lectures/Week-04.1-Maximum-Likelihood-I.ipynb

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    "# Maximum Likelihood and Applications in Astronomy, Part I\n",
    "\n",
    "M. Juric (with material from G. Richards 2016, Ivezic [Sections 4.0, 4.1, 4.2.1-6, 4.3], Bevington, and Leighly)."
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    "## Statistical Modeling\n",
    "\n",
    "A statistical model is a mathematical model that captures assumptions about hown the data were generated (i.e., both the physical and the measurement process that generated them).\n",
    "\n",
    "It is a formal (quantitative) representation of the theory.\n",
    "\n",
    "Example: the data below follow a linear x-y relationship ($y_{true}(x) = ax+b$) and are measured by an instrument that adds some random, Gaussian, error (i.e. $y = y_{true}(x) + r$, where $r$ has been drawn from $N(0, \\sigma)$). The parameters of this model would be $a, b$, and $\\sigma$.\n",
    "\n",
    "Astronomical example: imagine time is $x_1$ below, $y$ is a star's flux, and $\\sigma$ is the error in flux estimate. The parameters $a$ and $b$ would correspond to the rate of change of flux with time, and the flux at $t=0$, respectivelly.\n",
    "\n",
    "![SlideGrab](figures/AnscombeQuartet.jpg)"
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    "## Statistical Inference\n",
    "\n",
    "Statistical *inference* is about drawing conclusions from data, specifically determining the properties of a population by data sampling.\n",
    "\n",
    "Three examples of inference are:\n",
    "1. What is the best estimate for a model parameter?\n",
    "2. How confident we are about the obtained value?\n",
    "3. Is the particular model/hypothesis consistent with the data?"
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    "## Statistical Inference Process\n",
    "\n",
    "We can break down the inference process (loop) into a few somewhat overlapping conceptual steps:\n",
    "\n",
    "1. **Hypothesis**: Formulate a model, a *hypothesis*, about how the data are generated. For example, this could be a statement that the data are a measurement of some quantity that come with Gaussian random errors (i.e., each measurement is equal to the true value, plus a deviation randomly drawn from the normal distribution). Models are typically written as an equation parametrized by a set of model parameters $\\boldsymbol{\\theta}$, denoted as $\\boldsymbol{M}(\\boldsymbol{\\theta})$. Note that both $\\boldsymbol{\\theta}$ and/or $\\boldsymbol{M}$ can be vectors (e.g. for a simple one-dimensional $y(x)$ linear dependence, $\\boldsymbol{\\theta} = \\{a, b\\}$ and $\\boldsymbol{M} = \\{ y \\}$.\n",
    "2. **Parameter Estimation**: Search for the \"best\" model parameters $\\boldsymbol{\\theta}$, usually by maximizing or minimizing a particular objective function (more soon).\n",
    "3. **Quantifying Estimate Uncertainty**: Determine what uncertainty remains about the values of model parameters -- the confidence region for model.\n",
    "4. **Hypothesis Testing**: Test whether our hypothesis $\\boldsymbol{M}(\\boldsymbol{\\theta})$ can be ruled out by the data (does the data (not) fit?). Possibly GOTO #1.\n",
    "\n",
    "These steps represent a blueprint for the frequentist approach in general. The **likelihood** (to be introduced soon) is just one of many possible so-called objective functions (also called fitness functions, or cost functions); other possibilities are explored briefly in §4.2.8 of the textbook."
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    "## Some Terminology\n",
    "\n",
    "* We typically study the properties of some ***population*** by measuring ***samples*** from that population. The population doesn't have to refer to different objects. E.g., we may be (re)measuring the position of an object at rest; the population is the (imagined) distribution of an infinite number of measurements smeared by the error, and the sample are the measurement we've actually taken.\n",
    "* A ***statistic*** is any function of the sample. For example, the sample mean is a statistic. But also, \"the value of the first measurement\" is also a statistic (though not a particularly good one!).\n",
    "* To conclude something about the population from the sample, we develop ***estimators***. An estimator is any statistic used as a rule for calculating an estimate of a given quantity based on the observed data: thus the rule (the *estimator*), the quantity of interest (the *estimand*) and its result (the *estimate*) can be distinguished. Sometimes *estimator* and the *estimate* are used interchangeably in everyday (scientific) language. Much of frequentist statistics concerns itself with development of different estimation rules -- different estimators -- and proving their properties.\n",
    "* There are ***point*** and ***interval estimators***. The point estimators yield single-valued results (example: the position of an object), while with an interval estimator, the result would be a range of plausible values (example: confidence interval for the position of an object)."
