Change API to make more flexible in practice (#33)

* total change

* format

* format

* format

* oh well then

* more

* pre-commit

* format

* add more examples

Co-authored-by: Nathan Simpson <phinate@protonmail.com>

Author: Nathan Simpson

.pre-commit-config.yaml

@@ -1,6 +1,6 @@
 repos:
 - repo: https://github.com/psf/black
-  rev: 22.1.0
+  rev: 22.3.0
   hooks:
   - id: black-jupyter
 

demo.ipynb

@@ -0,0 +1,3 @@
+celluloid
+git+http://github.com/scikit-hep/pyhf.git@make_difffable_model_ctor
+plothelp

examples/ap000.gif

@@ -0,0 +1,234 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import jax\n",
+    "import jax.numpy as jnp\n",
+    "import matplotlib.pyplot as plt\n",
+    "import optax\n",
+    "from jaxopt import OptaxSolver\n",
+    "import relaxed\n",
+    "from celluloid import Camera\n",
+    "from functools import partial\n",
+    "import matplotlib.lines as mlines\n",
+    "\n",
+    "# matplotlib settings\n",
+    "plt.rc(\"figure\", figsize=(6, 3), dpi=220, facecolor=\"w\")\n",
+    "plt.rc(\"legend\", fontsize=6)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Optimising a simple one-bin analysis with `relaxed`\n",
+    "\n",
+    "Let's define an analysis with a predicted number of signal and background events, with some uncertainty on the background estimate. We'll abstract the analysis configuration into a single parameter $\\phi$ like so:\n",
+    "\n",
+    "$$s = 15 + \\phi $$\n",
+    "$$b = 45 - 2 \\phi $$\n",
+    "$$\\sigma_b = 0.5 + 0.1*\\phi^2 $$\n",
+    "\n",
+    "Note that $s \\propto \\phi$ and $\\propto -2\\phi$, so increasing $\\phi$ corresponds to increasing the signal/backround ratio. However, our uncertainty scales like $\\phi^2$, so we're also going to compromise in our certainty of the background count as we do that. This kind of tradeoff between $s/b$ ratio and uncertainty is important for the discovery of a new signal, so we can't get away with optimising $s/b$ alone.\n",
+    "\n",
+    "To illustrate this, we'll plot the discovery significance for this model with and without uncertainty."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# model definition\n",
+    "def yields(phi, uncertainty=True):\n",
+    "    s = 15 + phi\n",
+    "    b = 45 - 2 * phi\n",
+    "    db = (\n",
+    "        0.5 + 0.1 * phi**2 if uncertainty else jnp.zeros_like(phi) + 0.001\n",
+    "    )  # small enough to be negligible\n",
+    "    return jnp.asarray([s]), jnp.asarray([b]), jnp.asarray([db])\n",
+    "\n",
+    "\n",
+    "# our analysis pipeline, from phi to p-value\n",
+    "def pipeline(phi, return_yields=False, uncertainty=True):\n",
+    "    y = yields(phi, uncertainty=uncertainty)\n",
+    "    # use a dummy version of pyhf for simplicity + compatibility with jax\n",
+    "    model = relaxed.dummy_pyhf.uncorrelated_background(*y)\n",
+    "    nominal_pars = jnp.array([1.0, 1.0])\n",
+    "    data = model.expected_data(nominal_pars)  # we expect the nominal model\n",
+    "    # do the hypothesis test (and fit model pars with gradient descent)\n",
+    "    pvalue = relaxed.infer.hypotest(\n",
+    "        0.0,  # value of mu for the alternative hypothesis\n",
+    "        data,\n",
+    "        model,\n",
+    "        test_stat=\"q0\",  # discovery significance test\n",
+    "        lr=1e-3,\n",
+    "        expected_pars=nominal_pars,  # optionally providing MLE pars in advance\n",
+    "    )\n",
+    "    if return_yields:\n",
+    "        return pvalue, y\n",
+    "    else:\n",
+    "        return pvalue\n",
+    "\n",
+    "\n",
+    "# calculate p-values for a range of phi values\n",
+    "phis = jnp.linspace(0, 10, 100)\n",
+    "\n",
+    "# with uncertainty\n",
+    "pipe = partial(pipeline, return_yields=True, uncertainty=True)\n",
+    "pvals, ys = jax.vmap(pipe)(phis)  # map over phi grid\n",
+    "# without uncertainty\n",
+    "pipe_no_uncertainty = partial(pipeline, uncertainty=False)\n",
+    "pvals_no_uncertainty = jax.vmap(pipe_no_uncertainty)(phis)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, axs = plt.subplots(2, 1, sharex=True)\n",
+    "axs[0].plot(phis, pvals, label=\"with uncertainty\", color=\"C2\")\n",
+    "axs[0].plot(phis, pvals_no_uncertainty, label=\"no uncertainty\", color=\"C4\")\n",
+    "axs[0].set_ylabel(\"$p$-value\")\n",
