{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# **Statistics lecture 4 Hands-on session : solutions notebook**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the companion notebook to lecture 4 in the statistical course series, covering the following topics:\n", "1. Parameter estimation using simple tools\n", "2. Parameter estimation using the profile likelihood\n", "3. Limit-setting, using simple tools\n", "4. Limit-setting using the profile likelihood\n", "\n", "First perform the usual imports:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import scipy.stats\n", "from matplotlib import pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Parameter estimation using simple techniques" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the previous lecture, we have applied hypothesis testing to a simple counting experiment, using the observed count $n$ as discriminant.\n", "Recall the example we used:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "nbins = 10\n", "x = np.linspace(0.5, nbins - 0.5, nbins)\n", "# The background follows a linear shape\n", "b_yields = np.array([ (1 - i/2/nbins) for i in range(0, nbins) ])\n", "b_yields *= b_yields/np.sum(b_yields)\n", "# The signal shape is a peak\n", "s_yields = np.zeros(nbins)\n", "s_yields[3:7] = [ 0.1, 0.4, 0.4, 0.1 ]\n", "# Now generate some data\n", "s = 10\n", "b = 500\n", "s_and_b = s*s_yields + b*b_yields\n", "b_only = b*b_yields\n", "np.random.seed(1) # make sure we always generate the same\n", "data = [ np.random.poisson(s*s_yield + b*b_yield) for s_yield, b_yield in zip(s_yields, b_yields) ]\n", "s30_and_b = 30*s_yields + b*b_yields\n", "plt.bar(x, s30_and_b, color='orange', label='s=30')\n", "plt.bar(x, b_only, color='b', yerr=np.sqrt(b_only), label='b only')\n", "plt.scatter(x, data, zorder=10, color='k')\n", "plt.xlabel('bin number')\n", "plt.ylabel('Total expected yield')\n", "plt.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are some fluctuations in the data, but it seems that the $s$ is fairly small compared to the $s=30$ shown in orange (note that we cheated and generated the data for $s=10$, so this is not unexpected!). We can use the tools defined in the previous lecture to find the best-fit $s$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "best-fit s = 3.523759577550049\n" ] } ], "source": [ "def lambda_s(s_hypo, dataset) :\n", " return -2*sum( [ np.log(scipy.stats.poisson.pmf(n, s_hypo*s_yield + b*b_yield)) for n, s_yield, b_yield in zip(dataset, s_yields, b_yields) ] )\n", "\n", "from scipy.optimize import minimize_scalar\n", "s_hat = minimize_scalar(lambda s_hypo : lambda_s(s_hypo, data), (0, 100)).x\n", "print('best-fit s =', s_hat)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So it's indeed pretty small. However we also want an uncertainty to go with the central value -- this allows for instance to estimate the compatibility with a reference value such as $s=0$ or $s=30$.\n", "\n", "We'll see how to do this properly in the next section, but we'll try a simpler situation first.\n", "\n", "For this, only consider a measurement in one bin, and assume it is Gaussian:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sb4 = s30_and_b[4]\n", "b4 = b_only[4]\n", "n4 = data[4]\n", "plt.bar(x[4], sb4, color='orange', label='s=30')\n", "plt.bar(x[4], b4, color='b', yerr=np.sqrt(b_only[4]), label='b only')\n", "plt.scatter(x[4], n4, zorder=10, color='k')\n", "plt.xlabel('bin number')\n", "plt.ylabel('Total expected yield')\n", "plt.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this simple model, the best-fit value is" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2.70967741935484\n" ] } ], "source": [ "s4 = n4 - b4\n", "print(s4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And more importantly, we know that the uncertainty on this value is" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "6.6332495807108\n" ] } ], "source": [ "sigma4 = np.sqrt(n4)\n", "print(sigma4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So we have our result: $s_4 = 2.7 \\pm 6.6$.