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    "## Motivating Example: Measuring the Position of a Quasar\n",
    "\n",
    "Let's assume we wish to estimate the true position $x_{true} \\equiv \\mu_{true}$ of a quasar from a series of astrometric measurements $\\{ x_i \\}$. For simplicity, we look at a simplified 1D case (a single coordinate).\n",
    "\n",
    "1. We adopt a hypothesis (model) where a) the observed quasar does not move, and b) individual measurement errors are drawn from a normal (Gaussian) distribution.\n",
    "2. We derive the expression for the likelihood of there being a quasar at some position $x = \\mu$ that gives rise to our individual measurements $\\{x_i\\}$. We find the value of $\\hat \\mu$ for which our observations are maximally likely (maximize the likelihood function).\n",
    "3. We determine the error bars (confidence intervals) on our measurement.\n",
    "4. We test whether what we've observed is consistent with our adopted model. For example, is it possible that the quasar was really a misidentified star with measurable proper motion?\n"
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    "### The Probability of Obtaining a Particular Measurement\n",
    "\n",
    "As our measurement errors are Gaussian, the probability density of each measurement (assuming the quasar's true location is $\\mu$, and the precision of our apparatus is $\\sigma$) is given by:\n",
    "\n",
    "$$p(x \\mid \\boldsymbol{M(\\theta)}) = p(x|\\mu + N(0, \\sigma)) = p(x|\\mu,\\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left(\\frac{-(x-\\mu)^2}{2\\sigma^2}\\right).$$\n",
    "\n",
    "This is the probability of a measurement being between $x, x+dx$ given the model parameter values $\\mu, \\sigma$.\n",
    "\n",
    "Note the notational sloppiness in the second equality sign above, where we dropped $N()$: it's typical to just name the parameters when writing out PDFs, and leave the mathematical details of the model implied from the context.\n",
    "\n",
    "Here's a visualization with $\\mu = 5$, and $\\sigma = 1$:"
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\n",
      "text/plain": [
       "<Figure size 720x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Execute this cell\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# plot the distributions\n",
    "fig, ax = plt.subplots(figsize=(10, 7.5))\n",
    "dist = norm(5, 1)\n",
    "x = np.linspace(0, 10, 1000)\n",
    "plt.plot(x, dist.pdf(x), c='black',label=r'$\\mu=5,\\ \\sigma=1$')\n",
    "\n",
    "plt.xlim(0, 10)\n",
    "plt.ylim(0, 0.5)\n",
    "\n",
    "plt.xlabel('$x$')\n",
    "plt.ylabel(r'$p(x|\\mu=5,\\sigma=1)$')\n",
    "plt.title('Probability of $x$')\n",
    "\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### The Probability of the Dataset (of the Sample)\n",
    "\n",
    "But we (usually) have more than one measurement -- a set of measurements, ${x_i}$.\n",
    "\n",
    "If we want to know the *joint* probability of obtaining that entire set, we compute the *product* of all the individual probabilities:\n",
    "\n",
    "$$p(\\{x_i\\}|\\boldsymbol{M(\\theta)}) = \\prod_{i=1}^n p(x_i|\\boldsymbol{M(\\theta)}),$$\n",
    "\n",
    "Why? Because for independent events (and we assume the results of individual measurements are independent), the joint probability is simply the product of individual probabilities -- i.e., $p(A, B) = p(A)p(B)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "For our quasar-with-Gaussian-errors example, this equates to:\n",
    "\n",
    "$$p(\\{x_i\\}|N(\\mu, \\sigma)) = \\prod_{i=1}^n \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left(-\\frac{(x_i-\\mu)^2}{2\\sigma^2}\\right),$$\n",
    "\n",
    "which we can rewrite as:\n",
    "\n",
    "$$p(\\{x_i\\}|N(\\mu, \\sigma)) = \\left( \\prod_{i=1}^n \\frac{1}{\\sigma\\sqrt{2\\pi}} \\right) \\exp\\left( -\\frac{1}{2} \\sum \\left[\\frac{(x_i-\\mu)}{\\sigma} \\right]^2 \\right),$$\n",
    "\n",
    "where we have written the product of the exponentials as the exponential of the sum of the arguments."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Introducing Maximum Likelihood Estimation (MLE)\n",
    "\n",
    "We now have a way to compute the joint probability to obtain a set of measurements $\\{x_i\\}$ if the quasar's true position is known to be $\\mu$, and the measurement apparatus' precision was $\\sigma$:\n",
    "\n",
    "$$p(\\{x_i\\}|N(\\mu, \\sigma))$$\n",
    "\n",
