{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Reporting summaries of the posterior\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
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\\n\"+\n", " \"\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
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i < css_urls.length; i++) {\n", " var url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " }\n", "\n", " for (var i = 0; i < js_urls.length; i++) {\n", " var url = js_urls[i];\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };var element = document.getElementById(\"1001\");\n", " if (element == null) {\n", " console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n", " return false;\n", " }\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " \n", " var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.4.0.min.js\"];\n", " var css_urls = [];\n", " \n", "\n", " var inline_js = [\n", " function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", " function(Bokeh) {\n", " \n", " \n", " }\n", " ];\n", "\n", " function run_inline_js() {\n", " \n", " if (root.Bokeh !== undefined || force === true) {\n", " \n", " for (var i = 0; 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\\n\"+\n \"

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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded() {\n var el = document.getElementById(\"1001\");\n if (el != null) {\n el.textContent = \"BokehJS is loading...\";\n }\n if (root.Bokeh !== undefined) {\n if (el != null) {\n el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(display_loaded, 100)\n }\n }\n\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (var i = 0; i < js_urls.length; i++) {\n var url = js_urls[i];\n var element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error;\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };var element = document.getElementById(\"1001\");\n if (element == null) {\n console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n return false;\n }\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n \n var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.4.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.4.0.min.js\"];\n var css_urls = [];\n \n\n var inline_js = [\n function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\n function(Bokeh) {\n \n \n }\n ];\n\n function run_inline_js() {\n \n if (root.Bokeh !== undefined || force === true) {\n \n for (var i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n if (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import pandas as pd\n", "\n", "import arviz as az\n", "\n", "import bokeh_catplot\n", "\n", "import bokeh.io\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Reporting summaries of MCMC samples\n", "\n", "Performing MCMC calculations gives you full information about the posterior, which you can summarize in beautiful and informative corner plots. But very often, we wish to summarize the posterior in a few simple numbers. In particular, we wish to report a **credible interval** for a single parameter. This is computed from the marginalized posterior. The credible interval is a region in parameter space where we might expect a parameter value to lie. This credible interval is often reported and plotted as an error bar.\n", "\n", "We will consider three commonly used ways of plotting a value plus error bar. We will use a 95% credible interval for this demonstration.\n", "\n", "1. **mean ± standard deviation**: The most commonly used credible interval is $\\mu \\pm k\\sigma$, where $k$ is chosen to give the appropriate credible interval, approximating the marginalized posterior as Normal. We'll do 95%, which means $k = 1.96$.\n", "2. **median with quantile**: The posterior need not be Normal. If it is not, we would like a more robust way to summarize it. A simple method is to report the median, and then give lower and upper bounds to the error bar based on quantile. In our case, we would report the 2.5th percentile and the 97.5th percentile.\n", "3. **mode with HPD**: This method uses the **highest posterior density** interval, or HPD. If we're considering a 95% credible interval, the HPD interval is the shortest interval that contains 95% of the probability mass of the posterior. So, we report the mode (the most probable parameter value) and then the bounds on the HPD interval.\n", "4. **median with HPD**: We report the HPD, but report the median parameter value from our samples instead of the mode.\n", "\n", "To illustrate the relative merits of these reporting schemes, we'll draw samples samples out of some artificial distributions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Some distributions to sample\n", "\n", "We will generate some distributions to sample. We will consider an exponential distribution, a Gaussian, the sum of two Normals, and a distribution with a long tail. We choose these to illustrate how various choices of the credible interval will be reported.\n", "\n", "Let's define the models and draw samples. MCMC is not necessary here for this illustrative exercise, so I will just draw the samples using Numpy." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Parametrize models\n", "mu = 1.0\n", "sigma = 0.25\n", "mu_2 = np.array([mu, 3.0])\n", "sigma_2 = np.array([sigma, 0.5])\n", "\n", "# Draw out of the distributions, start with exponential\n", "x_expon = np.random.exponential(mu, size=15000)\n", "\n", "# Normal\n", "x_norm = np.random.normal(mu, sigma, size=15000)\n", "\n", "# Bimodal mixture of Normals\n", "which_norm = np.random.choice([0, 1], 15000)\n", "x_2norm = np.random.normal(mu_2[which_norm], sigma_2[which_norm])\n", "\n", "# Heavy tailed Pareto\n", "x_heavytail = np.random.pareto(1.2, size=15000)\n", "\n", "# Store samples in a list for easy access\n", "samples = [x_expon, x_norm, x_2norm, x_heavytail]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's look at what we got by plotting the histograms." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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tail\"]\n", "plots = [\n", " bokeh_catplot.histogram(\n", " x,\n", " density=True,\n", " kind='step',\n", " frame_width=250,\n", " frame_height=200,\n", " x_axis_label=\"x\",\n", " y_axis_label=\"pdf\",\n", " title=t,\n", " line_kwargs={\"line_width\": 2, \"line_color\": \"gray\"},\n", " )\n", " for x, t in zip(samples[:-1], names[:-1])\n", "]\n", "\n", "# Make plot for heavy tail (needs different bins because of heavy tail)\n", "plots.append(\n", " bokeh_catplot.histogram(\n", " samples[-1],\n", " bins=np.arange(21),\n", " density=True,\n", " kind='step',\n", " frame_width=250,\n", " frame_height=200,\n", " x_axis_label=\"x\",\n", " y_axis_label=\"pdf\",\n", " title=\"Heavy tail\",\n", " line_kwargs={\"line_width\": 2, \"line_color\": \"gray\"},\n", " )\n", ")\n", "\n", "# Show in a grid\n", "bokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Summarizing the \"MCMC\" results with error bars\n", "\n", "We'll start by computing the mean, standard deviation, median, and quantiles, since these are easy to compute with NumPy." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# DataFrame to store summary stats\n", "df_summary = pd.DataFrame(\n", " index=[\"mean\", \"std\", \"2.5\", \"median\", \"97.5\", \"mode\", \"hpd_low\", \"hpd_high\"],\n", " columns=names,\n", ")\n", "\n", "for x, name in zip(samples, names):\n", " df_summary.loc[\"mean\", name] = np.mean(x)\n", " df_summary.loc[\"std\", name] = np.std(x)\n", " df_summary.loc[[\"2.5\", \"median\", \"97.5\"], name] = np.percentile(x, [2.5, 50, 97.5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Computation of the HPD is a little trickier. The idea is that we rank-order the MCMC trace. We know that the number of samples that are included in the HPD is 0.95 times the total number of MCMC samples. We then consider all intervals that contain that many samples and find the shortest one. This is accomplished using the ArviZ function `az.hpd()` function." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "for x, name in zip(samples, names):\n", " df_summary.loc[['hpd_low', 'hpd_high'], name] = az.hpd(x, credible_interval=0.95)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we want to add the mode of our samples to our summary statistics. We could do that from MCMC samples using the log posterior values, and find the samples for which the log posterior is maximal. But since we quickly generated our samples using Numpy, we do not have those, so I will hand-code the known modes." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "df_summary.loc['mode', :] = [0, mu, mu, 1] " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can take a look at our various summary statistics. (Of course, this data frame is not tidy, but remember, organization for data for display and for analysis are two different things. This data frame is organized for display.)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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ExponentialGaussianTwo GaussiansHeavy tail
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" ], "text/plain": [ " Exponential Gaussian Two Gaussians Heavy tail\n", "mean 1.00477 1.0029 2.00153 5.01804\n", "std 1.00847 0.253494 1.07954 91.1717\n", "2.5 0.0235762 0.498385 0.589259 0.02152\n", "median 0.691216 1.00456 1.70622 0.783385\n", "97.5 3.73934 1.4988 3.83332 21.1252\n", "mode 0 1 1 1\n", "hpd_low 0.000117043 0.50544 0.506971 5.39473e-05\n", "hpd_high 3.00807 1.50197 3.72021 11.7915" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_summary" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is easier to visualize these summaries as error bars on plots. There's a bit of code below to generate the plots, but bear with me." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "5061" } }, "output_type": "display_data" } ], "source": [ "# y_values for bars on plots\n", "y_vals = [\n", " [0.8, 0.6, 0.4, 0.2], # Exponential\n", " [1.33, 1, 0.66, 0.33], # Gaussian\n", " [0.6, 0.45, 0.3, 0.15], # Two Gaussians\n", " [0.2, 0.15, 0.1, 0.05], # Heavy tail\n", "]\n", "\n", "# Color scheme\n", "colors = bokeh.palettes.Category10[10]\n", "\n", "\n", "def plot_errorbar(x, y, barx, color, legend_label, p):\n", " p.circle([x], [y], size=10, color=color, legend_label=legend_label)\n", " p.line(barx, [y, y], line_width=4, color=color)\n", "\n", "\n", "legends = [\"mean ± std\", \"quantile\", \"mode/HPD\", \"median/HPD\"]\n", "for p, name, y in zip(plots, names, y_vals):\n", " # Mean ± std\n", " x = df_summary.loc[\"mean\", name]\n", " barx = x + np.array([1, -1]) * 1.96 * df_summary.loc[\"std\", name]\n", " plot_errorbar(x, y[0], barx, colors[0], legends[0], p)\n", "\n", " # Median with central 95% interval\n", " x = df_summary.loc[\"median\", name]\n", " barx = df_summary.loc[[\"2.5\", \"97.5\"], name]\n", " plot_errorbar(x, y[1], barx, colors[1], legends[1], p)\n", "\n", " # Mode with HPD\n", " x = df_summary.loc[\"mode\", name]\n", " barx = df_summary.loc[[\"hpd_low\", \"hpd_high\"], name]\n", " plot_errorbar(x, y[2], barx, colors[2], legends[2], p)\n", "\n", " # Median with HPD\n", " x = df_summary.loc[\"median\", name]\n", " barx = df_summary.loc[[\"hpd_low\", \"hpd_high\"], name]\n", " plot_errorbar(x, y[3], barx, colors[3], legends[3], p)\n", "\n", "plots[0].legend.visible = True\n", "\n", "bokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Relative merits of each method\n", "\n", "* The **mean/std** does not respect bounds on the posterior, nor any asymmetry. Unless we are going for speed and using a MAP finder/Gaussian approximation, there is no need for this method or summarizing the posterior.\n", "\n", "
\n", "\n", "* The primary advantage of the **quantile approach** is that it is very easy to interpret, especially for the researcher uninitiated to Bayesian statistics. It does not suffer from the problems that the mean/std method does. It does not rely on any approximations.\n", "\n", "
\n", "\n", "* The **mode/HPD** method gives just that: where the parameter value is most likely to fall, which is not necessarily the interquantile region with the median at its center. It is also nice to know the most probable parameter value (the MAP). The drawback is the possible difficulty of interpretability for the uninitiated. Furthermore, I think it places too much emphasis on the MAP. This is not always the most relevant thing to know.\n", "\n", "
\n", "\n", "* The **median/HPD** method is my personal favorite. It gives the HPD (with the advantages I just stated) and also the median, which to me is a more useful statistic than the mode.\n", "\n", "
\n", "\n", "In any case, attempting to describe a multi-modal posterior with an error bar is misleading and futile. A distribution with a long tail can also be deceiving. Even if you report a 95% credible interval, there is still a 5% chance the parameter value would be *reeeeally* big." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### How to display the summary in text.\n", "\n", "One issue that may be worrying you is how to report the asymmetric error bars in text. This is best seen by example. For the example of the Exponential, we may report the median with HPD as $0.71^{+2.31}_{-0.70}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPython 3.7.6\n", "IPython 7.11.1\n", "\n", "numpy 1.18.1\n", "pandas 0.24.2\n", "arviz 0.6.1\n", "bokeh 1.4.0\n", "bokeh_catplot 0.1.7\n", "jupyterlab 1.2.5\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,pandas,arviz,bokeh,bokeh_catplot,jupyterlab" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 4 }