+    "# plot vertical dotted line at minimum of p-values + s/b\n",
+    "best_phi = phis[jnp.argmin(pvals)]\n",
+    "axs[0].axvline(x=best_phi, linestyle=\"dotted\", color=\"C2\", label=\"optimal p-value\")\n",
+    "axs[0].axvline(\n",
+    "    x=phis[jnp.argmin(pvals_no_uncertainty)],\n",
+    "    linestyle=\"dotted\",\n",
+    "    color=\"C4\",\n",
+    "    label=r\"optimal $s/b$\",\n",
+    ")\n",
+    "axs[0].legend(loc=\"upper left\", ncol=2)\n",
+    "s, b, db = ys\n",
+    "s, b, db = s.ravel(), b.ravel(), db.ravel()  # everything is [[x]] for pyhf\n",
+    "axs[1].fill_between(phis, s + b, b, color=\"C9\", label=\"signal\")\n",
+    "axs[1].fill_between(phis, b, color=\"C1\", label=\"background\")\n",
+    "axs[1].fill_between(phis, b - db, b + db, facecolor=\"k\", alpha=0.2, label=r\"$\\sigma_b$\")\n",
+    "axs[1].set_xlabel(\"$\\phi$\")\n",
+    "axs[1].set_ylabel(\"yield\")\n",
+    "axs[1].legend(loc=\"lower left\")\n",
+    "plt.suptitle(\"Discovery p-values, with and without uncertainty\")\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Using gradient descent, we can optimise this analysis in an uncertainty-aware way by directly optimising $\\phi$ for the lowest discovery p-value. Here's how you do that:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The fast way!\n",
+    "# use the OptaxSolver wrapper from jaxopt to perform the minimisation\n",
+    "# set a couple of tolerance kwargs to make sure we don't get stuck\n",
+    "solver = OptaxSolver(pipeline, opt=optax.adam(1e-3), tol=1e-8, maxiter=10000)\n",
+    "pars = 9.0  # random init\n",
+    "result = solver.run(pars).params\n",
+    "print(\n",
+    "    f\"our solution: phi={result:.5f}\\ntrue optimum: phi={phis[jnp.argmin(pvals)]:.5f}\\nbest s/b:     phi=10\"\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The longer way (but with plots)!\n",
+    "pipe = partial(pipeline, return_yields=True, uncertainty=True)\n",
+    "solver = OptaxSolver(pipe, opt=optax.adam(1e-1), has_aux=True)\n",
+    "pars = 9.0\n",
+    "state = solver.init_state(pars)  # we're doing init, update steps instead of .run()\n",
+    "\n",
+    "plt.rc(\"figure\", figsize=(6, 3), dpi=220, facecolor=\"w\")\n",
+    "plt.rc(\"legend\", fontsize=8)\n",
+    "fig, axs = plt.subplots(1, 2)\n",
+    "cam = Camera(fig)\n",
+    "steps = 5  # increase me for better results! (100ish works well)\n",
+    "for i in range(steps):\n",
+    "    pars, state = solver.update(pars, state)\n",
+    "    s, b, db = state.aux\n",
+    "    val = state.value\n",
+    "    ax = axs[0]\n",
+    "    cv = ax.plot(phis, pvals, c=\"C0\")\n",
+    "    cvs = ax.plot(phis, pvals_no_uncertainty, c=\"green\")\n",
+    "    current = ax.scatter(pars, val, c=\"C0\")\n",
+    "    ax.set_xlabel(r\"analysis config $\\phi$\")\n",
+    "    ax.set_ylabel(\"p-value\")\n",
+    "    ax.legend(\n",
+    "        [\n",
+    "            mlines.Line2D([], [], color=\"C0\"),\n",
+    "            mlines.Line2D([], [], color=\"green\"),\n",
+    "            current,\n",
+    "        ],\n",
+    "        [\"p-value (with uncert)\", \"p-value (without uncert)\", \"current value\"],\n",
+    "        frameon=False,\n",
+    "    )\n",
+    "    ax.text(0.3, 0.61, f\"step {i}\", transform=ax.transAxes)\n",
+    "    ax = axs[1]\n",
+    "    ax.set_ylim((0, 80))\n",
+    "    b1 = ax.bar(0.5, b, facecolor=\"C1\", label=\"b\")\n",
+    "    b2 = ax.bar(0.5, s, bottom=b, facecolor=\"C9\", label=\"s\")\n",
+    "    b3 = ax.bar(\n",
+    "        0.5, db, bottom=b - db / 2, facecolor=\"k\", alpha=0.5, label=r\"$\\sigma_b$\"\n",
+    "    )\n",
+    "    ax.set_ylabel(\"yield\")\n",
+    "    ax.set_xticks([])\n",
+    "    ax.legend([b1, b2, b3], [\"b\", \"s\", r\"$\\sigma_b$\"], frameon=False)\n",
+    "    plt.tight_layout()\n",
+    "    cam.snap()\n",
+    "\n",
+    "ani = cam.animate()\n",
+    "# uncomment this to save and view the animation!\n",
+    "# ani.save(\"ap00.gif\", fps=9)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "interpreter": {
+   "hash": "22d6333b89854cd01c2018f3ca2f5a59a2cde2765fbca789ff36cfad48ca629b"
+  },
+  "kernelspec": {
+   "display_name": "Python 3.9.12 ('venv': venv)",
+   "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": "3.9.12"
+  },
+  "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}