    "But our situation is the reverse: it's the data that are known, and we want to estimate the parameters, $\\mu$ and $\\sigma$. So we come to an idea: ***let us vary and pick the values of $\\mu$ and $\\sigma$ that result in the largest probability that the dataset $\\{ x_i \\}$ would be measured***. Intuitively that makes sense (and mathematically it can be shown that this leads to estimators with many good properties).\n",
    "\n",
    "In other words, we take the expression above, and now **think of the data as fixed**, and it being a ***function of the unknown model parameters***:\n",
    "\n",
    "$$L(\\mu, \\sigma) \\equiv p(\\{x_i\\}|N(\\mu, \\sigma))$$\n",
    "\n",
    "or, more generally:\n",
    "\n",
    "$$L(\\boldsymbol \\theta) \\equiv p(\\boldsymbol D|M(\\boldsymbol \\theta))$$\n",
    "\n",
    "and ***find the value of $\\boldsymbol \\theta$ that maximizes $L$***:\n",
    "\n",
    "$$\\underset{\\boldsymbol{\\theta}_{ML}}{\\max}\\,\\,  L(\\boldsymbol{\\theta}) \\equiv p(\\boldsymbol D|M(\\boldsymbol \\theta)) $$\n",
    "\n",
    "Those will be our **maximum likelihood estimates (MLEs)** for the true values of the particular model. In the notation above, $D$ stands for the set of collected data.\n",
    "\n",
    "(note: this is $\\S 4.2$ in the textbook)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Quasar example (w. known, homoscedastic, measurement errors)\n",
    "\n",
    "We've already assumed in our problem that the precision of our instrument is always the same (we used the same $\\sigma$). This is known as the measurement having **homoscedastic errors** -- errors drawn from the same distribution with fixed parameters. Note this does not mean the individual errors are the same -- just that they've been drawn from the same distribution.\n",
    "\n",
    "We'll next further simplify our example by assuming the precision of our instrument is also a-priori known. This makes the math easier, but results will generalize.\n",
    "\n",
    "Therefore, we now have only one unknown parameter in our model, the true position of the quasar $\\mu$:\n",
    "\n",
    "$$L(\\mu) = \\prod_{i=1}^N \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left(\\frac{-(x_i-\\mu)^2}{2\\sigma^2}\\right).$$\n",
    "\n",
    "."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Visualizing this\n",
    "\n",
    "Let's simulate 150 measurements ${x_i}$ drawn from a Gaussian with $\\mu = 5$ (quasar position) and $\\sigma = 1$ (measurement error), and draw the histogram of measurements:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Execute this cell\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "np.random.seed(seed=42)\n",
    "Nsamples=150\n",
    "mu_true = 5.\n",
    "measurements = np.random.normal(mu_true, 1, Nsamples)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "ax.hist(measurements, bins='fd');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now overplot and eyeball the model for some value of $\\mu$. Does it seem like a \"good fit\"?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def overplot_model(mu):\n",
    "    fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "\n",
    "    ax.hist(measurements, bins='fd', density=True);\n",
    "    x = np.linspace(2, 8, 1000)\n",
    "    ax.plot(x, norm(mu, 1).pdf(x))\n",
    "\n",
    "overplot_model(mu=4.9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "6d9d9b27f1e84e04a3c136e911497614",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=4.5, description='mu', max=8.0, min=2.0), Output()), _dom_classes=('wi…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import ipywidgets as widgets\n",
    "from ipywidgets import interact, fixed\n",
    "\n",
    "interact(overplot_model, mu=widgets.FloatSlider(min=2, max=8, step=0.1, value=4.5));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "How does the likelihood behave as we explore different potential values of $\\mu$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5deae155cc314fe0a733186c0a1202d2",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=4.5, description='mu', max=7.0, min=3.0), Output()), _dom_classes=('wi…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import ipywidgets as widgets\n",
    "from ipywidgets import interact, fixed\n",
    "\n",
    "def overplot_model_and_L(mu):\n",
    "    fig, (ax1, ax2) = plt.subplots(ncols=1, nrows=2, figsize=(12, 7.5))\n",
    "\n",
    "    ax1.hist(measurements, bins='fd', density=True);\n",
    "    x = np.linspace(2, 8, 1000)\n",
    "    ax1.plot(x, norm(mu, 1).pdf(x))\n",
    "    ax1.set_xlabel('$x$')\n",
    "    ax1.set_ylabel(r'$p_{\\rm histogram}(x)$')\n",
    "    ax1.set_xlim(2, 8)\n",
    "\n",
    "    #------------------------------------------------------------\n",
    "    # plot the likelihood distributions\n",
    "    mu_est = np.average(measurements)\n",
