diff --git a/poster/.gitignore b/poster/.gitignore new file mode 100644 index 0000000..528f902 --- /dev/null +++ b/poster/.gitignore @@ -0,0 +1 @@ +*.otf \ No newline at end of file diff --git a/poster/Manifest.toml b/poster/Manifest.toml new file mode 100644 index 0000000..5264242 --- /dev/null +++ b/poster/Manifest.toml @@ -0,0 +1,385 @@ +# This file is machine-generated - editing it directly is not advised + +[[ArgCheck]] +deps = ["Random"] +git-tree-sha1 = "dab25d711a1dedb707a55dbc1eb9fd578f76ff32" +uuid = "dce04be8-c92d-5529-be00-80e4d2c0e197" +version = "1.0.1" + +[[Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[BinDeps]] +deps = ["Compat", "Libdl", "SHA", "URIParser"] +git-tree-sha1 = "12093ca6cdd0ee547c39b1870e0c9c3f154d9ca9" +uuid = "9e28174c-4ba2-5203-b857-d8d62c4213ee" +version = "0.8.10" + +[[BinaryProvider]] +deps = ["Libdl", "SHA"] +git-tree-sha1 = "29995a7b317bbd06be147e1974a3541ce2502dca" +uuid = "b99e7846-7c00-51b0-8f62-c81ae34c0232" +version = "0.5.7" + 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"JSON", "Missings", "Printf", "Reexport", "Unicode"] +git-tree-sha1 = "45101c4d0df3946acb6e9bfcfd3a8c32abbd421b" +uuid = "324d7699-5711-5eae-9e2f-1d82baa6b597" +version = "0.7.1" + +[[Compat]] +deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"] +git-tree-sha1 = "ed2c4abadf84c53d9e58510b5fc48912c2336fbb" +uuid = "34da2185-b29b-5c13-b0c7-acf172513d20" +version = "2.2.0" + +[[Crayons]] +deps = ["Test"] +git-tree-sha1 = "f621b8ef51fd2004c7cf157ea47f027fdeac5523" +uuid = "a8cc5b0e-0ffa-5ad4-8c14-923d3ee1735f" +version = "4.0.0" + +[[DataAPI]] +git-tree-sha1 = "674b67f344687a88310213ddfa8a2b3c76cc4252" +uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a" +version = "1.1.0" + +[[DataFrames]] +deps = ["CategoricalArrays", "Compat", "DataAPI", "InvertedIndices", 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b/poster/Project.toml @@ -0,0 +1,7 @@ +[deps] +CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b" +CalibrationPaper = "5e6d2ce1-a021-4f7d-b692-0c7af47336b0" +DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" +PGFPlotsX = "8314cec4-20b6-5062-9cdb-752b83310925" +Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" +StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" diff --git a/poster/README.md b/poster/README.md new file mode 100644 index 0000000..164844f --- /dev/null +++ b/poster/README.md @@ -0,0 +1,45 @@ +# Poster + +This folder contains the source code of the poster for the paper +"Calibration tests in multi-class classification: A unifying framework" +by Widmann, Lindsten, and Zachariah which is going to be presented +at NeurIPS 2019. + +## Generate the figures + +Open a terminal in the current directory and install all required Julia packages +by running +```shell +julia --project=. -e "using Pkg; Pkg.instantiate()" +``` +Afterwards start a Julia REPL +```shell +julia --project=. +``` +and include the file `figures.jl` with + +``` julia +julia> include("figures.jl") +``` + +## Fonts + +If available, the non-free font Berling Antikva is used which is part +of the official design of Uppsala University. The corresponding OTF +fonts have to be available in the subfolder `fonts` as `Berling.otf`, +`Berling-Italic.otf`, `Berling-Bold.otf`, and `Berling-BoldItalic.otf`. +Unfortunately, the official fonts that can be +[downloaded](https://mp.uu.se/documents/432512/911262/18350_Typsnitt_PC+v2.zip/24599197-7993-f965-e58d-0e375939d144) +from Uppsala University's staff portal can not be used with older +version of LuaLaTeX (the problem is explained in a +[question on StackExchange](https://tex.stackexchange.com/questions/430837/why-does-lualatex-have-problem-with-a-ttf-that-xelatex-accepts/430872#comment1161846_430837). +However, suitable OTF fonts can be generated from these files according +to the +[instructions provided on StackExchange]((https://tex.stackexchange.com/a/430872). + +## Compile the poster + +The poster can be compiled as a PDF file by running +```shell +arara neurips.tex +``` diff --git a/poster/figures.jl b/poster/figures.jl new file mode 100644 index 0000000..938a164 --- /dev/null +++ b/poster/figures.jl @@ -0,0 +1,197 @@ +using CalibrationPaper +using CSV +using DataFrames +using PGFPlotsX +using StatsBase + +using Statistics + +function errors_comparison() + # initialize group plot + @pgf plt = GroupPlot( + { + group_style = + { + group_size = "3 by 4", + xlabels_at = "edge bottom", + ylabels_at = "edge left", + horizontal_sep = raw"0.1\linewidth", + vertical_sep = raw"0.05\linewidth", + }, + no_markers, + tick_label_style = { font = raw"\tiny" }, + grid = "major", + title_style = { align = "center" }, + width = raw"0.23\linewidth", + height = raw"0.155\linewidth", + "every x tick scale label/.style" = {at = "{(1,0)}", anchor = "west"}, + ylabel_style = { font = raw"\small" } + }) + + # define displayed experiments and corresponding labels + experiments = ["ECE_uniform", "SKCEb_median", "SKCEuq_median", "SKCEul_median"] + labels = [raw"$\widehat{\ECE}$", raw"$\biasedestimator$", + raw"$\unbiasedestimator$", raw"$\linearestimator$"] + + # define displayed models + models = [CalibrationPaperModel(10, 0.1, 0.0, false), + CalibrationPaperModel(10, 0.1, 0.5, true), + CalibrationPaperModel(10, 0.1, 1.0, false)] + + # directory with all experimental results + datadir = joinpath(@__DIR__, "..", "experiments", "data", "errors") + + # for all studied experiments and models + for (i, experiment) in enumerate(experiments), (j, model) in enumerate(models) + # obtain full path + file = joinpath(datadir, "$experiment.csv") + + # load estimates + estimates = collect_estimates(CSV.read(file), model) + + # compute histogram + hist = fit(Histogram, estimates, closed = :left) + + # create axis object with histogram + @pgf ax = Axis(PlotInc({ ybar_interval, fill = "Dark2-A!30!white" }, Table(hist)), + VLine({ solid, thick, "Dark2-B" }, mean(estimates))) + + # add titles + if i == 1 + ax["title"] = "\$\\symbf{M$j}\$" + end + + # add labels + if j == 1 + ax["ylabel"] = labels[i] + end + + # compute true value + if startswith(experiment, "ECE") + analytic = CalibrationPaper.analytic_ece(model) + else + # take mean of quadratic unbiased estimates + if startswith(experiment, "SKCEuq") + analytic = mean(estimates) + else + file_uq = replace(file, r"(SKCEb|SKCEul)" => "SKCEuq") + estimates_uq = collect_estimates(CSV.read(file_uq), model) + analytic = mean(estimates_uq) + end + end + + # plot true value + @pgf push!