From 1a8084b930b4803ec94ea70cfbd2bcfba191dbd0 Mon Sep 17 00:00:00 2001
From: pat-alt <altmeyerpat@gmail.com>
Date: Wed, 22 Mar 2023 13:44:27 +0100
Subject: [PATCH] progress

---
 paper/paper.pdf | Bin 245887 -> 248984 bytes
 paper/paper.tex |  15 ++++++++++++++-
 2 files changed, 14 insertions(+), 1 deletion(-)

diff --git a/paper/paper.pdf b/paper/paper.pdf
index c2073ea05e680bdb3d380fc0146d21a59edc7ba2..df78c2736dc7c33f5e831bc52f4429fbfdfb3211 100644
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diff --git a/paper/paper.tex b/paper/paper.tex
index bb65d607..faa5f70f 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -31,12 +31,16 @@
 \usepackage{microtype}      % microtypography
 \usepackage{xcolor}         % colors
 \usepackage{amsmath}
+\usepackage{amsthm}
 \usepackage{graphicx}
 
 % Bibliography
 \bibliographystyle{plainnat}
 \setcitestyle{numbers,square,comma}
 
+% Numbered Environments:
+\newtheorem{definition}{Definition}[section]
+
 
 \title{Conformal Counterfactual Explanations}
 
@@ -123,7 +127,16 @@ Solutions to Equation~\ref{eq:general} are considered valid as soon as the predi
 
 The crucial difference between Adversarial Examples (AE) and Counterfactual Explanations is one of intent. While an AE is intended to go unnoticed, a CE should have certain desirable properties. The literature has made this explicit by introducing various so-called \textit{desiderata}. To properly serve both AI practitioners and individuals affected by AI decision-making systems, counterfactuals should be sparse, proximate~\citep{wachter2017counterfactual}, actionable~\citep{ustun2019actionable}, diverse~\citep{mothilal2020explaining}, plausible~\citep{joshi2019realistic,poyiadzi2020face,schut2021generating}, robust~\citep{upadhyay2021robust,pawelczyk2022probabilistically,altmeyer2023endogenous} and causal~\citep{karimi2021algorithmic} among other things. Researchers have come up with various ways to meet these desiderata, which have been surveyed in~\citep{verma2020counterfactual} and~\citep{karimi2020survey}. 
 
-To fulfil this latter goal, researchers have come up with a myriad of ways. @joshi2019realistic were among the first to suggest that instead of searching counterfactuals in the feature space, we can instead traverse a latent embedding learned by a surrogate generative model. Similarly, @poyiadzi2020face use density ... Finally, @karimi2021algorithmic argues that counterfactuals should comply with the causal model that generates them [CHECK IF WE CAN PHASE THIS LIKE THIS]. Other related approaches include ... All of these different approaches have a common goal: they aim to ensure that the generated counterfactuals comply with the (learned) data-generating process (DGB). 
+Finding ways to generate \textit{plausible} counterfactuals has been one of the primary concerns. To this end, \citet{joshi2019realistic} were among the first to suggest that instead of searching counterfactuals in the feature space $\mathcal{X}$, we can instead traverse a latent embedding $\mathcal{Z}$ that implicitly codifies the data generating process (DGP) of $x\sim\mathcal{X}$. To learn the latent embedding, they introduce a surrogate model. In particular, they propose to use the latent embedding of a Variational Autoencoder (VAE) trained to generate samples $x^*\sim \mathcal{G}(z)$ where $\mathcal{G}$ denotes the decoder part of the VAE. Provided the surrogate model is well-trained, their proposed approach can yield compelling counterfactual explanations like the one in the centre panel of Figure~ref{fig:vae}. 
+
+Others have proposed similar approaches. \citet{dombrowski2021diffeomorphic} traverse the base space of a normalizing flow to solve Equation~\ref{eq:general}, essentially relying on a different surrogate model for the generative task. \citet{poyiadzi2020face} use density estimators ($\hat{p}: \mathcal{X} \mapsto [0,1]$) to constrain the counterfactual paths. \citet{karimi2021algorithmic} argue that counterfactuals should comply with the causal model that generates the data. All of these different approaches share a common goal: ensuring that the generated counterfactuals comply with the true and unobserved DGP. To summarize this broad objective, we propose the following definition:
+
+\begin{definition}[Plausible Counterfactuals]
+  \label{def:plausible}
+  Let $\mathcal{X}|t$ denote the true conditional distribution of samples in the target class. Then for $x^{\prime}$ to be considered a plausible counterfactual, we need: $x^{\prime} \sim \mathcal{X}|t$.
+\end{definition}
+
+Note that Definition~\ref{def:plausible} subsumes the notion of plausible counterfactual paths, since we can simply apply it to each counterfactual state along the path.
 
 \begin{figure}
   \centering
-- 
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