\n", "\n", "One thing to understand about this result: it does not mean the true value is necessarily between $2.7 - 6.6$ and $2.7 + 6.6$. The interval is to be taken in the sense of a Gaussian distribution: there is a $68\\%$ chance that the true value is indeed somewhere in this interval, but it can also fall outside.\n", "\n", "This can be illustrated as follows, where we draw a large number of random values for n4 and compute the interval for each case:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "fraction of intervals containing true value: 0.8\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Generate 100 random values of n4, for a true signal of 10\n", "ntoys = 100\n", "s_true = 10\n", "toy_sigma4 = np.sqrt(10 + b4)\n", "toy_n4s = [ np.random.normal(b4 + s_true, toy_sigma4) for i in range(0, ntoys) ]\n", "\n", "# Plot the intervals\n", "s_range = np.arange(s_true - 3*toy_sigma4, s_true + 3*toy_sigma4,0.1)\n", "plt.plot(s_range, scipy.stats.norm.pdf(s_range, loc=s_true, scale=toy_sigma4))\n", "plt.ylim(0,0.1)\n", "plt.xlabel('best_fit_s')\n", "plt.ylabel('P(best_fit_s)')\n", "plt.errorbar(x = toy_n4s - b4, y = np.linspace(0, 0.10, ntoys), xerr=[ toy_sigma4 ] *ntoys, capsize=3, marker='o');\n", "\n", "# Compute the best-fit s4 and the interval for each toy, then count the number of intervals that contain the true value\n", "toy_best_s4s = [ toy_n4 - b4 for toy_n4 in toy_n4s ]\n", "toy_intervals = [ (toy_s4 - toy_sigma4, toy_s4 + toy_sigma4) for toy_s4 in toy_best_s4s ]\n", "n_good_intervals = sum([ s_true > lo and s_true < hi for (lo, hi) in toy_intervals ])\n", "print('fraction of intervals containing true value: ', n_good_intervals/ntoys)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So fraction of intervals containing the true value is 68\\% as expected (you can increase the number of toys to get a better estimate, although the plot will look more cluttered...) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Parameter estimation using likelihood methods" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above method is only applicable to single-bin measurements, so we need something better for our multi-bin case.\n", "\n", "To test some value $s$, we use the likelihood-based discriminant\n", "$$\n", "t(s) = -2 \\log \\frac{L(s)}{L(\\hat{s})}\n", "$$\n", "following the same principle as for discovery. Define it:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "def t_s(s_test) :\n", " return lambda_s(s_test, data) - lambda_s(s_hat, data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now plot it:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "s_vals = np.arange(-10,20,1)\n", "plt.plot(s_vals, t_s(s_vals));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It follows a parabola, which is expected for a near-Gaussian case. A short computation shows that for a Gaussian likelihood $\\hat{s} \\sim G(s, \\sigma_s)$, one has\n", "$$\n", "t(s) = \\left(\\frac{s - \\hat{s}}{\\sigma_s}\\right)^2\n", "$$\n", "which explains the parabola.\n", "\n", "In this Gaussian case, the $\\pm 1\\sigma$ uncertainties are reached for $t(s)=1$. We will assume that this still holds true in cases which are not exactly Gaussian, so that we **define** the uncertainties using the crossings with $t(s)=1$:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "14.616612758932273 -6.82979837729991\n", "3.523759577550049 + 11.092853181382225 - 10.35355795484996\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(s_vals, t_s(s_vals))\n", "plt.xlabel('s')\n", "plt.ylabel('t(s)')\n", "plt.axhline(1, linestyle='--')\n", "\n", "# Find the crossings with t(s) = 1\n", "s_pos = scipy.optimize.root_scalar(lambda s : t_s(s) - 