{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Modern machine learning methods
for trustworthy science\n", "\n", "Tom Charnock\n", "\n", "Institut d'Astrophysique de Paris\n", "


\n", "\n", "
\n", "
\n", " \"Sorbonne\n", "
\n", "
\n", " \"ANR\"\n", "
\n", "
\n", " \"IAP\"\n", "
\n", "
\n", " \"CNRS\"\n", "
\n", "
\n", " \"Aquila\"\n", "
\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Why neural networks don't work
(and how to use them)\n", "\n", "Tom Charnock\n", "\n", "Institut d'Astrophysique de Paris\n", "


\n", "\n", "
\n", "
\n", " \"Sorbonne\n", "
\n", "
\n", " \"ANR\"\n", "
\n", "
\n", " \"IAP\"\n", "
\n", "
\n", " \"CNRS\"\n", "
\n", "
\n", " \"Aquila\"\n", "
\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Why neural networks don't work

\n", "\n", "Tom Charnock\n", "\n", "Institut d'Astrophysique de Paris\n", "


\n", "\n", "
\n", "
\n", " \"Sorbonne\n", "
\n", "
\n", " \"ANR\"\n", "
\n", "
\n", " \"IAP\"\n", "
\n", "
\n", " \"CNRS\"\n", "
\n", "
\n", " \"Aquila\"\n", "
\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Apologies about the term bias\n", "\n", "- when something is intrinsically unknowable it is biased\n", "\n", "- if there is some offset, which could in principle be corrected, it is biased" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "### I (almost always) mean the top one" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "
\n", "
\n", "$$\\mathbb{NN}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}) : {\\bf d}\\rightarrow\\boldsymbol{\\tau}$$\n", "An approximation to a model, $\\mathcal{M} : {\\bf d}\\rightarrow{\\bf t}$\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## A crazy likelihood surface of how likely we are to get targets from data\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## What are we actually interested in?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$$\\mathcal{P}({\\bf t}|{\\bf d}) = \\int d\\boldsymbol{\\omega}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", " $\\mathcal{P}({\\bf t}|{\\bf d})$ - Posterior predictive density
\n", " How likely are the true targets given some data?

\n", "
\n", "

\n", "
\n", " $\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$ - Likelihood
\n", " How likely are the targets to be generated by a particular network?\n", "
\n", "
\n", " $\\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$ - Probability density
\n", " What is the probability of obtaining a particular network with particular parameter values?\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$$\\mathcal{P}({\\bf t}|{\\bf d})= \\int d\\boldsymbol{\\omega}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$$\n", "


\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Where does this information about the weights and hyperparameters come from?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Training and validation data" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Training data and targets: $\\small\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train}\\equiv\\{{\\bf d}^\\textrm{train}_i, {\\bf t}^\\textrm{train}_i|i\\in[1,n_\\textrm{train}]\\}$

\n", "Validation data and targets: $\\small\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{val}\\equiv\\small\\{{\\bf d}^\\textrm{val}_i, {\\bf t}^\\textrm{val}_i|i\\in[1,n_\\textrm{val}]\\}$\n", "#### Posterior distribution of weights and hyperparameters\n", "$$\\begin{align*}\n", " &\\small\\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train},\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{val})\\propto\\\\\n", " &\\small\\phantom{hello}\\mathcal{L}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train},\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{val})p(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## The failing of traditional training" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "

\n", " $\\mathcal{M} : {\\bf d} \\to {\\bf t}$ approximator
\n", " $$\\mathbb{NN}(\\boldsymbol{\\omega}, \\boldsymbol{\\alpha}) : {\\bf d}\\to\\boldsymbol{\\tau}$$
\n", " Cost function and likelihood\n", " $$\\boldsymbol{\\Lambda}({\\bf t}, \\boldsymbol{\\tau}) = -\\ln\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega}^*,\\boldsymbol{\\alpha}^*)$$\n", "
\n", "

\n", "
\n", "
\n", " $\\boldsymbol{\\Lambda}({\\bf t},\\boldsymbol{\\tau})$
smooth and convex

\n", " $\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$
complex and non-convex in $\\boldsymbol{\\omega}$ and $\\boldsymbol{\\alpha}$\n", "
\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Optimising (or training) a network" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "#### What are the maximum likelihood estimates of the weights?\n", "
\n", "\n", "\n", "$$\\boldsymbol{\\omega}^\\textrm{MLE}=\\underset{\\boldsymbol{\\omega}}{\\textrm{argmax}}\\left[\\mathcal{L}(\\boldsymbol{\\{}{\\bf t}\\boldsymbol{\\}}^\\textrm{train}|\\boldsymbol{\\{}{\\bf d}\\boldsymbol{\\}}^\\textrm{train}, \\boldsymbol{\\omega}, \\boldsymbol{\\alpha}^*)\\right]$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Local maximum likelihood estimates\n", "
\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## The main problem..." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## We degenerate the posterior\n", "
\n", "$$\\begin{align*}\n", " \\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train})\\propto&~\\mathcal{L}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train})p(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\\\\\n", " \\to&~\\delta(\\boldsymbol{\\omega}-\\boldsymbol{\\omega}^\\textrm{MLE},\\boldsymbol{\\alpha}-\\boldsymbol{\\alpha}^*)\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### All predictions are (probably incorrect) estimates

