text:
\section{Introduction}
\label{sec:intro}

\begin{figure}[t!]
\centering
\includegraphics[width=0.8\linewidth]{figures/sim.png}
\caption{\textbf{Simulatability Metric.} Given task input $\mathbf{r}_i$ and the task LM's predicted label $\mathbf{\hat{y}}_i$, simulatability measures rationale ($\mathbf{r}_i$) quality as how accurately a simulator (\textit{i.e., } either LM or human observer) predicts $\mathbf{\hat{y}}_i$ using both task $\mathbf{x}_i$ and $\mathbf{r}_i$ as input, compared to using only $\mathbf{x}_i$.}
\label{fig:sim}
\end{figure}

Neural language models (LMs) perform well on many NLP tasks \citep{devlin2018bert, liu2019roberta}, but their complex behavior can be hard to explain \citep{rudin2019stop, lipton2018mythos, bender2021dangers}.
Unlike extractive rationales \citep{denil2014extraction, sundararajan2017axiomatic, chan2022unirex} which are limited to input token scoring, \textit{free-text rationales} are designed to explain LMs' decisions more intuitively and flexibly via natural language (Fig. \ref{fig:f_model}) \citep{narang2020wt5, lakhotia2020fid, kumar2020nile, rajani2019explain, camburu2018snli, park2018multimodal, hendricks2016generating}.
Free-text rationales are especially needed if the LM's reasoning involves latent knowledge beyond the task input (Fig. \ref{fig:f_model}) --- \textit{e.g., } in commonsense reasoning \citep{rajani2019explain, wiegreffe2021measuring}.
To meaningfully explain LM behavior, rationales should be \emph{faithful} (reflects LM's actual behavior) \citep{jacovi2020towards} and \emph{plausible} (convincing to humans) \citep{deyoung2019eraser}.
Thus, it is important to have metrics for evaluating free-text rationales w.r.t. these desiderata.

Given an LM's (\textit{i.e., } task LM's) predicted label (in text classification) and a rationale for this prediction, existing free-text rationale metrics \cite{hase2020leakage, wiegreffe2021measuring} are defined w.r.t. rationale-label association.
This is implemented using \textit{simulatability}: How much does the rationale help a simulator (\textit{i.e., } observer) predict the task LM's predicted label \cite{doshi2017towards, hase2020evaluating}?
In existing simulatability metrics, faithfulness and plausibility are measured via LM and human simulators, respectively.
Since the simulator itself is a complex system, simulatability (\textit{i.e., } change in simulator accuracy) can be susceptible to confounding variables (\textit{e.g., } task model's task performance).
In particular, simulators are pretrained on external corpora by default, yet there is no standard protocol for assessing whether this hurts simulatability metrics' reliability.

In light of this, we propose \textbf{F}ree-Text \textbf{R}ationale Simul\textbf{A}tability \textbf{M}eta-\textbf{E}valuation (\textbf{\textsc{FRAME}}), a framework for evaluating simulatability metrics for free-text rationales.
\textsc{FRAME}\xspace is based on three axioms: (1) good metrics should yield highest scores for \textit{reference rationales}, which maximize rationale-label association by construction; (2) good metrics should be appropriately sensitive to semantic perturbation of rationales; and (3) good metrics should be robust to variation in the task LM's task performance.
For (1), we simply define the reference rationale as the task LM's predicted label.
This provides a powerful invariance for analyzing the quality of free-text rationales, whose unconstrained nature typically makes such analysis difficult.
For (2), we test if the metric responds appropriately to equivalent (should not change meaning) and contrastive (should change meaning) perturbations of reference rationales.
For (3), we compute simulatability as a function of the task LM's task performance, by varying factors like the task LM's number of train instances, number of noisy train instances, and capacity.
We also separately compute simulatability on correctly and incorrectly predicted instances, to see if it is consistent across these two subpopulations.