    "    L = lambda mu: np.prod( norm(measurements, 1).pdf(mu) )\n",
    "\n",
    "    L_mu = L(mu)\n",
    "    ax2.scatter([mu], [L_mu])\n",
    "\n",
    "    ax2.set_xlim(2, 8)\n",
    "    ax2.set_ylim(0, 1.1*L(mu_est))\n",
    "    ax2.set_xlabel('$\\mu$')\n",
    "    ax2.set_ylabel(r'$L(\\mu)$')\n",
    "\n",
    "interact(overplot_model_and_L, mu=widgets.FloatSlider(min=3, max=7, step=0.1, value=4.5));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above allows us to explore all values of $\\mu$ and find the maximum! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Finding the Maximum Analytically\n",
    "\n",
    "For practical (and some theoretical) reasons, it's better to work with the natural logarithm of the likelihood.\n",
    "\n",
    "We define the *log-likelihood function* as ${\\rm lnL} = \\ln[L(\\theta)]$.  The maximum of this function happens at the same place as the maximum of $L$. We therefore have:\n",
    "\n",
    "$${\\rm ln}\\,L = {\\rm constant} - \\sum_{i=1}^N \\frac{(x_i - \\mu)^2}{2\\sigma^2}.$$\n",
    "\n",
    "Take a second and make sure that you understand how we got there.  It might help to remember that above, we wrote:\n",
    "\n",
    "$$L(\\mu) = \\left( \\prod_{i=1}^n \\frac{1}{\\sigma\\sqrt{2\\pi}} \\right) \\exp\\left( -\\frac{1}{2} \\sum \\left[\\frac{(x_i-\\mu)}{\\sigma} \\right]^2 \\right).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We then determine the maximum in the same way that we always do.  It is the parameter set for which the derivative of ${\\rm lnL}$ is zero:\n",
    "\n",
    "$$\\frac{d\\;{\\rm lnL}(\\mu)}{d\\mu}\\Biggr\\rvert_{\\hat \\mu} \\equiv 0.$$\n",
    "\n",
    "That gives $$ \\sum_{i=1}^N \\frac{x_i - \\hat \\mu}{\\sigma^2} = 0.$$\n",
    "\n",
    "(note: we should also check that the $2^{\\rm nd}$ derivative is negative, to ensure this is the *maximum* of $L$)\n",
    "\n",
    "(also note: any constants in $\\ln L$ disappear when differentiated, so constant terms can typically be ignored.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Since $\\sigma = {\\rm constant}$, that says \n",
    "\n",
    "$$\\sum_{i=1}^N x_i = \\sum_{i=1}^N \\hat \\mu = N \\hat \\mu.$$\n",
    "\n",
    "Thus we find that\n",
    "\n",
    "$$\\hat \\mu = \\frac{1}{N}\\sum_{i=1}^N x_i \\equiv \\bar x,$$\n",
    "\n",
    "which is just the arithmetic mean of all the measurements."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The Sample Mean is an ML Estimator\n",
    "\n",
    "We have now shown that the maximum-likelihood estimator for a normal distribution's $\\mu$ parameter (assuming fixed $\\sigma$) is just the arithmetic mean of all observations.\n",
    "\n",
    "Remember that the arithmetic mean was **defined** as:\n",
    "\n",
    "$$ \\bar x \\equiv \\mathbf{E}[\\hat h] $$\n",
    "\n",
    "and because:\n",
    "\n",
    "$$ \\mu = \\mathbf{E}[N(\\mu, \\sigma)] $$\n",
    "\n",
    "we **hoped** it would be a good estimator of $\\mu$.\n",
    "\n",
    "Now that we **know** that the arithmetic mean is actually a Maximum Likelihood (ML) estimator ***for a homoscedastic Gaussian*** with known errors, we **know** it has all of the good ML estimator properties."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Properties of ML Estimators\n",
    "\n",
    "Assuming the data truly are drawn from the model, ML estimators have the following useful properties:\n",
    "\n",
    "* **They are consistent estimators**; that is, they can be proven to converge to the true parameter value as the number of data points increases.\n",
    "* **They are asymptotically normal estimators**. The distribution of the parameter estimate, as the number of data points increases to infinity, approaches a normal distribution, centered at the MLE, with a certain spread. This spread can often be easily calculated and used as a confidence band around the estimate, as discussed below.\n",
    "* **They asymptotically achieve the theoretical minimum possible variance, called the Cramér–Rao bound**. In other words, they achieve the best possible error given the data at hand; that is, no other estimator can do better in terms of efficiently using each data point to reduce the total error of the estimate (see eq. 3.33 in the textbook).\n",
    "\n",
    "Note: ML estimators are *not* guarenteed to be unbiased! Uncorrected standard deviation is the ML estimator for population standard deviation, yet as we've seen it's a biased estimator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Quantifying Estimate Uncertainty\n",
    "\n",
    "Let us look at what the full likelihood function looks like for our heteroscedastic Gaussian case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu_est = np.average(measurements)\n",