(ax, VLine({ dashed, thick, "Dark2-C" }, analytic)) + # hack so that limits are updated as well + @pgf push!(ax, PlotInc({ draw = "none" }, Coordinates([analytic], [0]))) + + # add axis to group plot + push!(plt, ax) + end + + # add labels + picture = TikzPicture(plt, + raw"\node[anchor=north] at ($(group c1r4.west |- group c1r4.outer south)!0.5!(group c3r4.east |- group c3r4.outer south)$){calibration error estimate};", + raw"\node[anchor=south, rotate=90, yshift=1ex] at ($(group c1r1.north -| group c1r1.outer west)!0.5!(group c1r4.south -| group c1r4.outer west)$){\# runs};") + + # save histogram + figuresdir = joinpath(@__DIR__, "figures") + isdir(figuresdir) || mkdir(figuresdir) + pgfsave(joinpath(figuresdir, "errors_comparison.tex"), picture; include_preamble = false) + + nothing +end + +function pvalues_comparison() + # initialize group plot + @pgf plt = GroupPlot( + { + group_style = + { + group_size = "3 by 6", + xlabels_at = "edge bottom", + ylabels_at = "edge left", + horizontal_sep = raw"0.1\linewidth", + vertical_sep = raw"0.015\linewidth", + xticklabels_at = "edge bottom", + }, + no_markers, + tick_label_style = { font = raw"\tiny" }, + grid = "major", + title_style = { align = "center" }, + width = raw"0.23\linewidth", + height = raw"0.155\linewidth", + "every x tick scale label/.style" = {at = "{(1,0)}", anchor = "west"}, + ylabel_style = { font = raw"\small" }, + xmin = 0, xmax = 1, ymin=-0.1, ymax=1.1 + }) + + # define displayed experiments and corresponding labels + experiments = ["ECE_uniform", "SKCEb_median_distribution_free", + "SKCEuq_median_distribution_free", + "SKCEul_median_distribution_free", + "SKCEuq_median_asymptotic", + "SKCEul_median_asymptotic"] + labels = [raw"$\symbf{C}$", raw"$\symbf{D}_{\mathrm{b}}$", + raw"$\symbf{D}_{\mathrm{uq}}$", raw"$\symbf{D}_{\mathrm{l}}$", + raw"$\symbf{A}_{\mathrm{uq}}$", + raw"$\symbf{A}_{\mathrm{l}}$"] + + # define displayed models + models = [CalibrationPaperModel(10, 0.1, 0.0, false), + CalibrationPaperModel(10, 0.1, 0.5, true), + CalibrationPaperModel(10, 0.1, 1.0, false)] + + # define range of significance levels + αs = 0:0.01:1 + + # directory with all experimental results + datadir = joinpath(@__DIR__, "..", "experiments", "data", "pvalues") + + # for all studied experiments and models + for (i, experiment) in enumerate(experiments), (j, model) in enumerate(models) + # obtain full path + file = joinpath(datadir, "$experiment.csv") + + # load p-value estimates + pvalues = collect_estimates(CSV.read(file), model) + + # compute empirical CDF + empiricalCDF = ecdf(pvalues) + + if iszero(model.π) + # if the model is calibrated we plot the empirical estimate of + # P[p(T) < \alpha | H_0] together with the diagonal of the unit square + @pgf ax = Axis(PlotInc({ thick }, Table(αs, empiricalCDF.(αs))), + PlotInc({ dashed, thick }, Coordinates([0, 1], [0, 1]))) + else + # otherwise we plot the empirical estimate for P[p(T) > \alpha | H_1] + @pgf ax = Axis(PlotInc({ thick }, Table(αs, 1 .- empiricalCDF.(αs)))) + end + + # add titles + if i == 1 + ax["title"] = "\$\\symbf{M$j}\$" + end + + # add labels + if j == 1 + ax["ylabel"] = labels[i] + end + + # add axis to group plot + push!(plt, ax) + end + + # add labels + picture = TikzPicture(plt, + raw"\node[anchor=north] at ($(group c1r6.west |- group c1r6.outer south)!0.5!(group c3r6.east |- group c3r6.outer south)$){significance level};", + raw"\node[anchor=south, rotate=90] at ($(group c1r1.north -| group c1r1.outer west)!0.5!(group c1r6.south -| group c1r6.outer west)$){empirical test error};") + + # save histogram + figuresdir = joinpath(@__DIR__, "figures") + isdir(figuresdir) || mkdir(figuresdir) + pgfsave(joinpath(figuresdir, "pvalues_comparison.tex"), picture; include_preamble = false) + + nothing +end diff --git a/poster/figures/barrier.svg b/poster/figures/barrier.svg new file mode 100644 index 0000000..6b02b0e --- /dev/null +++ b/poster/figures/barrier.svg @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/poster/figures/bear.svg b/poster/figures/bear.svg new file mode 100644 index 0000000..145002e --- /dev/null +++ b/poster/figures/bear.svg @@ -0,0 +1,71 @@ + + + +image/svg+xml + + + + + + + + + + + + \ No newline at end of file diff --git a/poster/figures/car.svg b/poster/figures/car.svg new file mode 100644 index 0000000..b8fe910 --- /dev/null +++ b/poster/figures/car.svg @@ -0,0 +1,17 @@ + + + + + + + + + image/svg+xml + + + + + + + + \ No newline at end of file diff --git a/poster/figures/car0.svg b/poster/figures/car0.svg new file mode 100644 index 0000000..54dc3b5 --- /dev/null +++ b/poster/figures/car0.svg @@ -0,0 +1,89 @@ + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + diff --git a/poster/figures/car1.svg b/poster/figures/car1.svg new file mode 100644 index 0000000..3be8735 --- /dev/null +++ b/poster/figures/car1.svg @@ -0,0 +1,89 @@ + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + diff --git