On three text classification datasets --- e-SNLI \citep{camburu2018snli}, CoS-E v1.0 \citep{rajani2019explain, talmor2018commonsenseqa}, and CoS-E v1.11 \citep{rajani2019explain, talmor2018commonsenseqa} --- we show that existing metrics cannot satisfy all three \textsc{FRAME}\xspace axioms.
Since they involve pretraining simulators on external corpora, existing metrics struggle to isolate the rationale's contribution to simulator accuracy from the pretraining knowledge's, hence muddling the metric's signal.
Therefore, we introduce a non-pretraining simulatability metric to address this issue.
For faithfulness, our proposed metric improves performance on (1) and (3) by an average of 41.7\% and 42.9\%, respectively, while performing competitively on (2) (Sec. \ref{sec:exp:faith}).
For plausibility, we conduct a user study of (1), showing that our metric improves performance on (1) by 47.7\% (Sec. \ref{sec:exp:plaus}).
Finally, we use \textsc{FRAME}\xspace to analyze the effects of label leakage on simulatability (Sec. \ref{sec:exp:leakage}).

\begin{figure}[t!]
\centering
\includegraphics[width=0.8\linewidth]{figures/f_model.png}
\caption{\textbf{Self-Rationalizing Task LM.} Given $\mathbf{x}_i$ as input, the self-rationalizing task LM ($\mathcal{F}$) is trained to jointly generate gold label $\mathbf{\dot{y}}_i$ and gold rationale $\mathbf{\dot{r}}_i$ as a single output sequence.}
\label{fig:f_model}
\end{figure}
\section{Background}
\label{sec:background}

\subsection{Free-Text Rationale Generation}
\label{sec:background:gen}
Let $\mathcal{F}_{\text{task}}$ denote a task LM for $m$-class text classification.
For task instance $i$, let $\mathbf{x}_i = [x_{i}^{t}]_{t=1}^{n_x}$ be the $n_x$-token input sequence (\textit{e.g., } a sentence), $\dot{y}_i$ be the gold label, and $\hat{y}_i$ be $\mathcal{F}_{\text{task}}$'s predicted label.
Let $\mathbf{\dot{y}}_i = [\dot{y}_{i}^{t}]_{t=1}^{n_{\dot{y}}}$ and $\mathbf{\hat{y}}_i = [\hat{y}_{i}^{t}]_{t=1}^{n_y}$ be token sequence representations of $\dot{y}_i$ and $\hat{y}_i$, respectively.
A \textit{free-text rationale} $\mathbf{r}_i = [r_{i}^{t}]_{t=1}^{n_r}$ is an $n_r$-token sequence that uses natural language to explain the reasoning process behind $\mathcal{F}_{\text{task}}$ predicting $\hat{y}_i$ (Fig. \ref{fig:f_model}-\ref{fig:h_model}).
Unlike extractive rationales, which are constrained to scoring tokens in $\mathbf{x}_i$, free-text rationales can have arbitrary content, style, and length (\textit{i.e., } $n_r$) \citep{narang2020wt5, camburu2018snli}.
Free-text rationales can be generated via any means, but are typically generated by an explainer LM $\mathcal{F}_{\text{expl}}$.
In general, we use $\mathbf{r}_i$ to denote a free-text rationale with no assumptions about its generation process, but we specifically denote $\mathcal{F}_{\text{expl}}$'s predicted rationale as $\mathbf{\hat{r}}_i$.
While $\mathcal{F}_{\text{task}}$ and $\mathcal{F}_{\text{expl}}$ can be separate \citep{kumar2020nile, rajani2019explain, camburu2018snli}, \textit{self-rationalizing LMs} combine $\mathcal{F}_{\text{task}}$ and $\mathcal{F}_{\text{expl}}$ into a single LM $\mathcal{F}$, which jointly generates the task and rationale outputs \citep{narang2020wt5, lakhotia2020fid, do2020snli, liu2018towards} (Fig. \ref{fig:f_model}).
In particular, text-to-text self-rationalizing LMs (\textit{e.g., } T5 \cite{raffel2019exploring}) output an ($n_y+n_r$)-token sequence $[\mathbf{\hat{y}}_i, \mathbf{\hat{r}}_i]$.
Following recent works \citep{hase2020leakage, wiegreffe2021measuring}, we focus on T5-based self-rationalizing LMs as $\mathcal{F}$.