\n", "
\n", "$$\\mathcal{P}({\\bf t}|{\\bf d})=\\delta(\\boldsymbol{\\tau})$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### There is no way to interpret how close $\\boldsymbol{\\tau}$ is to ${\\bf t}$..." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "### Because the likelihood is non-interpretably complex" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Are there better methods?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Variational inference\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$$\\small\\mathcal{P}({\\bf t}|{\\bf d})=\\int d\\boldsymbol{\\omega}d\\boldsymbol{\\nu}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\\mathcal{Q}(\\boldsymbol{\\omega}|\\boldsymbol{\\nu}, \\boldsymbol{\\alpha}, \\boldsymbol{\\{}{\\bf d}, {\\bf t}\\boldsymbol{\\}}^\\textrm{train})p(\\boldsymbol{\\nu},\\boldsymbol{\\alpha})$$\n", "
\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "#### Still depends on fixed weights in the complex likelihood surface
and choice of variational distribution\n", "
\n", "\n", "$$\\begin{align*}\n", "\\mathcal{P}({\\bf t}|{\\bf d})&=\\int d\\boldsymbol{\\omega}d\\boldsymbol{\\nu}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega},\\boldsymbol{\\alpha})\\mathcal{Q}(\\boldsymbol{\\omega}|\\boldsymbol{\\nu}, \\boldsymbol{\\alpha}, \\boldsymbol{\\{}{\\bf d}, {\\bf t}\\boldsymbol{\\}}^\\textrm{train})\\\\\n", "&\\phantom{hellohello}\\times\\delta(\\boldsymbol{\\nu}-\\boldsymbol{\\nu}^\\textrm{MLE}, \\boldsymbol{\\alpha}-\\boldsymbol{\\alpha}^*)\\\\\n", "&=\\int d\\boldsymbol{\\omega}~\\mathcal{L}({\\bf t}|{\\bf d},\\boldsymbol{\\omega}, \\boldsymbol{\\alpha}^*)\\mathcal{Q}(\\boldsymbol{\\omega}|\\boldsymbol{\\nu}^\\textrm{MLE}, \\boldsymbol{\\alpha}^*, \\boldsymbol{\\{}{\\bf d}, {\\bf t}\\boldsymbol{\\}}^\\textrm{train}).\n", "\\end{align*}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Bayesian neural networks" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$$\\begin{align*}\n", "\\small\\mathcal{P}({\\bf t}|{\\bf d}) =&\\small~\\int d\\boldsymbol{\\omega}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d}, \\boldsymbol{\\omega}, \\boldsymbol{\\alpha})\\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d}, {\\bf t}\\boldsymbol{\\}}^\\textrm{train})\\\\\n", " \\propto&~\\int d\\boldsymbol{\\omega}d\\boldsymbol{\\alpha}~\\mathcal{L}({\\bf t}|{\\bf d}, \\boldsymbol{\\omega}, \\boldsymbol{\\alpha})\\\\\n", " &\\phantom{hello}\\times \\prod_i^{n_\\textrm{train}}\\mathcal{L}({\\bf t}^\\textrm{train}_i|{\\bf d}_i^\\textrm{train}, \\boldsymbol{\\omega},\\boldsymbol{\\alpha})p(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}).\\phantom{hello}\n", "\\end{align*}\n", "$$\n", "\n", "### Sample the likelihood of the training data" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "# Still dependent on the training data!\n", "\n", "
\n", "\n", "Classical network : $\\tiny \\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train}) \\to \\delta(\\boldsymbol{\\omega}-\\boldsymbol{\\omega}^\\textrm{MLE},\\boldsymbol{\\alpha}-\\boldsymbol{\\alpha}^*)$\n", "\n", "Variational inference : $\\tiny \\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train}) = \\mathcal{Q}(\\boldsymbol{\\omega}|\\boldsymbol{\\nu}^\\textrm{MLE}, \\boldsymbol{\\alpha}^*, \\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train})$\n", "\n", "Bayesian networks : $\\tiny \\mathcal{P}(\\boldsymbol{\\omega},\\boldsymbol{\\alpha}|\\boldsymbol{\\{}{\\bf d},{\\bf t}\\boldsymbol{\\}}^\\textrm{train}) =\\prod_i^{n_\\textrm{train}}\\mathcal{L}({\\bf t}^\\textrm{train}_i|{\\bf d}^\\textrm{train}_i,\\boldsymbol{\\omega},\\boldsymbol{\\alpha})p(\\boldsymbol{\\omega},\\boldsymbol{\\alpha})$\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Problems with physical models..." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## How can we use a neural network then?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Build it into the physical model" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Method 1 :

Infer the data, physics and the neural network" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Method 2 :

Understand the likelihood (using neural physical engines)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "
\n", " \n", "
\n", "

\n", "
\n", "

\n", "
\n", "
\n", " \n", "
\n", "

\n", "
\n", " \n", "

" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Method 3 :

Likelihood-free inference" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "Compare distance between observed summaries and simulation summaries and select results within $\\epsilon$

\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Conclusions" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "### Neural networks are not to be trusted\n", "\n", "### They can make trusty companions - when the correct framework is introduced\n", "\n", "### Using statistics we can build neural networks into the forward model to get unbiased results" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## For more information read my new blog

\n", "\n", "
\n", " bit.ly/ProbNN\n", "
\n" ] } ], "metadata": { "celltoolbar": "Diaporama", "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": "3.7.4" }, "rise": { "leap_motion": { "naturalSwipe": true, "pointerColor": "#d80000", "pointerOpacity": 0.5 } } }, "nbformat": 4, "nbformat_minor": 4 }