a/poster/figures/car2.svg b/poster/figures/car2.svg new file mode 100644 index 0000000..e8ad59b --- /dev/null +++ b/poster/figures/car2.svg @@ -0,0 +1,89 @@ + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + diff --git a/poster/figures/car3.svg b/poster/figures/car3.svg new file mode 100644 index 0000000..e7e6ae5 --- /dev/null +++ b/poster/figures/car3.svg @@ -0,0 +1,89 @@ + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + diff --git a/poster/figures/car4.svg b/poster/figures/car4.svg new file mode 100644 index 0000000..f0c4308 --- /dev/null +++ b/poster/figures/car4.svg @@ -0,0 +1,89 @@ + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + diff --git a/poster/figures/comparison_estimates.tex b/poster/figures/comparison_estimates.tex deleted file mode 100644 index bec2e80..0000000 --- a/poster/figures/comparison_estimates.tex +++ /dev/null @@ -1,341 +0,0 @@ -\begin{tikzpicture} -\begin{groupplot}[group style={group size={3 by 4}, xlabels at={edge bottom}, ylabels at={edge left}, horizontal sep={0.1\linewidth}, vertical sep={0.04\linewidth}}, no markers, tick label style={font={\footnotesize}}, grid={major}, title style={align={center}}, width={0.22\linewidth}, height={0.1\linewidth}, every x tick scale label/.style={at={{(1,0)}}, anchor={west}}, ylabel style={font={\small}}] - \nextgroupplot[title={\textbf{M1}}, ylabel={$\ECE$}] - \addplot+[ybar interval, fill={blue!25}] - table[row sep={\\}] - { - \\ - 0.17 7.0 \\ - 0.18 55.0 \\ - 0.19 229.0 \\ - 0.2 796.0 \\ - 0.21 1782.0 \\ - 0.22 2550.0 \\ - 0.23 2307.0 \\ - 0.24 1389.0 \\ - 0.25 667.0 \\ - 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index 868480b..1f0f8a4 100644 --- a/poster/neurips.tex +++ b/poster/neurips.tex @@ -1,33 +1,44 @@ % arara: lualatex: { shell: true } -% arara: biber % arara: lualatex: { shell: true } -% arara: lualatex: { shell: true } -\documentclass{article} +% arara: lualatex: { shell: true, synctex: true } + +% A2 : 420 x 594 mm | +% 2A0 : 1189 x 1682 mm > Factor 2.83 => 11pt ~ 31 pt +% 1m : 1000 x 1414 > Factor 2.38 => 12pt ~ 28 pt + +% A3 : 297 x 420 mm | +% 2A0 : 1189 x 1682 mm > Factor 4 => 11pt ~ 44 pt +% 1m : 1000 x 1414 > Factor 3.36 => 10pt ~ 34 pt +\PassOptionsToPackage{force}{filehook} % see https://tex.stackexchange.com/questions/513051/filehook-error-with-memoir-after-update-texlive-2019-in-oct-15 +\documentclass[10pt]{article} \usepackage{luatex85} % layout -\usepackage[a0paper,portrait]{geometry} +\usepackage[a3paper,landscape]{geometry} -% Math packages +% math support \usepackage{mathtools,amssymb} -% Use Fira fonts -\RequirePackage[factor=0]{microtype} % No protrusion -\usepackage[no-math]{fontspec} +% fonts +\RequirePackage[factor=0]{microtype} % no protrusion +\usepackage{unicode-math} \defaultfontfeatures{Ligatures=TeX} -\setsansfont[ - BoldFont={Fira Sans SemiBold}, - ItalicFont={Fira Sans BookItalic}, - BoldItalicFont={Fira Sans SemiBold Italic} -]{Fira Sans Book} -\setmonofont[BoldFont={Fira Mono Medium}]{Fira Mono} - -% Use sans serif fonts -\renewcommand{\familydefault}{\sfdefault} - -% Add sans serif math fonts -\usepackage[scaled]{newtxsf} +\IfFileExists{fonts/Berling.otf}{% + % load fonts of official UU design + \setmainfont{Berling}[% + Path=./fonts/, + Extension=.otf, + BoldFont=*-Bold, + ItalicFont=*-Italic, + BoldItalicFont=*-BoldItalic] +}{% + \setmainfont{Libertinus Serif} +} +\setsansfont{Libertinus Sans} +\setmonofont{Libertinus Mono} +\setmathfont{Libertinus Math} + \usepackage{bm} % language support @@ -39,13 +50,15 @@ \usepackage{booktabs} % colors -\usepackage[RGB]{xcolor} +\usepackage[CMYK]{xcolor} \usepackage{UUcolorPantone} \newcommand{\hl}[1]{\begingroup\bfseries\boldmath\color{uured}#1\endgroup} % graphics -\usepackage{graphics} +\usepackage{graphicx} +\usepackage{svg} +\svgpath{{./figures/}} % captions \usepackage{caption,subcaption} @@ -53,279 +66,403 @@ % fancy lists \usepackage{enumitem} +\setlist{leftmargin=*,itemsep=0pt} \setlist[itemize,1]{label={\color{uured}$\blacktriangleright$}} % hyperlinks \usepackage{hyperref} % boxes -\usepackage[poster,xparse]{tcolorbox} +\usepackage[poster,xparse,raster]{tcolorbox} % poster settings \tcbposterset{ - coverage = {spread,top=0pt,bottom=0pt}, - boxes = {enhanced, - % Colors - coltext=black,coltitle=black,colback=white,colbacktitle=uulightgrey, - % Fonts - fonttitle=\bfseries\Large\scshape,fontupper=\large,fontlower=\large, - % Margins - boxsep=0pt,left=\tcbpostercolspacing,right=\tcbpostercolspacing, - top=\tcbposterrowspacing,toptitle=\tcbposterrowspacing, - bottom=\tcbposterrowspacing,bottomtitle=\tcbposterrowspacing, - % Subtitles - subtitle style={opacityframe=0,opacityback=0.5,fontupper=\large\bfseries}, + coverage = + { + spread, + interior style={color=white}, + }, + poster = + { + columns=10, + rows=1, + }, + boxes = + { + enhanced standard jigsaw, + sharp corners=downhill, + arc=3pt, + boxrule=1pt, + lower separated=false, + % colors + coltext=black, + colback=white, + colframe=black, + coltitle=black, + colbacktitle=uulightgrey, + % fonts + fonttitle=\bfseries\large, + % subtitles + subtitle style= + { + frame empty, + hbox, + rounded corners=east, + arc=8pt, + coltext=white!50!uulightgrey, + colback=black!10!uudarkgrey, + }, } } % plots \usepackage{pgfplots,pgfplotstable} \pgfplotsset{compat=1.16} - -% general settings for plots -\pgfplotsset{grid style=dashed} -\pgfplotsset{enlargelimits=auto} - -\usepgfplotslibrary{groupplots,colorbrewer} -\usetikzlibrary{plotmarks,calc} +\usetikzlibrary{positioning,arrows,arrows.meta,calc,decorations.markings,intersections,patterns} + +\usepgfplotslibrary{groupplots,fillbetween} +\usetikzlibrary{plotmarks} + +% Use colorbrewer +\usepgfplotslibrary{colorbrewer} +\pgfplotsset{ + % initialize Dark2-8: + cycle list/Dark2-8, + % combine it with ’mark list*’: + cycle multiindex* list={ + mark list*\nextlist + Dark2-8\nextlist + }, +} % plotting options -\pgfplotsset{table/search path={data/}} -\pgfplotsset{max space between ticks=150} -\pgfplotsset{every axis/.append style={axis background style={fill=gray!10}},tick label style={font={\footnotesize}}, label style={font={\small}}} -\pgfplotsset{every axis plot/.append style={thick}} -\pgfplotsset{every axis legend/.append style={font=\small, fill=none}} - -% bibliography -\usepackage[backend=biber,sortcites,doi=false,isbn=false,url=false,giveninits=true,maxbibnames=10,style=alphabetic]{biblatex} -\addbibresource{references.bib} -\AtBeginBibliography{\scriptsize} % small font size -\AtEveryBibitem{% -\ifentrytype{inproceedings}{ - \clearfield{review}% - \clearfield{editor}% - \clearfield{series}%% -}{} -} +\pgfplotsset{every axis/.append style={axis background style={fill=gray!10}}} % automatic references \usepackage{cleveref} % some abbreviations -\newcommand{\Prob}{\mathbb{P}} -\newcommand*{\E}{\mathbb{E}} -\DeclareMathOperator*{\argmax}{arg\,max} -\DeclareMathOperator{\ECE}{ECE} -\DeclareMathOperator{\biasedskce}{SKCE_b} -\DeclareMathOperator{\unbiasedskce}{SKCE_{uq}} -\DeclareMathOperator{\linearskce}{SKCE_{ul}} -\DeclareMathOperator{\asympunbiasedskce}{aSKCE_{uq}} -\DeclareMathOperator{\asymplinearskce}{aSKCE_{ul}} -\DeclareMathOperator{\measure}{CE} -\DeclareMathOperator{\kernelmeasure}{KCE} -\DeclareMathOperator{\squaredkernelmeasure}{SKCE} -\DeclareMathOperator{\Expect}{\mathbb{E}} -\DeclareMathOperator{\Dir}{Dir} -\DeclareMathOperator{\Categorical}{Cat} - -% Define number of columns and rows -\tcbposterset{poster={columns=2,rows=8}} - -% Define split box -\DeclareTColorBox{splitbox}{ O{} }{standard jigsaw,sharp corners, - sidebyside,sidebyside align=center,opacityfill=0,#1} +\newcommand*{\Prob}{\mathbb{P}} +\newcommand*{\Expect}{\mathbb{E}} +\newcommand*{\transpose}[1]{{#1}^{\mathsf{T}}} +\newcommand*{\ECE}{\mathup{ECE}} +\newcommand*{\measure}{\mathup{CE}} +\newcommand*{\kernelmeasure}{\mathup{KCE}} +\newcommand*{\squaredkernelmeasure}{\mathup{SKCE}} +\newcommand*{\biasedestimator}{\widehat{\mathup{SKCE}}_{\mathup{b}}} +\newcommand*{\unbiasedestimator}{\widehat{\mathup{SKCE}}_{\mathup{uq}}} +\newcommand*{\linearestimator}{\widehat{\mathup{SKCE}}_{\mathup{ul}}} +\newcommand*{\Dir}{\mathup{Dir}} +\newcommand*{\Categorical}{\mathup{Cat}} % metadata -\title{Calibration tests in multi-class classification:\\A unifying framework} -\author{David Widmann$^*$ Fredrik Lindsten$^\dagger$ Dave Zachariah$^*$} -\def\institute{$^*$Department of Information Technology, Uppsala University\\ $^\ddagger$Division of Statistics and Machine Learning, Linköping University} -\date{} -\def\contact{david.widmann@it.uu.se fredrik.lindsten@liu.se dave.zachariah@it.uu.se} +\title{Calibration tests in multi-class classification:\\ A unifying framework} +\author{David Widmann$^\star$ Fredrik Lindsten$^\ddagger$ Dave Zachariah$^\star$} +\date{\today} +\makeatletter +\pgfkeys{% + /my poster/.cd, + title/.initial=\@title, + author/.initial=\@author, + institute/.initial={}, + contact/.initial={}, + date/.initial=\@date, +} +\makeatother + +\pgfkeys{% + /my poster/.cd, + institute={$^\star$Department of Information Technology, Uppsala University $^\ddagger$Division of Statistics and Machine Learning, Linköping University}, + contact={david.widmann@it.uu.se fredrik.lindsten@liu.se dave.zachariah@it.uu.se}, +} -% Default settings \pagestyle{empty} \begin{document} -\begin{tcbposter}[fontsize=26pt] - \makeatletter - \posterbox[blankest,width=\paperwidth,interior engine=path,interior style={color=white},% - top=\tcbposterrowspacing,bottom=\tcbposterrowspacing,left=0.15\paperwidth,right=0.15\paperwidth,% - height=0.15\paperwidth,height plus=0.15\paperwidth,halign=flush center,valign=center, % - underlay={% - \node[right,inner sep=0pt,outer sep=0pt,text width=0.15\paperwidth,align=center] at (frame.west) {\includegraphics[width=0.12\paperwidth]{figures/logos/UU.pdf}};% - \node[left,inner sep=0pt,outer sep=0pt,text width=0.15\paperwidth,align=center] at (frame.east) {\includegraphics[width=0.12\paperwidth]{figures/logos/LiU.pdf}};% - }]{name=title,column=1,below=top,xshift=-\tcbpostercolspacing}{% - \vspace*{2ex}% - {\bfseries\fontsize{4.5ex}{5.2ex}\selectfont{\@title}}\\[2ex]% - {\LARGE{\@author}}\\[1ex]% - {\large{\institute}}% +\begin{tcbposter} + + % title + \posterbox[blankest,interior engine=path,halign=left,valign=center,right=4cm, + underlay = + {% + \node[left,inner sep=0pt,outer sep=0pt,align=center] at (frame.east) {\includegraphics[width=2cm]{figures/logos/UU.pdf}\\[1ex]\includegraphics[width=3cm]{figures/logos/LiU.pdf}};% + }]{name=title,column=1,span=6,below=top}{% + \Huge\textbf{\pgfkeysvalueof{/my poster/title}}\\[1ex] + \large\pgfkeysvalueof{/my poster/author}\\[1ex] + \normalsize\pgfkeysvalueof{/my poster/institute}% }% - \makeatother % footline - \makeatletter - \posterbox[blankest,width=\paperwidth,% - top=5pt,bottom=5pt,left=\tcbpostercolspacing,right=\tcbpostercolspacing,% - valign=center,fontupper=\ttfamily\small,interior engine=path, interior style={color=uumidgrey}% - ]{name=footline,column=1,above=bottom,xshift=-\tcbpostercolspacing}{% - \@date\hfill\contact% + \posterbox[blankest,top=2pt,bottom=2pt,valign=center,fontupper=\ttfamily\small,interior engine=path,interior style={color=uumidgrey}% + ]{name=footline,column=1,span=10,above=bottom}{% + \pgfkeysvalueof{/my poster/date}\hfill\pgfkeysvalueof{/my poster/contact}% }% - \makeatother - % references - \posterbox[adjusted title=References]{name=references,column=1,above=footline,span=2}{ - \printbibliography[heading=none] - } + \posterbox[adjusted title={Motivation - what is a calibrated model?