    "mu_est_err = 1./np.sqrt(Nsamples)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "L = lambda mu: np.prod( norm(measurements, 1).pdf(mu) )\n",
    "\n",
    "#------------------------------------------------------------\n",
    "# plot the likelihood distributions\n",
    "fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "mu = np.linspace(4, 6, 21)\n",
    "L_mu = np.vectorize(L)(mu)\n",
    "plt.scatter(mu, L_mu, c='black', label=r'$L(\\mu) \\equiv p(\\{x_i\\}|\\mu,\\sigma=1)$')\n",
    "\n",
    "# overplot the gaussian of the same width\n",
    "v = np.linspace(0, 10, 5000)\n",
    "C = L(mu_est) / norm(mu_est, mu_est_err).pdf(mu_est)\n",
    "plt.plot(v, C*norm(mu_est, mu_est_err).pdf(v), c='red', label=r'$\\propto N(\\mu=%.2f,\\ \\sigma=%.2f)$' % (mu_est, mu_est_err))\n",
    "plt.axvline(5)\n",
    "\n",
    "plt.xlim(4, 6)\n",
    "plt.ylim(0, 1.1*L(mu_est))\n",
    "plt.xlabel('$\\mu$')\n",
    "#plt.ylabel()\n",
    "plt.title('Likelihood of $\\mu$, given the measurement dataset $\\{x_i\\}$')\n",
    "\n",
    "# plot legend in the requested order (work around a matplot bug/feature)\n",
    "handles, labels = plt.gca().get_legend_handles_labels()\n",
    "order = [1, 0]\n",
    "plt.legend([handles[idx] for idx in order],[labels[idx] for idx in order]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This is just a Gaussian with the mean at $\\hat \\mu$ and standard deviation $\\frac{\\sigma}{\\sqrt N}$!\n",
    "\n",
    "Proof:\n",
    "\n",
    "$$L(\\mu) = \\left( \\prod_{i=1}^n \\frac{1}{\\sigma\\sqrt{2\\pi}} \\right) \\exp\\left( - \\frac{1}{2} \\sum \\left[\\frac{(x_i-\\mu)}{\\sigma} \\right]^2 \\right) \\propto \\exp\\left( - \\frac{1}{2\\sigma} \\sum (x_i-\\hat x + \\hat x  -\\mu)^2 \\right) =\\, ... $$\n",
    "$$ = \\exp\\left( - \\frac{N}{2\\sigma^2} \\sum (\\hat x  -\\mu)^2 \\right).$$\n",
    "\n",
    "It tells us how **relatively likely** any other value of $\\mu$ is. The $(\\hat \\mu - \\sigma_\\mu, \\hat \\mu + \\sigma_\\mu)$ range gives us a **confidence interval**. If we repeated the same measurement a hundred times, we'd find for 68 experiments the true value was within their computed confidence intervals ($1 \\sigma$ errors).\n",
    "\n",
    "Note: This also confirms that the standard error of the mean is equal to $\\frac{\\sigma}{\\sqrt N}$. I stated this previously when discussing the Central Limit Theorem, but now ***you*** *can prove this* starting with the expression for $L(\\mu)$ for the homoscedastic Gaussian model (after some straightforward but tedious manipulation of the sums under the exponential, above)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Generalizing the Quantification of Estimate Uncertainty\n",
    "\n",
    "For any likelihood distribution, with any number of parameters, we *define* the uncertainties on our MLEs as the second (partial) derivatives of the log-likelihood:\n",
    "\n",
    "$$\\sigma^{-2}_{jk} = - \\frac{d^2}{d\\theta_j} \\frac{\\ln L}{d\\theta_k} \\Biggr\\rvert_{\\theta=\\hat \\theta}$$\n",
    "\n",
    "Taken together, these entries are know as the **covariance matrix**.\n",
    "\n",
    "The marginal error bars for each parameter, $\\theta_i$ are given by the square roots of the diagonal elements, $\\sigma_{ii} = \\sigma^2_i$. These are the \"error bars\" that are typically quoted with each measurement."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "For our Gaussian example: \n",
    "$$\\sigma_{\\mu} = \\left( - \\frac{d^2\\ln L(\\mu)}{d\\mu^2}\\Biggr\\rvert_{\\hat \\mu}\\right)^{-1/2}$$\n",
    "\n",
    "we find\n",
    "$$\\frac{d^2\\ln L(\\mu)}{d\\mu^2}\\Biggr\\rvert_{\\hat \\mu} = - \\sum_{i=1}^N\\frac{1}{\\sigma^2} = -\\frac{N}{\\sigma^2},$$\n",
    "since, again, $\\sigma = {\\rm constant}$. Therefore:\n",
    "\n",
    "$$\\sigma_{\\mu} = \\frac{\\sigma}{\\sqrt{N}}.$$\n",
    "\n",
    "So, our estimator of $\\mu$ is $\\overline{x}\\pm\\frac{\\sigma}{\\sqrt{N}}$ as discussed above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### What is $\\pm \\sigma$? Interpretation as a Gaussian Approximation to the Likelihood Function\n",
    "\n",
    "The result for $\\sigma_{\\mu}$ has been derived by expanding $\\ln L$ in a Taylor series around the maximum and retaining terms up to second order (essentially, $\\ln L$ is approximated by a parabola, or an ellipsoidal surface in multidimensional cases, around its maximum). If this expansion is exact (as is the case for a Gaussian error distribution), then we've completely captured the uncertainty information.\n",
    "\n",