1, bracket=(10,20)).root\n", "s_neg = scipy.optimize.root_scalar(lambda s : t_s(s) - 1, bracket=(-10,0)).root\n", "print(s_pos, s_neg)\n", "print (s_hat, '+', s_pos - s_hat, '-', s_hat - s_neg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This defines a type of confidence intervals called *likelihood intervals* which are not quite exact in the non-Gaussian case but are an excellent approximation to the exact ones. Non-Gaussian effects are accounted for in the expression of $t(s)$, which here uses a Poisson expression, so this is a better approximation than just assuming that everything is Gaussian. This type of technique is usually referred to as an *asymptotic approximation*, and we'll see a similar example below for the limits." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Limit-setting using simple tools\n", "\n", "Measuring the signal is useful when it is large, but when it is very small it can be more interesting to set an upper limit on it. This means computing the value of the signal yield that is large enough to be excluded with a given confidence level, typically 95%.\n", "\n", "\n", "This happens to be the case for our measurement, so we can apply the limit-setting here. To start with, we consider only the measurement in bin 4.\n", "\n", "This \"95%\" means that if we would repeat the experiment many times, the true signal would be below the limit 95% of the time. This is also represented in the picture below, with the shaded area corresponding to 5% of the integral." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sigma4 = np.sqrt(n4)\n", "ns = np.arange(35,85,0.1)\n", "plt.plot(ns, scipy.stats.norm.pdf(ns, loc=n4 + 1.64*sigma4, scale=sigma4))\n", "plt.ylim(0,0.1)\n", "plt.xlabel('n4')\n", "plt.ylabel('P(n4)')\n", "\n", "up = 1\n", "shaded_ns = np.arange(35, n4, 0.1)\n", "plt.fill_between(shaded_ns, scipy.stats.norm.pdf(shaded_ns, loc=n4 + 1.64*sigma4, scale=sigma4), alpha=0.5, color='g')\n", "plt.axvline(x=n4 + 1.64*sigma4, linestyle='--', color='k')\n", "plt.axvline(x=n4, linestyle='--', color='k')\n", "plt.text(n4 + 1.64*sigma4 + 2, 0.08, 's95 + b4')\n", "plt.text(n4 - 11, 0.08, 'observed n4');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now compute the limit in the single-bin case, using the values of $s_4$, $b_4$ $\\sigma_4$ and $n_4$ defined in the previous sections." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.6448536269514729\n", "95% CL limit = 13.620402050661333\n" ] } ], "source": [ "# First find the position on the Gaussian corresponding to a 5% quantile\n", "z_5 = scipy.stats.norm.isf(0.05)\n", "print(z_5)\n", "# and the limit\n", "s95 = n4 + z_5*sigma4 - b4\n", "print('95% CL limit = ', s95)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. Limit-setting in the general case" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's apply limit-setting on the multi-bin example. This is justified here, since from the previous section we know the signal is not large." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.bar(x, s30_and_b, color='orange', label='s=30')\n", "plt.bar(x, b_only, color='b', yerr=np.sqrt(b_only), label='b only')\n", "plt.scatter(x, data, zorder=10, color='k')\n", "plt.xlabel('bin number')\n", "plt.ylabel('Total expected yield')\n", "plt.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As usual, we use the likelihood-based discriminant\n", "$$\n", "t(s) = -2 \\log \\frac{L(s)}{L(\\hat{s})}\n", "$$\n", "\n", "The only difference compared to parameter measurement is that we will only consider the positive side of the distribution, $s > \\hat{s}$, since we are looking for an *upper* limit. \n", "\n", "As we saw before, for the Gaussian case we have\n", "$$\n", "t(s) = \\left(\\frac{s - \\hat{s}}{\\sigma_s}\\right)^2\n", "$$\n", "so that the limit is reached for $t(s) = 1.64^2$. Of course our model is not exactly Gaussian, but we can still use the Gaussian relation $t(s) = 1.64^2$ to define our limit, using the exat non-Gaussian expression for $t(s)$. This still accounts for non-Gaussianity through the expression of $t(s)$, and usually provides an excellent approximation to the true limit. This technique is usually referred to as the *asymptotic* approximation.