely it is to observe our given sample with the model in mind."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Method of moments and Maximum likelihood - "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So, break it down like this.\n",
      "\n",
      "(1) A distribution is parameterized by some values:\n",
      "\n",
      "* For gamma distribution, it's $\\alpha$ and $\\beta$.\n",
      "\n",
      "(2) For a distribution, there is a link between the parameters and the mean ($\\mu$), variance ($\\sigma$):\n",
      "\n",
      "* For gamma distribution, $\\mu = \\alpha \\beta$ and $\\sigma^2 = \\alpha \\beta^2$\n",
      "\n",
      "(3) For a given sample of data, we can compute sample statistics ($\\mu = x$, $\\sigma = s$).\n",
      "\n",
      "(4) We can then use the sample statistics to compute the distribution paramters and fit it!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$ \\alpha = \\frac{\\sigma^2}{\\beta^2}$\n",
      "\n",
      "$ \\beta = \\frac{\\mu}{\\alpha} $ \n",
      "\n",
      "$ \\beta^2 = \\frac{\\mu^2}{\\alpha^2} $ \n",
      "\n",
      "$ \\alpha = \\frac{\\sigma^2 \\alpha^2}{\\mu^2}$\n",
      "\n",
      "$ \\alpha = \\frac{\\mu^2}{\\sigma^2}$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This notebook has a nice review of both:\n",
      "    \n",
      "http://nbviewer.ipython.org/gist/fonnesbeck/5850483"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}