    "In general, this is not the case and the error bars give a ***Gaussian approximation*** to the actual likelihood surface. The true likelihood surface can significantly deviate from a smooth Gaussian shape. Furthermore, it often happens in practice that the likelihood surface is multimodal.\n",
    "\n",
    "It is always a good idea to visualize the likelihood surface when in doubt (see examples in §5.6 in the textbook)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### MLE applied to a Heteroscedastic Gaussian\n",
    "\n",
    "Now let's look at a case where the errors are heteroscedastic.  For example, the weather is different each time we measure the position of our quasar, and therefore the astrometric uncertainty $\\sigma_i$ is different for each measurement, but still known. Since $\\sigma$ is not a constant, then following what we previously wrote, we have:\n",
    "\n",
    "$$\\ln L = {\\rm constant} - \\sum_{i=1}^N \\frac{(x_i - \\mu)^2}{2\\sigma_i^2}.$$\n",
    "\n",
    "Taking the derivative:\n",
    "$$\\frac{d\\;{\\rm lnL}(\\mu)}{d\\mu}\\Biggr\\rvert_{\\hat \\mu} = \\sum_{i=1}^N \\frac{(x_i - \\hat \\mu)}{\\sigma_i^2} = 0,$$\n",
    "then simplifying:\n",
    "\n",
    "$$\\sum_{i=1}^N \\frac{x_i}{\\sigma_i^2} = \\sum_{i=1}^N \\frac{\\hat \\mu}{\\sigma_i^2},$$\n",
    "\n",
    "yields a MLE solution of \n",
    "$$\\hat \\mu = \\frac{\\sum_i^N (x_i/\\sigma_i^2)}{\\sum_i^N (1/\\sigma_i^2)},$$\n",
    "\n",
    "with uncertainty\n",
    "$$\\sigma_{\\mu} = \\left( \\sum_{i=1}^N \\frac{1}{\\sigma_i^2}\\right)^{-1/2}.$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Goodness of Fit\n",
    "\n",
    "The MLE approach tells us what the \"best\" model parameters are, but not how good the fit is in absolute terms.  If the model is wrong, \"best\" might not be particularly revealing !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Goodness of Fit and $\\chi^2$\n",
    "\n",
    "What does it mean that \"the fit is good\"? **We can quantify it by observing how likely it is we would draw a dataset with this particular $\\ln L(\\hat \\mu)$, if the model is correct, and seeing whether that's very unlikely (in which case, the hypothesis is probably incorrect)**.\n",
    "\n",
    "We can define a quantity $\\chi^2$:\n",
    "\n",
    "$$\\chi^2 \\equiv \\sum_{i=1}^N \\left({x_i-\\mu \\over \\sigma_i}\\right)^2,$$\n",
    "\n",
    "and then for the Gaussian case we've been studying we obtain:\n",
    "\n",
    "$$\\ln L = {\\rm constant} - \\frac{1}{2}\\sum_{i=1}^N z^2 = {\\rm constant} - \\frac{1}{2}\\chi^2.$$\n",
    "\n",
    "Note that there's a 1:1 relationship between the $\\chi^2$ and $\\ln L$. We can therefore also look at how likely is it that a particular $\\chi^2$ would occur when drawing a sample. (this is how it's traditionally done, and has a few other benefits)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The distribution of $\\chi^2$ values\n",
    "\n",
    "Let's compute the $\\chi^2$s for (say) $5,000$ samples of (say) 5 measurements, and see what the distribution looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe698b51150>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "np.random.seed(seed=42)\n",
    "Nsamples=5\n",
    "mu_true = 5.\n",
    "sigma = 1.\n",
    "\n",
    "Ndraws = 5000\n",
    "chi2vals = np.empty(Ndraws)\n",
    "\n",
    "for i in range(Ndraws):\n",
    "    measurements = np.random.normal(mu_true, sigma, Nsamples)\n",
    "    mu_est = np.average(measurements)\n",
    "    chi2vals[i] = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "ax.hist(chi2vals, bins='fd', density=True);\n",
    "\n",
    "from scipy.stats import chi2\n",
    "v = np.linspace(*ax.get_xlim(), 1000)\n",
    "ax.plot(v, chi2.pdf(v, Nsamples - 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Our computed $\\chi^2$ values are distributed as the **$\\chi^2$ distribution (with $k=N-N_{par}$ degrees of freedom).**\n",
    "\n",
    "The *number of degrees of freedom*, $k$ is given by the number of data points, $N$, minus any constraints (the number of model parameters, $N_{par}$). Given those two, the $\\chi^2$ distribution is defined by:\n",
    "\n",
    "$$p(Q \\mid k)\\equiv \\chi^2(Q \\mid k) = \\frac{1}{2^{k/2}\\Gamma(k/2)}Q^{k/2-1}\\exp(-Q/2),$$\n",
    "\n",
    "where $Q \\equiv \\chi^2 >0$. Note that the $\\Gamma$ function would just be the usual factorial function if we were dealing with integers, but here we have half integers.\n",
    "\n",
    "This is ugly, but it is really just a formula like anything else.  Note that the shape of the distribution *only* depends on the sample size $N=N-N_{par}$ and not on either $\\mu$ or $\\sigma$!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Testing for goodness of fit\n",
    "\n",
    "The main idea is to see how probable is it that, if this is the correct model, our particular value of $\\chi^2$ would arise. If the probability is low (discuss: how low?), we may need to rethink our hypothesis.\n",
    "\n",
    "Example: we now draw a sample from a uniform distribution, but still assume it's just a single value + Gaussian errors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "χ^2(uniform sample) = 1.796\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7fe680acd510>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x540 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "np.random.seed(seed=42)\n",
    "Nsamples=5\n",
    "mu_true = 5.\n",
    "sigma = 1.\n",
    "\n",
    "Ndraws = 5000\n",
    "chi2vals = np.empty(Ndraws)\n",
    "\n",
    "for i in range(Ndraws):\n",
    "    measurements = np.random.normal(mu_true, sigma, Nsamples)\n",
    "    mu_est = np.average(measurements)\n",
    "    chi2vals[i] = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "ax.hist(chi2vals, bins='fd', density=True);\n",
    "\n",
    "from scipy.stats import chi2\n",
    "v = np.linspace(*ax.get_xlim(), 1000)\n",
    "ax.plot(v, chi2.pdf(v, Nsamples - 1))\n",
    "\n",
    "# Test with a sample _not_ drawn from the gaussian model (say, uniform)\n",
    "measurements = np.random.uniform(mu_true - sigma, mu_true + sigma, Nsamples)\n",
    "mu_est = np.average(measurements)\n",
    "chi2uni = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "print(\"χ^2(uniform sample) = %.3f\" % chi2uni)\n",
    "ax.axvline(chi2uni, color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let's play with this a bit more..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5757b489831a4944b3c4135c7271c5a5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, continuous_update=False, description='Nsamples', max=500, min=2), Flo…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "np.random.seed(seed=42)\n",
    "Nsamples=5\n",
    "mu_true = 5.\n",
    "sigma = 1.\n",
    "\n",
    "import ipywidgets as widgets\n",
    "from ipywidgets import interact, fixed\n",
    "\n",
    "def make_chi2_plot(Nsamples, mu_true, sigma):\n",
    "    Ndraws = 5000\n",
    "    chi2vals = np.empty(Ndraws)\n",
    "\n",
    "    for i in range(Ndraws):\n",
    "        measurements = np.random.normal(mu_true, sigma, Nsamples)\n",
    "        mu_est = np.average(measurements)\n",
    "        chi2vals[i] = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "\n",
    "    fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "    ax.hist(chi2vals, bins='fd', density=True);\n",
    "\n",
    "    from scipy.stats import chi2\n",
    "    v = np.linspace(*ax.get_xlim(), 1000)\n",
    "    c = chi2.pdf(v, Nsamples - 1)\n",
    "    ax.plot(v, c)\n",
    "    \n",
    "    # Test with a sample _not_ drawn from the gaussian model (say, uniform)\n",
    "    measurements = np.random.uniform(mu_true - sigma, mu_true + sigma, Nsamples)\n",
    "    mu_est = np.average(measurements)\n",
    "    chi2uni = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "    print(\"χ^2(uniform sample) = %.3f\" % chi2uni)\n",
    "    ax.axvline(chi2uni, color='red')\n",
    "\n",
    "    ax.set_xlim(chi2.ppf(0.001, Nsamples - 1), chi2.ppf(0.999, Nsamples - 1))\n",
    "    ax.set_ylim(0, 1.1*c.max())\n",
    "\n",
    "\n",
    "interact(make_chi2_plot,\n",
    "         Nsamples=widgets.IntSlider(min=2, max=500, step=1, value=5, continuous_update=False),\n",
    "         mu_true=widgets.FloatSlider(min=0, max=10, step=1, value=5, continuous_update=False),\n",
    "         sigma=widgets.FloatSlider(min=0.1, max=6, step=0.1, value=1, continuous_update=False),\n",
    "        );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that:\n",
    "* For small `Nsamples`, it's difficult to accept the hypothesis that the uniform has not been drawn from our model\n",
    "* The shape of the distribution is independent of $\\mu$ and $\\sigma$\n",
    "* The more samples there are (larger `Nsamples`), the more the $\\chi^2$ distribution looks like a Gaussian!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Gaussian Approximation: $\\chi^2$ per degree of freedom\n",
    "\n",
    "For large $k$ (say, $k > 100$ or so), $\\chi^2$-distribution becomes well approximated by the Normal distribution (Gaussian):\n",
    "\n",
    "$$ p(\\chi^2 \\mid k) \\sim N(\\chi^2 \\mid k, \\sqrt{2k}) $$\n",
    "\n",
    "In practice we frequently divide $\\chi^2$ by the number of degrees of freedom, and work with:\n",
    "\n",
    "$$\\chi^2_{dof} = \\frac{1}{N-N_{par}} \\sum_{i=1}^N \\left(\\frac{x_i-\\overline{x}}{\\sigma}\\right)^2$$\n",