\subsection{Free-Text Rationale Evaluation}
\label{sec:background:eval}

Given $\mathbf{x}_i$ and $\mathcal{F}$, existing free-text rationale metrics evaluate $\mathbf{r}_i$ by measuring the association between $\mathbf{r}_i$ and $\mathbf{\hat{y}}_i$, denoted as $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i, \mathbf{x}_i; \mathcal{F})$.
$\rho$ is computed via \textit{simulatability} (sim), which measures how predictive $\mathbf{r}_i$ is of $\mathbf{\hat{y}}_i$ \citep{doshi2017towards, hase2020evaluating}.
That is, how accurately does a \textit{simulator} (\textit{i.e., } observer) predict $\mathbf{\hat{y}}_i$ using both $\mathbf{x}_i$ and $\mathbf{r}_i$ as input, compared to using only $\mathbf{x}_i$?
Let $\Phi$ be a sim metric and $\mathcal{S}$ be a simulator.
Then, $\Phi(\mathbf{r}_i, \mathbf{\hat{y}}_i, \mathbf{x}_i; \mathcal{F}, \mathcal{S}) = \mathbbm{1}_{\mathcal{S}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) - \mathbbm{1}_{\mathcal{S}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$, where $0 \leq \mathbbm{1}_{c}(b \hspace{0.5mm} | \hspace{0.5mm} a) \leq 100$ is $c$'s accuracy in predicting $b$ given $a$.
$\mathbbm{1}_{\mathcal{S}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$ and $\mathbbm{1}_{\mathcal{S}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i)$ are the \textit{control} and \textit{treatment} terms, respectively, with higher $\Phi$ indicating higher $\rho$ (\textit{i.e., } $\mathbf{r}_i$ quality).

\subsubsection{Simulator Configuration}
The rationale desideratum (\textit{e.g., } faithfulness, plausibility) evaluated with $\Phi$ is determined by the choice of $\mathcal{S}$.
We discuss these choices below.

\paragraph{Faithfulness}
For faithfulness, we can set $\mathcal{S}$ as another LM \cite{hase2020leakage, wiegreffe2021measuring}.
However, in order to do so, prior works implicitly make two assumptions.
First, they assume $\rho$ is sufficiently informative about $\mathcal{F}$'s reasoning process w.r.t. $\mathbf{r}_i$, $\mathbf{\hat{y}}_i$, and $\mathbf{x}_i$.
Note that $\rho$ (\textit{i.e., } rationale-label association) only constitutes one aspect of faithfulness.
Thus, a rationale that maximizes $\rho$ may not necessarily explain $\mathcal{F}$'s entire reasoning process, but can still provide useful signal about $\mathcal{F}$'s behavior.
This assumption is also the basis of existing extractive rationale metrics \cite{deyoung2019eraser, pruthi2020evaluating, hooker2018benchmark, shrikumar2017learning}.
Second, despite them being different LMs, they assume $\mathcal{S}$'s reasoning process is sufficiently similar to $\mathcal{F}$'s w.r.t. the given task, such that $\mathcal{S}$'s behavior reflects $\mathcal{F}$'s.
This is needed because using $\mathcal{F}$ as the simulator would trivially result in perfect $\mathbf{\hat{y}}_i$ prediction accuracy for the control term, rendering $\Phi$ meaningless.
To support this assumption, prior works use the same LM architecture, training process, and training data for both $\mathcal{F}$ and $\mathcal{S}$ \cite{hase2020leakage, wiegreffe2021measuring}.

In practice, it can be difficult for $\mathcal{S}$ to jointly approximate the $P(\hspace{0.5mm} \cdot \hspace{0.5mm} |\mathbf{x}_i)$ and $P( \hspace{0.5mm} \cdot \hspace{0.5mm} | \mathbf{x}_i, \mathbf{r}_i)$ distributions.
To address this issue, \citet{wiegreffe2021measuring} decomposes $\mathcal{S}$ into two LMs: $\mathcal{G}$ for $P( \hspace{0.5mm} \cdot \hspace{0.5mm} | \mathbf{x}_i)$ and $\mathcal{H}$ for $P(\hspace{0.5mm} \cdot \hspace{0.5mm} |\mathbf{x}_i, \mathbf{r}_i)$.
Meanwhile, \citet{hase2020leakage} trains $\mathcal{S}$ to approximate both $P(\hspace{0.5mm} \cdot \hspace{0.5mm} |\mathbf{x}_i)$ and $P( \hspace{0.5mm} \cdot \hspace{0.5mm} | \mathbf{x}_i, \mathbf{r}_i)$ by randomly dropping out either $\mathbf{x}_i$ or $\mathbf{r}_i$ from the input during training.
In this paper, we follow the first approach, since it enables us to use the same standard training process for all LMs.
We call $\mathcal{G}$ (Fig. \ref{fig:g_model}) and $\mathcal{H}$ (Fig. \ref{fig:h_model}) the \textit{control LM} and \textit{treatment LM}, respectively.
Then, $\Phi$ is computed as: $\Phi(\mathbf{r}_i, \mathbf{\hat{y}}_i, \mathbf{x}_i; \mathcal{F}, \mathcal{G}, \mathcal{H}) = \mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) - \mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$.
Unless $\Phi$ is specifically not defined w.r.t. $\mathbf{\hat{y}}_i$, $\mathbf{x}_i$, $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$, we abbreviate $\Phi(\mathbf{r}_i, \mathbf{\hat{y}}_i, \mathbf{x}_i; \mathcal{F}, \mathcal{G}, \mathcal{H})$ as $\Phi(\mathbf{r}_i)$.