\n", "\n", "Another way to find the limit is to note that the p-value is given by $1 - \\Phi\\left(\\sqrt{t(s)}\\right)$, assuming the Gaussian/asymptotic approximation. This allows to compute the p-value as a function of $s$. Then to compute the limit, we check when this p-value reaches 5% (for a 95% CL limit)\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ " converged: True\n", " flag: 'converged'\n", " function_calls: 9\n", " iterations: 8\n", " root: 22.168063096101502" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "s_vals = np.linspace(10, 50, 100)\n", "plt.plot(s_vals, scipy.stats.norm.sf(np.sqrt(t_s(s_vals))))\n", "plt.xlabel('s')\n", "plt.ylabel('p-value')\n", "plt.axhline(0.05, linestyle='--', color='k')\n", "scipy.optimize.root_scalar(lambda s : scipy.stats.norm.sf(np.sqrt(t_s(s))) - 0.05, bracket=(20,30) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 5. Expected limits" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can compute *expected* limits for a given scenario, for instance $s=0$. This can be obtained using 2 methods:\n", "* By generating \"toy datasets\" (pseudo-experiments), computing the limit for each one and histogramming the result\n", "* by fitting a special \"Asimov\" dataset that is constructed to precisely correspond to this hypothesis. For instance here one would build a dataset consisting only of background.\n", "\n", "First, we investigate the difference between the two:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "toy_data1 = np.random.poisson(b_only)\n", "toy_data2 = np.random.poisson(b_only)\n", "toy_data3 = np.random.poisson(b_only)\n", "plt.bar(x, s30_and_b, color='orange', label='s=30')\n", "plt.bar(x, b_only, color='b', yerr=np.sqrt(b_only), label='b only')\n", "plt.scatter(x, toy_data1, zorder=10, color='k', label='toy 1')\n", "plt.scatter(x, toy_data2, zorder=10, color='lime', label='toy 2')\n", "plt.scatter(x, toy_data3, zorder=10, color='m', label='toy 3')\n", "plt.xlabel('bin number')\n", "plt.ylabel('Total expected yield')\n", "plt.legend()\n", "plt.figure()\n", "asimov = b_only\n", "plt.bar(x, s30_and_b, color='orange', label='s=30')\n", "plt.bar(x, b_only, color='b', yerr=np.sqrt(b_only), label='b only')\n", "plt.scatter(x, asimov, zorder=10, color='k', label='Asimov')\n", "plt.xlabel('bin number')\n", "plt.ylabel('Total expected yield')\n", "plt.legend();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The toys fluctuate around the b-only expectation, while the Asimov perfectly lines up with it by definition.\n", "\n", "To get Asimov-based results, one also includes the range of variations of these limits du to the fluctuations of $\\hat{s}$. Since\n", "$$\n", "t(s) = \\left(\\frac{s - \\hat{s}}{\\sigma_s}\\right)^2\n", "$$\n", "the $\\pm 1\\sigma$ variations in the limit are given by\n", "$$\n", "\\sqrt{t(s)} \\to \\sqrt{t(s)} \\pm 1 \\text{ for the } \\pm 1\\sigma \\text{ variation}\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def lambda_gamma_s(s_hypo, data) :\n", " return -2*sum( [ np.log(scipy.stats.gamma.pdf(n+1, s_hypo*s_yield + b*b_yield)) for n, s_yield, b_yield in zip(data, s_yields, b_yields) ] )\n", "\n", "def t_s_asimov(s_test) :\n", " return lambda_gamma_s(s_test, asimov) - lambda_gamma_s(0, asimov)\n", "\n", "s_vals = np.linspace(10, 60, 100)\n", "observed = scipy.stats.norm.sf(np.sqrt(t_s(s_vals)))\n", "expected = scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)))\n", "exp_pos1 = scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) + 1)\n", "exp_neg1 = scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) - 1)\n", "exp_pos2 = scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) + 2)\n", "exp_neg2 = scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) - 2)\n", "plt.fill_between(s_vals, exp_pos2, exp_neg2, color='y', label='2sigma')\n", "plt.fill_between(s_vals, exp_pos1, exp_neg1, color='lightgreen', label='1sigma')\n", "plt.plot(s_vals, observed, label='observed')\n", "plt.plot(s_vals, expected, color='k', linestyle='--', label='expected')\n", "plt.axhline(0.05, linestyle=':', color='k')\n", "plt.legend();\n", "#plt.fill_between(s_vals, scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) + 2), scipy.stats.norm.sf(np.sqrt(t_s_asimov(s_vals)) - 2), color='y')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As before, we can find the limit values by looking at the intersection with $p=0.05$\n", "\n", "## 6. $CL_s$ limits\n", "\n", "We now add one last wrinkle: the $CL_s$ procedure. This is an additional correction on top of the limit setting procedure described above, to avoid the limit going negative. Just to see why this is needed, recall how we set the limit in the 1-bin case:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "n4 = 44 , b4 = 41.29032258064516\n", "95% CL limit = 13.620402050661333\n" ] } ], "source": [ "print('n4 =', n4, ', b4 =', b4)\n", "s95 = n4 + z_5*sigma4 - b4\n", "print('95% CL limit = ', s95)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now imagine we had a negative fluctuation in $n_4$, and repeat the computation" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "n4 = 30 , b4 = 41.29032258064516\n", "95% CL limit = -0.37959794933866675\n" ] } ], "source": [ "n4 = 30\n", "print('n4 =', n4, ', b4 =', b4)\n", "s95 = n4 + z_5*sigma4 - b4\n", "print('95% CL limit = ', s95)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This requires a pretty large fluctuation, but it can happen. Of course we know that the signal is positive, so we know that this is due to a fluctuation: basically we know we are in those 5% of experiments where the limit isn't valid.\n", "\n", "One way to do this is to correct the p-value for $s \\sim 0$ to avoid setting an exclusion in this region. This is what $CL_s$ does, by using\n", "$$\n", "p_{CL_s} = \\frac{p(s)}{p(s=0)}\n", "$$\n", "as a modified p-value. If there is good exclusion near 0 ($p(0) \\ll 1$) then the denominator is small, which inflates $p_{CL_s}$ and makes the exclusion weaker.\n", "We can modify our previous example to do this:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "observed_cls = observed / [ scipy.stats.norm.sf(np.sqrt(t_s(s)) - np.sqrt(t_s_asimov(s))) for s in s_vals ]\n", "expected_cls = expected / [ scipy.stats.norm.sf(np.sqrt(t_s_asimov(s)) - np.sqrt(t_s_asimov(s))) for s in s_vals ]\n", "exp_pos1_cls = exp_pos1 / [ scipy.stats.norm.sf(np.sqrt(t_s_asimov(s)) - np.sqrt(t_s_asimov(s)) + 1) for s in s_vals ]\n", "exp_neg1_cls = exp_neg1 / [ scipy.stats.norm.sf(np.sqrt(t_s_asimov(s)) - np.sqrt(t_s_asimov(s)) - 1) for s in s_vals ]\n", "exp_pos2_cls = exp_pos2 / [ scipy.stats.norm.sf(np.sqrt(t_s_asimov(s)) - np.sqrt(t_s_asimov(s)) + 2) for s in s_vals ]\n", "exp_neg2_cls = exp_neg2 / [ scipy.stats.norm.sf(np.sqrt(t_s_asimov(s)) - np.sqrt(t_s_asimov(s)) - 2) for s in s_vals ]\n", "plt.fill_between(s_vals, exp_pos2_cls, exp_neg2_cls, color='y', label='2sigma')\n", "plt.fill_between(s_vals, exp_pos1_cls, exp_neg1_cls, color='lightgreen', label='1sigma')\n", "plt.plot(s_vals, observed_cls, label='observed')\n", "plt.plot(s_vals, expected_cls, color='k', linestyle='--', label='expected')\n", "plt.axhline(0.05, linestyle=':', color='k')\n", "plt.legend()\n", "plt.ylim(0,1);\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that the exclusion indeed becomes weaker near 0, which is now not excluded anymore by the $2\\sigma$ band." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }