{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Reporting summaries of the posterior\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "nbsphinx": "hidden", "tags": [] }, "outputs": [], "source": [ "# Colab setup ------------------\n", "import os, sys, subprocess\n", "if \"google.colab\" in sys.modules:\n", " cmd = \"pip install --upgrade iqplot colorcet arviz watermark\"\n", " process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n", " stdout, stderr = process.communicate()\n", " data_path = \"https://s3.amazonaws.com/bebi103.caltech.edu/data/\"\n", "else:\n", " data_path = \"../data/\"\n", "# ------------------------------" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/bois/opt/anaconda3/lib/python3.9/site-packages/colorcet/__init__.py:74: UserWarning: Trying to register the cmap 'cet_gray' which already exists.\n", " register_cmap(\"cet_\"+name, cmap=cm[name])\n", "/Users/bois/opt/anaconda3/lib/python3.9/site-packages/colorcet/__init__.py:74: UserWarning: Trying to register the cmap 'cet_gray_r' which already exists.\n", " register_cmap(\"cet_\"+name, cmap=cm[name])\n" ] }, { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
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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 (also known as the **HDI** for **highest density interval**). 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 Normal, 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": 3, "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", "# Instantiate RNG\n", "rg = np.random.default_rng(seed=3252)\n", "\n", "# Draw out of the distributions, start with exponential\n", "x_expon = rg.exponential(mu, size=15000)\n", "\n", "# Normal\n", "x_norm = rg.normal(mu, sigma, size=15000)\n", "\n", "# Bimodal mixture of Normals\n", "which_norm = rg.choice([0, 1], 15000)\n", "x_2norm = rg.normal(mu_2[which_norm], sigma_2[which_norm])\n", "\n", "# Heavy tailed Pareto\n", "x_heavytail = rg.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": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Normals\", \"Heavy tail\"]\n", "plots = [\n", " iqplot.histogram(\n", " x,\n", " density=True,\n", " rug=False,\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", " iqplot.histogram(\n", " samples[-1],\n", " bins=np.arange(21),\n", " density=True,\n", " rug=False,\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": 5, "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.hdi()` function." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "for x, name in zip(samples, names):\n", " df_summary.loc[['hpd_low', 'hpd_high'], name] = az.hdi(x, hdi_prob=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": 7, "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": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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ExponentialNormalTwo NormalsHeavy tail
mean1.0078431.0014472.0040054.134071
std1.0053510.249141.0738544.580247
2.50.0248750.5115520.5904380.02307
median0.7010921.0006981.7679750.79814
97.53.6495411.492053.80497720.613608
mode01.01.01
hpd_low0.0000880.5318630.5613620.000057
hpd_high2.9965491.5073143.75565311.296449
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" ], "text/plain": [ " Exponential Normal Two Normals Heavy tail\n", "mean 1.007843 1.001447 2.004005 4.134071\n", "std 1.005351 0.24914 1.07385 44.580247\n", "2.5 0.024875 0.511552 0.590438 0.02307\n", "median 0.701092 1.000698 1.767975 0.79814\n", "97.5 3.649541 1.49205 3.804977 20.613608\n", "mode 0 1.0 1.0 1\n", "hpd_low 0.000088 0.531863 0.561362 0.000057\n", "hpd_high 2.996549 1.507314 3.755653 11.296449" ] }, "execution_count": 8, "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": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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[\n", " [0.8, 0.6, 0.4, 0.2], # Exponential\n", " [1.33, 1, 0.66, 0.33], # Normal\n", " [0.6, 0.45, 0.3, 0.15], # Two Normals\n", " [0.5, 0.4, 0.3, 0.2], # Heavy tail\n", "]\n", "\n", "# Color scheme\n", "colors = bokeh.palettes.Category10[10]\n", "\n", "\n", "def plot_interval(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_interval(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_interval(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_interval(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_interval(x, y[3], barx, colors[3], legends[3], p)\n", "\n", "plots[0].legend.visible = True\n", "for i in [1, 2, 3]:\n", " plots[i].legend.visible = False\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/Normal 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 may also be nice to know the most probable parameter value (the MAP), though this is seldom as useful as expectations. 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": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.9.7\n", "IPython version : 7.29.0\n", "\n", "numpy : 1.20.3\n", "pandas : 1.3.5\n", "arviz : 0.11.4\n", "bokeh : 2.3.3\n", "iqplot : 0.2.4\n", "jupyterlab: 3.2.1\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,pandas,arviz,bokeh,iqplot,jupyterlab" ] } ], "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": "3.9.7" } }, "nbformat": 4, "nbformat_minor": 4 }