}, colback=blondsvag]{name=calibration,column=3,span=4,below=title}{ + \begin{tcolorbox}[colback=blondstark] + \begin{center} + A \hl{calibrated model} yields predictions consistent with empirically observed frequencies. + \end{center} + \end{tcolorbox} - \posterbox[adjusted title=Summary of our work]{name=summary,column=1,below=title}{ - \begin{itemize} - \item We propose a \hl{unifying framework} for quantifying calibration error of probabilistic classifiers that encompasses several existing error measures. - \item We introduce a new \hl{kernel calibration error} (KCE), for which we derive \hl{unbiased and consistent estimators}. - \item We show how the transfer of calibration error estimates to \hl{probabilities of false rejection} makes them interpretable and statistically commensurable. - \item We provide \hl{Julia packages} for calibration estimation. - \end{itemize} - } + \tcbsubtitle{Collision detection system} - \posterbox[adjusted title=General setup]{name=setup,column=1,below=summary}{ - \begin{itemize} - \item Let $X$ be random inputs (features) of $m$ classes $1,\ldots,m$, denoted by $Y$. - \item Consider a \hl{probabilistic model} $g$ that predicts a probability distribution of classes $g(X) \in \Delta^{m}$ for input $X$, where $\Delta^m \coloneqq \{z \in \mathbb{R}^m_{\geq 0} : \|z\|_1 = 1\}$ denotes the $(m-1)$-dimensional probability simplex. - \item Ideally $g$ predicts $g_y(X) = \Prob[Y = y \,|\,X]$ for all classes $y$. - \end{itemize} - } + Consider a model that predicts if there is an object, a human, or an animal ahead of a car. - \posterbox[adjusted title={Calibration}]{name=calibration,column=1,below=setup}{ - Although in practice the model will never be ideal but at most close to it, we can strive to satisfy other desirable statistical properties such as \hl{calibration}. - \begin{tcolorbox}[colback=blondsvag, halign=center] - Informally, in the long run every prediction should match the relative frequencies of the observed classes. - \end{tcolorbox} - \vspace{-\topsep} - \begin{itemize} - \item Mathematically, a model \(g\) is calibrated if + \begin{minipage}[c]{0.6\linewidth} + \begin{center} + \begin{tikzpicture} + \node[draw, inner sep=2mm] (image) at (0, 0) {\includesvg[height=8mm]{car}}; + \node[above=2mm of image, anchor=base, font=\scriptsize] {Input $X$}; + + \node[draw, fill=gronskasvag, right=0.75cm of image, inner sep=2mm] (model) + {\includesvg[height=8mm]{gear}}; + \node[above=2mm of model, anchor=base, font=\scriptsize] {Model $g$}; + \draw [->] (image) -- (model); + + \node[draw, right=0.75cm of model, minimum height=1.2cm, font=\scriptsize, align=center] (prediction) + {\begin{tabular}{@{}ccc@{}} + \includesvg[width=6mm]{barrier} & \includesvg[width=6mm]{pedestrian} & \includesvg[width=6mm]{bear} \\ + 80\% & 0\% & 20\% \\ + \end{tabular}}; + \node[above=2mm of prediction, anchor=base, font=\scriptsize] {Prediction $g(X) \in \Delta^m$}; + \draw [->] (model) -- (prediction); + \end{tikzpicture} + \end{center} + \end{minipage}% + \begin{minipage}[c]{0.4\linewidth} + We use $m$ for the number of classes, and + $\Delta^m \coloneqq \{ z \in [0,1]^m \colon \|z\|_1 = 1\}$ for the + $(m-1)$-dimensional probability simplex. + \end{minipage}\vspace*{\baselineskip} + + If the model is calibrated we know that for all inputs with this + prediction there is an object ahead 80\% of the time, a human 0\% + of the time, and an animal 20\% of the time. + + \begin{center} + \begin{tikzpicture} + \node[minimum height=1.2cm, inner sep=2mm] (image) at (0, 0) + {\begin{tabular}{@{}ccc@{}} + \includesvg[height=3mm]{car0} & \includesvg[height=3mm]{car1} & \includesvg[height=3mm]{car2} \\ + \includesvg[height=3mm]{car3} & \includesvg[height=3mm]{car4} & $\cdots$ \\ + \end{tabular}}; + + \node[draw, fill=gronskasvag, right=0.75cm of image, inner sep=2mm] (model) + {\includesvg[height=8mm]{gear}}; + \draw [->] (image) -- (model); + + \node[draw, right=0.75cm of model, minimum height=1.2cm, font=\scriptsize, align=center] (prediction) + {\begin{tabular}{@{}ccc@{}} + \includesvg[width=6mm]{barrier} & \includesvg[width=6mm]{pedestrian} & \includesvg[width=6mm]{bear} \\ + 80\% & 0\% & 20\% \\ + \end{tabular}}; + \draw [->] (model) -- (prediction); + + \node[right=1cm of prediction] (empirical) + {\begin{tabular}{@{}cccccc@{}} \toprule + \multicolumn{4}{c}{\includesvg[width=3mm]{barrier}} & \includesvg[width=3mm]{pedestrian} & \includesvg[width=3mm]{bear} \\ \midrule + \includesvg[height=3mm]{car0} & \includesvg[height=3mm]{car2} & \includesvg[height=3mm]{car3} & \includesvg[height=3mm]{car4} & & \includesvg[height=3mm]{car1} \\ + $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & & $\vdots$ \\ \bottomrule + \end{tabular}}; + \node[above=2mm of empirical, anchor=base, font=\scriptsize] (A) {Empirical frequency $r(g(X)) \in \Delta^m$}; + \node[font=\scriptsize] at (prediction |- A) {Prediction $g(X) \in \Delta^m$}; + + \path (prediction) -- node [font=\boldmath\Huge, color=uured, align=center, midway] {$=$} (empirical); + \end{tikzpicture} + \end{center} +} + + \posterbox[adjusted title={Quantifying calibration - a unifying framework}, colback=gryningmellan]{name=error,column=1,span=3,between=calibration and footline}{ + \begin{tcolorbox}[colback=blondstark] + We define the \hl{calibration error}~($\measure$) of model $g$ with respect to a class $\mathcal{F}$ of functions $f \colon \Delta^m \to \mathbb{R}^m$ as \begin{equation*} - g_y(X) = \Prob[Y = y \,|\, g(X)] \quad \text{almost surely for all classes } y, + \measure[\mathcal{F}, g] \coloneqq \sup_{f \in \mathcal{F}} \Expect\left[\transpose{(r(g(X)) - g(X))} f(g(X)) \right]. \end{equation*} - or equivalently if - \begin{equation}\label{eq:calibration} - g(X) = r(g(X)) \coloneqq (\mathbb{P}[Y = 1 \,|\, g(X)], \ldots, \mathbb{P}[Y = m \,|\, g(X)]) \quad \text{almost surely}. - \end{equation} + \end{tcolorbox} - \item There are many calibrated models \parencite{vaicenavicius19_evaluat}, so there is hope to find one. - \begin{tcolorbox}[colback=blondsvag] - Consider three equally likely classes with triangular distributions of inputs: - \begin{center} - \pgfplotsset{width=0.25\textwidth,height=0.1\textwidth} - \input{figures/triangular_pdf_x_y.tex} - \end{center} - The following models are all calibrated but only the first one is ideal: - \begin{center} - \pgfplotsset{width=0.25\linewidth,height=0.1\textwidth,ytick={0,0.5,1}} - \input{figures/triangular_models.tex} - \end{center} - \end{tcolorbox} - \end{itemize} - } + By design, if model $g$ is calibrated then the $\measure$ is zero, regardless of $\mathcal{F}$. - \posterbox[adjusted title={Calibration error}]{name=error,column=1,below=calibration}{ - We propose the following general measure of miscalibration which arises naturally from the definition of calibration in \cref{eq:calibration}. + \tcbsubtitle{Kernel calibration error} - \begin{tcolorbox}[colback=sandstark] - The \hl{calibration error}~($\measure$) of model $g$ w.r.t.\ a class $\mathcal{F}$ of functions $f \colon \Delta^m \to \mathbb{R}^m$ is + \begin{tcolorbox}[colback=blondstark] + We define the \hl{kernel calibration error} ($\kernelmeasure$) + of model $g$ with respect to a matrix-valued kernel + $k \colon \Delta^m \times \Delta^m \to \mathbb{R}^{m \times m}$ as \begin{equation*} - \measure[\mathcal{F}, g] \coloneqq \sup_{f \in \mathcal{F}} \Expect\left[{(r(g(X)) - g(X))}^\intercal f(g(X)) \right]. + \kernelmeasure[k, g] \coloneqq \measure[\mathcal{F}, g], \end{equation*} + where $\mathcal{F}$ is the unit ball in the reproducing kernel + Hilbert space corresponding to $k$. \end{tcolorbox} - \vspace{-\topsep} + If $k$ is a universal kernel, then the $\kernelmeasure$ is zero if + and only if model $g$ is calibrated. + + \tcbsubtitle{Relation to existing measures} \begin{itemize} - \item By design, if model $g$ is calibrated then the $\measure$ is zero, regardless of $\mathcal{F}$. - \item The $\measure$ is equal to the common expected calibration error ($\ECE$) + \item For common distances $d$ the expected calibration error ($\ECE$) \begin{equation}\label{eq:ece} \ECE[d, g] = \Expect[d(r(g(X)), g(X))] \end{equation} - for certain choices of $\mathcal{F}$ and distances $d$ such as the city block distance, the total variation distance, and the squared Euclidean distance. - \item The $\measure$ captures also the maximum mean calibration error \parencite{kumar18_train_calib_measur_neural_networ}. + can be formulated as a $\measure$. + + \item The framework captures the maximum mean calibration error as well. \end{itemize} } - \posterbox[adjusted title={Estimators of the calibration error}]{name=estimator,column=2,below=title}{ - Consider the task of estimating the $\measure$ of model $g$ using a validation set $\{(X_i, Y_i)\}_{i=1}^n$ of i.i.d.\ random pairs of inputs and labels that are distributed according to $(X,Y)$. - + \posterbox[adjusted title=The paper in 30 seconds, colback=blondmellan]{name=summary,column=1,span=2,between=title and error}{ \begin{itemize} - \item Standard estimators of the $\ECE$ are inconsistent and biased in many cases \parencite{vaicenavicius19_evaluat} and can scale poorly to large $m$. - - \begin{tcolorbox}[colback=blondsvag] - The main difficulty is the estimation of the function $r$ in \cref{eq:ece}. \Cref{eq:kce} shows that for $\kernelmeasure$ there is no explicit dependence on $r$! - \end{tcolorbox} - - \item For $i,j \in \{1,\ldots,n\}$ define - \begin{equation*} - h_{i,j} \coloneqq {(e_{Y_i} - g(X_i))}^\intercal k(g(X_i), g(X_j)) (e_{Y_j} - g(X_j)). - \end{equation*} + \item We propose a \hl{unifying framework} of calibration errors + that allows us to derive a new \hl{kernel calibration error} with + \hl{unbiased and consistent estimators}. + \item Calibration error estimates are not interpretable. Instead we + can conduct hypothesis tests of calibration. + \item In contrast to existing approaches, the KCE enables + well-founded bounds and approximations of the p-value for + calibration tests. + \end{itemize} - \begin{tcolorbox}[colback=sandstark] - If $\mathbb{E}[\|k(g(X),g(X))\|] < \infty$, then the following estimators are \hl{consistent estimators of the squared kernel calibration error} $\squaredkernelmeasure[k, g] \coloneqq \kernelmeasure[k,g]^2$. - \begin{center} - \begin{tabular}{llll} \toprule - Notation & Definition & Properties & Complexity\\ \midrule - $\biasedskce$ & $n^{-2} \sum_{i,j=1}^n h_{i,j}$ & biased & $O(n^2)$ \\ - $\unbiasedskce$ & $ {\binom{n}{2}}^{-1} \sum_{1 \leq i < j \leq n} h_{i,j}$ & unbiased & $O(n^2)$ \\ - $\linearskce$ & $ {\lfloor n/2\rfloor}^{-1} \sum_{i = 1}^{\lfloor n / 2\rfloor} h_{2i-1,2i}$ & unbiased & $O(n)$ \\ \bottomrule - \end{tabular} - \end{center} - \end{tcolorbox} + \tcbsubtitle{Take with you} + \begin{itemize} + \item Kernel calibration error (KCE) with unbiased and consistent estimators + \item Calibration errors have no meaningful unit or scale + \item Reliable calibrations tests with the KCE \end{itemize} } - \posterbox[adjusted title={Viewing estimators as test statistics}]{name=statistics,column=2,below=estimator}{ - In general, the $\measure$ does not have a meaningful unit or scale. This renders it difficult to interpret an estimated non-zero error and to compare different models. - - \begin{tcolorbox}[colback=sandstark] - For the consistent and unbiased estimators of the $\squaredkernelmeasure$ we derive \hl{bounds and approximations of the probability of false rejection} of a calibrated model. + \posterbox[adjusted title={Estimating the calibration error}, colback=gronskasvag]{name=estimation,column=4,span=3,below=calibration}{ + We want to estimate the $\measure$ of model $g$ using a validation + data set $\{(X_i, Y_i)\}_{i=1}^n$ of i.i.d.