    "\n",
    "which is distributed as\n",
    "\n",
    "$$ p(\\chi^2_{dof}) \\sim N\\left(\\chi^2_{dof} \\rvert 1, \\sqrt{\\frac{2}{N-N_{par}}}\\right) $$\n",
    "\n",
    "(where $k = N-N_{par}$). Therefore, we expect $\\chi^2_{dof}$ to be equal to 1, to within a few $\\sqrt{\\frac{2}{N-N_{par}}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Exploring the Gaussian approximation\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d001be622b6f4f8398036b56c58fdabe",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, continuous_update=False, description='Nsamples', max=1000, min=2), Fl…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "from astroML.plotting import setup_text_plots\n",
    "setup_text_plots(fontsize=15, usetex=False)\n",
    "\n",
    "#------------------------------------------------------------\n",
    "np.random.seed(seed=42)\n",
    "Nsamples=5\n",
    "mu_true = 5.\n",
    "sigma = 1.\n",
    "\n",
    "import ipywidgets as widgets\n",
    "from ipywidgets import interact, fixed\n",
    "\n",
    "def make_chi2_plot(Nsamples, mu_true, sigma):\n",
    "    Ndraws = 5000\n",
    "    chi2vals = np.empty(Ndraws)\n",
    "\n",
    "    for i in range(Ndraws):\n",
    "        measurements = np.random.normal(mu_true, sigma, Nsamples)\n",
    "        mu_est = np.average(measurements)\n",
    "        chi2vals[i] = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "\n",
    "    fig, ax = plt.subplots(figsize=(12, 7.5))\n",
    "    ax.hist(chi2vals, bins='fd', density=True);\n",
    "\n",
    "    from scipy.stats import chi2\n",
    "    v = np.linspace(*ax.get_xlim(), 1000)\n",
    "    c = chi2.pdf(v, Nsamples - 1)\n",
    "    ax.plot(v, c, label=r'$\\chi^2(k)$')\n",
    "\n",
    "    g = norm(Nsamples - 1, np.sqrt(2*(Nsamples - 1))).pdf(v)\n",
    "    ax.plot(v, g, color='gray', label=r'$N(k, \\sqrt{2k})$ approximation')\n",
    "    \n",
    "    # Test with a sample _not_ drawn from the gaussian model (say, uniform)\n",
    "    measurements = np.random.uniform(mu_true - sigma, mu_true + sigma, Nsamples)\n",
    "    mu_est = np.average(measurements)\n",
    "    chi2uni = np.sum(np.square((measurements - mu_est) / sigma))\n",
    "    print(\"χ^2(uniform sample) = %.3f\" % chi2uni)\n",
    "    ax.axvline(chi2uni, color='red')\n",
    "\n",
    "    ax.set_xlim(chi2.ppf(0.001, Nsamples - 1), chi2.ppf(0.999, Nsamples - 1))\n",
    "    ax.set_ylim(0, 1.1*c.max())\n",
    "\n",
    "    ax.legend()\n",
    "\n",
    "\n",
    "interact(make_chi2_plot,\n",
    "         Nsamples=widgets.IntSlider(min=2, max=1000, step=1, value=5, continuous_update=False),\n",
    "         mu_true=widgets.FloatSlider(min=0, max=10, step=1, value=5, continuous_update=False),\n",
    "         sigma=widgets.FloatSlider(min=0.1, max=6, step=0.1, value=1, continuous_update=False),\n",
    "        );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Goodness of Fit: An Example\n",
    "\n",
    "Physically, \"not using the correct model\" may come either from the object, or the misunderstanding of our measurement apparatus. For example, we can also get overly high or low values of $\\chi^2_{dof}$ if our errors are under- or over-estimated.\n",
    "\n",
    "<img src=\"http://www.astroml.org/_images/fig_chi2_eval_1.png\" width=800>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Summary\n",
    "\n",
    "So now, given a model, we have a prescription for:\n",
    "\n",
    "* Estimating its parameters (write the likelihood, find the maximum)\n",
    "* Estimating the (approximate) error bars (inverse derivatives of the log likelihood)\n",
    "* Assessing the goodness of fit (compare the obtained vs expected $\\chi^2$)\n",
    "\n",
    "and this is universally applicable (Gauss, Cauchy, Poisson, etc..), though analytically soluble only for a few distributions (e.g., Gauss)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Truncated/Censored Data and Other Cost Functions\n",
    "\n",
    "Note that dealing with missing data points (\"censored data\") adds complications that we don't have time to get into here, but see Textbook 4.2.7  Also maximum likelihood is just one possible \"cost function\", see Textbook, 4.2.8. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Next time:\n",
    "\n",
    "* Least-Squares as a Maximum Likelihood problem\n",
    "* Optimal photometry as a Maximum Likelihood problem\n",
    "* Confidence Estimates, particularly Bootstrap and Jackknife error estimates (Textbook $\\S$4.5)\n",
    "* Hypothesis testing (Textbook $\\S$4.7)"
   ]
  }
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