\ pairs of inputs and labels. + + \tcbsubtitle{Kernel calibration error} + + For $i,j \in \{1,\ldots,n\}$, let + $h_{i,j} \coloneqq \transpose{(e_{Y_i} - g(X_i))} k(g(X_i), g(X_j)) (e_{Y_j} - g(X_j))$, + where $e_i \in \Delta^m$ denotes the $i$th unit vector. + + \begin{tcolorbox}[colback=blondstark] + If $\mathbb{E}[\|k(g(X),g(X))\|] < \infty$, then \hl{consistent estimators} + of the squared kernel calibration error + $\squaredkernelmeasure[k, g] \coloneqq \kernelmeasure[k,g]^2$ are: + \begin{center} + \begin{tabular}{llll} \toprule + Notation & Definition & Properties & Complexity\\ \midrule + $\biasedestimator$ & $n^{-2} \sum_{i,j=1}^n h_{i,j}$ & biased & $O(n^2)$ \\ + $\unbiasedestimator$ & $ {\binom{n}{2}}^{-1} \sum_{1 \leq i < j \leq n} h_{i,j}$ & unbiased & $O(n^2)$ \\ + $\linearestimator$ & $ {\lfloor n/2\rfloor}^{-1} \sum_{i = 1}^{\lfloor n / 2\rfloor} h_{2i-1,2i}$ & unbiased & $O(n)$ \\ \bottomrule + \end{tabular} + \end{center} \end{tcolorbox} - \vspace{-\topsep} - \begin{itemize} - \item These results allow us to test the hypothesis that model $g$ is calibrated. - \item The bounds enable us to transfer unintuitive calibration error estimates to an \hl{intuitive and interpretable} probabilistic setting. - \end{itemize} - } + \tcbsubtitle{Relation to the expected calibration error} - \posterbox[adjusted title={Kernel calibration error}]{name=kce,column=1,between=error and references}{ - Let $\mathcal{F}$ be the unit ball in a reproducing kernel Hilbert space with matrix-valued kernel $k \colon \Delta^m \times \Delta^m \to \mathbb{R}^{m \times m}$. Then we define the \hl{kernel calibration error} ($\kernelmeasure$) with respect to kernel $k$ as $\kernelmeasure[k, g] \coloneqq \measure[\mathcal{F}, g]$. + Standard estimators of the $\ECE$ are usually biased and inconsistent. + The main difficulty is the estimation of the empirical frequencies + $r(g(X))$ in \cref{eq:ece}. For the $\kernelmeasure$ there is no need + to estimate them! + } - \begin{tcolorbox}[colback=blondsvag] - An example of a matrix-valued kernel is $k(a, b) = M \tilde{k}(a, b)$, where $M \in \mathbb{R}^{m\times m}$ is a positive semi-definite matrix and $\tilde{k} \colon \Delta^m \times \Delta^m \to \mathbb{R}$ is a real-valued kernel. - \end{tcolorbox} + \posterbox[adjusted title={Example: A simple matrix-valued kernel}, colback=sandsvag]{name=kernel,column=4,span=3,between=estimation and footline}{ + If $\tilde{k} \colon \Delta^m \times \Delta^m \to \mathbb{R}$ is a + real-valued kernel and $M \in \mathbb{R}^{m \times m}$ is positive semi-definite, + then $k = M \tilde{k}$ is a matrix-valued kernel. + } - If $k$ is a universal kernel, then the $\kernelmeasure$ is zero if and only if model $g$ is calibrated. + \posterbox[adjusted title={Is my model calibrated?}, colback=sandsvag]{name=statistics,column=7,span=4,below=top}{ + In general, calibration errors have no meaningful unit or scale. + This renders it difficult to interpret an estimated non-zero error. - \begin{tcolorbox}[colback=sandstark] - If $\Expect[\|k(g(X),g(X))\|] < \infty$, then - \begin{equation}\label{eq:kce} - \kernelmeasure[k,g] = {\bigg(\Expect[{(e_Y - g(X))}^{\intercal} k(g(X), g(X')) {(e_{Y'} - g(X'))}]\bigg)}^{1/2}, - \end{equation} - where $(X', Y')$ is an independent copy of $(X,Y)$ and $e_i$ denotes the $i$th unit vector. + \tcbsubtitle{Calibration tests} + \begin{minipage}[t]{0.35\linewidth} + \vspace*{0pt} + We can use the calibration error estimates to perform a + statistical test of the null hypothesis + \begin{equation*} + H_0 \coloneqq \text{\enquote{the model is calibrated}}. + \end{equation*} + \end{minipage} + \begin{minipage}[t]{0.65\linewidth} + \vspace*{0pt} + \begin{center} + \begin{tikzpicture}[ + declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));}, + declare function={binormal(\ma,\sa,\mb,\sb,\p)=(\p*normal(\ma,\sa)+(1-\p)*normal(\mb,\sb));} + ] + + \begin{axis}[ + domain = -0.1:0.2, + no marks, + xlabel = calibration error estimate, + ylabel = density, + grid=major, + ymin = 0, + tick label style={font=\tiny}, + label style={font=\small}, + width = 0.75\linewidth, + height = 0.33\linewidth, + legend pos=outer north east, + legend cell align=left, + legend style= + { + fill=none, + draw=none, + inner sep={0pt}, + font=\small, + align=left, + } + ] + + \draw [Dark2-A, thick] (0.07,\pgfkeysvalueof{/pgfplots/ymin}) -- (0.07,\pgfkeysvalueof{/pgfplots/ymax}) node [at end, above, anchor=south east, sloped, font=\small] {observed}; + + \draw[Dark2-B, thick] (0,\pgfkeysvalueof{/pgfplots/ymin}) -- (0,\pgfkeysvalueof{/pgfplots/ymax}) node [at end, above, anchor=south east, sloped, font=\small] {calibrated}; + + % mixture model of normal distributions + \addplot+ [color=Dark2-B, dashed, thick, samples=31, smooth, name path=A] {binormal(-0.05,0.01,0.05,0.03,0.5)}; + \addlegendentry{distribution\\ under $H_0$}; + + % indicate p-value + \path [name path=B] (\pgfkeysvalueof{/pgfplots/xmin},0) -- (\pgfkeysvalueof{/pgfplots/xmax},0); + \addplot+ [draw=Dark2-C, pattern color=Dark2-C, pattern={north east lines}] fill between [of=A and B, soft clip={domain=0.07:0.2}]; + \addlegendentry{p-value}; + + % add comment + \node[anchor=west, align=left, text=Dark2-C, font=\small] (annotation) at (0.075, 10) {reject $H_0$ if the \\p-value is small}; + \draw[->, >=stealth, thick, Dark2-C] (annotation) -- (0.08, 1); + \end{axis} + \end{tikzpicture} + \end{center} + \end{minipage} + + \begin{tcolorbox}[colback=blondstark] + We derive \hl{well-founded bounds and approximations} of the p-value + based on the $\squaredkernelmeasure$. \end{tcolorbox} } - \posterbox[adjusted title={Experiments}]{name=experiment,column=2,between=statistics and references}{ - We construct data sets $\{g(X_i), Y_i\}_{i=1}^{250}$ of three models with $10$ classes by sampling predictions $g(X_i) \sim \Dir(0.1, \dots, 0.1)$ and labels $Y_i$ conditionally on $g(X_i)$ from - \begin{align*} - \text{\textbf{M1: }} &\Categorical(g(X_i)), & - \text{\textbf{M2: }} &0.5\Categorical(g(X_i)) + 0.5\Categorical(1,0,\dots,0), & - \text{\textbf{M3: }} &\Categorical(0.1, \dots, 0.1). - \end{align*} - Model \textbf{M1} is calibrated, and models \textbf{M2} and \textbf{M3} are uncalibrated. + \posterbox[adjusted title={Experiments}, colback=gryningmellan]{name=experiment,column=7,span=4,between=statistics and footline}{ + We construct synthetic data sets $\{(g(X_i), Y_i)\}_{i=1}^{250}$ + from three generative models with $10$ classes by sampling + predictions $g(X_i) \sim \Dir(0.1, \dots, 0.1)$ and labels $Y_i$ + conditionally on $g(X_i)$ from + \begin{equation*} + \symbf{M1}\colon \, \Categorical(g(X_i)), \quad + \symbf{M2}\colon \, 0.5\Categorical(g(X_i)) + 0.5\Categorical(1,0,\dots,0), \quad + \symbf{M3}\colon \, \Categorical(0.1, \dots, 0.1). + \end{equation*} + Model $\symbf{M1}$ is calibrated, and models $\symbf{M2}$ and + $\symbf{M3}$ are uncalibrated. + + \tcbsubtitle{Calibration error estimates} \begin{center} - \input{figures/comparison_estimates.tex} - \captionof{figure}{Calibration error estimates of $10^4$ randomly sampled data sets. The solid black line indicates the mean of the calibration error estimates, and the dashed red line displays the true calibration error of the model.} + \input{figures/errors_comparison.tex} \end{center} + \tcbsubtitle{Empirical test errors} + \begin{center} - \input{figures/comparison_tests.tex} - \captionof{figure}{Test errors versus bounds/approximations of the probability of false rejection, evaluated on $500$ ($\asympunbiasedskce$) and $10^4$ (all other test statistics) randomly sampled data sets. For model \textbf{M1} the type I error is shown, for both uncalibrated models the type II error is plotted.} + \input{figures/pvalues_comparison.tex} \end{center} } \end{tcbposter} diff --git a/poster/references.bib b/poster/references.bib deleted file mode 100644 index 3040415..0000000 --- a/poster/references.bib +++ /dev/null @@ -1,68 +0,0 @@ -@InProceedings{kumar18_train_calib_measur_neural_networ, - author = {Kumar, Aviral and Sarawagi, Sunita and Jain, Ujjwal}, - title = {Trainable Calibration Measures for Neural Networks - from Kernel Mean Embeddings}, - booktitle = {Proceedings of the 35th International Conference on - Machine Learning}, - year = 2018, - volume = 80, - pages = {2805--2814}, - abstract = {Modern neural networks have recently been found to - be poorly calibrated, primarily in the direction of - over-confidence. Methods like entropy penalty and - temperature smoothing improve calibration by - clamping confidence, but in doing so compromise the - many legitimately confident predictions. We propose - a more principled fix that minimizes an explicit - calibration error during training. We present MMCE, - a RKHS kernel based measure of calibration that is - efficiently trainable alongside the negative - likelihood loss without careful hyper-parameter - tuning. Theoretically too, MMCE is a sound measure - of calibration that is minimized at perfect - calibration, and whose finite sample estimates are - consistent and enjoy fast convergence - rates. Extensive experiments on several network - architectures demonstrate that MMCE is a fast, - stable, and accurate method to minimize calibration - error while maximally preserving the number of high - confidence predictions.}, - pdf = - {http://proceedings.mlr.press/v80/kumar18a/kumar18a.pdf}, - series = {Proceedings of Machine Learning Research}, -} - -@InProceedings{vaicenavicius19_evaluat, - author = {Vaicenavicius, Juozas and Widmann, David and - Andersson, Carl and Lindsten, Fredrik and Roll, - Jacob and Sch\"{o}n, Thomas B.}, - title = {Evaluating model calibration in classification}, - booktitle = {Proceedings of Machine Learning Research}, - year = 2019, - volume = 89, - pages = {3459--3467}, - abstract = {Probabilistic classifiers output a probability - distribution on target classes rather than just a - class prediction. Besides providing a clear - separation of prediction and decision making, the - main advantage of probabilistic models is their - ability to represent uncertainty about - predictions. In safety-critical applications, it is - pivotal for a model to possess an adequate sense of - uncertainty, which for probabilistic classifiers - translates into outputting probability distributions - that are consistent with the empirical frequencies - observed from realized outcomes. A classifier with - such a property is called calibrated. In this work, - we develop a general theoretical calibration - evaluation framework grounded in probability theory, - and point out subtleties present in model - calibration evaluation that lead to refined - interpretations of existing evaluation - techniques. Lastly, we propose new ways to quantify - and visualize miscalibration in probabilistic - classification, including novel multidimensional - reliability diagrams.}, - month = 4, - series = {Proceedings of Machine Learning Research}, -} \ No newline at end of file