\paragraph{Plausibility}
For plausibility, $\mathcal{S}$ is considered as a human annotator, directly telling us if humans judge $\mathbf{r}_i$ as a convincing explanation for $\mathbf{\hat{y}}_i$ \cite{doshi2017towards, hase2020evaluating, chan2022unirex}.
Note that simulatability was originally defined in the context of plausibility \cite{doshi2017towards}, before being adapated for faithfulness \cite{hase2020leakage}.
Like in faithfulness, this implicitly assumes all humans share a sufficiently similar reasoning process w.r.t. the given task.
We use the same $\mathcal{S}$ for both $P(\hspace{0.5mm} \cdot \hspace{0.5mm} |\mathbf{x}_i, \mathbf{r}_i)$ and $P( \hspace{0.5mm} \cdot \hspace{0.5mm} | \mathbf{x}_i)$, since $\mathcal{S}$ generally does not need training.

\section{\textsc{FRAME}}
\label{sec:method}

\xiang{It is not very clear in the current writing what is the ``framework" for our ``meta evaluation" effort -- i.e., evaluating a metric. See Slack for other comments.}

\begin{figure}[t!]
\centering
\includegraphics[width=0.8\linewidth]{figures/g_model.png}
\caption{\textbf{Control LM.} Given task input $\mathbf{x}_i$, the control LM ($\mathcal{G}$) is an LM simulator trained to generate $\mathcal{F}$'s predicted label $\mathbf{\hat{y}}_i$.
}
\label{fig:g_model}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{figures/h_model.png}
\caption{\textbf{Treatment LM.} Given task input $\mathbf{x}_i$, augmented with some rationale (one of $\mathbf{r}_i$, $\mathbf{\dot{y}}_i$, $\mathbf{\dot{r}}_i$, or $\mathbf{\hat{y}}_i$) as input, the treatment LM ($\mathcal{H}$) is an LM simulator trained to generate $\mathcal{F}$'s predicted label $\mathbf{\hat{y}}_i$.}
\label{fig:h_model}
\end{figure}

In Sec. \ref{sec:background:eval}, we defined $\Phi$ as a sim metric for evaluating rationale quality via rationale-label association (\textit{i.e., } $\rho$).
Ideally, $\Phi$ should capture only $\rho$, without entangling other orthogonal information (\textit{e.g., } $\mathcal{F}$'s task performance).
While a number of $\Phi$ variants have been proposed \cite{hase2020leakage, wiegreffe2021measuring}, there is no standard protocol for evaluating $\Phi$, so it is unclear if such variants are behaving as desired.
Thus, we propose \textsc{FRAME}, a general framework for evaluating $\Phi$.
\textsc{FRAME}\xspace assesses $\Phi$ according to three axioms: (1) good metrics should yield highest scores for reference rationales, which maximize rationale-label association by construction; (2) good metrics should be appropriately sensitive to semantic perturbation of rationales; and (3) good metrics should be robust to variation in the task LM's task performance.
\textsc{FRAME}\xspace can be used to evaluate $\Phi$ w.r.t. both faithfulness and plausibility.
Still, in this work, we mainly focus on faithfulness evaluation, since it is fully automated via $\mathcal{G}$ and $\mathcal{H}$.
Sec. \ref{sec:method:axioms} introduces each axiom and explains how it is used as a meta-metric for $\Phi$, while Sec. \ref{sec:method:implementation} presents implementation details for each axiom's meta-metric.

\subsection{\textsc{FRAME}\xspace Axioms}
\label{sec:method:axioms}

\xiang{i'm thinking we could basically have one illustrative figure for each axiom, to convey the high level ideas readily. In these figures, we try to avoid referencing the notations, and abstract things as much as possible.}
\aaron{Redo axiom figures}

\paragraph{Axiom 1: Reference Rationale Upper Bound}

In certain respects, the quality of $\mathbf{r}_i$ can be measured as $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$, the association between $\mathbf{r}_i$ and  $\mathbf{\hat{y}}_i$ (Sec. \ref{sec:background:eval}).
Intuitively, $\argmax_{\mathbf{r}_i} \rho(\mathbf{r}_i, \mathbf{\hat{y}}_i) = \mathbf{\hat{y}}_i$, since $\mathbf{\hat{y}}_i$ should be maximally associated with itself.
It follows that $\Phi$ should yield a higher score for $\mathbf{\hat{y}}_i$ than for any other rationale, such that $\argmax_{\mathbf{r}_i} \Phi(\mathbf{r}_i) = \mathbf{\hat{y}}_i$.
We call $\mathbf{\hat{y}}_i$ the \textit{reference rationale}, which serves as an upper bound for $\Phi(\mathbf{r}_i)$.
Given the high complexity of NLM reasoning and the lack of constraints in free-text rationale generation (Sec. \ref{sec:background:gen}), it is difficult to establish any priors about $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$ for arbitrary $\mathbf{r}_i$.
To address this gap, the reference rationale upper bound provides a powerful invariance for analyzing free-text rationale quality.
Thus, \textsc{FRAME}\xspace evaluates $\Phi$ by checking that $\Phi(\mathbf{\hat{y}}_i) > \Phi(\mathbf{r}_i)$ for various $\mathbf{r}_i$ (\textit{e.g., } $\mathbf{\hat{r}}_i$) that are likely to differ in meaning from $\mathbf{\hat{y}}_i$.

\paragraph{Axiom 2: Rationale Perturbation Sensitivity}
\label{sec:method:axiom2}

Rationale $\mathbf{r}_i$ should yield high $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$ if $\mathbf{r}_i$ contains the same semantic information as $\mathbf{\hat{y}}_i$, even if $\mathbf{r}_i$ and $\mathbf{\hat{y}}_i$ do not have the same surface form.
Conversely, $\mathbf{r}_i$ should yield low $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$ if $\mathbf{r}_i$ differs greatly in meaning from $\mathbf{\hat{y}}_i$.
Thus, $\Phi$ should be appropriately sensitive when either \textit{equivalent} (does not change $\mathbf{r}_i$'s meaning) or \textit{contrastive} (changes $\mathbf{r}_i$'s meaning) perturbations are applied to $\mathbf{r}_i$.
We denote the perturbed rationale as $\Psi(\mathbf{r}_i)$, where $\Psi$ is a perturbation function.
While $\Phi$ should be properly perturbation-sensitive for any rationale $\mathbf{r}_i$, we focus on perturbing $\mathbf{\hat{y}}_i$ in this work.
This is because the value of $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$ for arbitrary $\mathbf{r}_i$ is unclear, whereas we know $\rho(\mathbf{\hat{y}}_i, \mathbf{\hat{y}}_i)$ is high.
See Sec. \ref{sec:app:method:implementation} for more details about equivalent and contrastive perturbations.

\paragraph{Axiom 3: Robustness to Variation in $\mathcal{F}$}
\label{sec:method:axiom3}

In general, $\rho(\mathbf{r}_i, \mathbf{\hat{y}}_i)$ should be independent of whether $\mathbf{\hat{y}}_i = \mathbf{\dot{y}}_i$  (\textit{i.e., } $\mathcal{F}$'s task performance).
By Axiom 1, $\Phi(\mathbf{\hat{y}}_i)$ should be high, regardless of how $\mathcal{F}$ generates $\mathbf{\hat{y}}_i$.
Given $\mathbf{\hat{y}}_i$ as an invariance, it follows that $\Phi(\mathbf{\hat{y}}_i)$ should be roughly constant w.r.t. changes in $\mathcal{F}$'s task performance.
Note that arbitrary $\mathbf{r}_i$ may not provide such an invariance.
For example, if we consider $\mathbf{\hat{r}}_i$ (\textit{i.e., } rationale outputted by $\mathcal{F}$), then $\mathbf{\hat{r}}_i$'s quality may depend largely on how $\mathcal{F}$ is trained.
Thus, in order to properly compare sim variants, we focus on analyzing sim variants w.r.t. $\mathbf{\hat{y}}_i$.
We vary $\mathcal{F}$'s performance in the following ways.

First, $\mathcal{F}$'s task performance can be changed by respectively varying the number of instances for training $\mathcal{F}$, number of noisy-labeled instances for training $\mathcal{F}$, and number of parameters in $\mathcal{F}$.
For each version of $\mathcal{F}$, we compute $\Phi(\mathbf{\hat{y}}_i)$.
A good $\Phi$ should yield similar $\Phi(\mathbf{\hat{y}}_i)$ across all $\mathcal{F}$ versions.

Second, we separately compute $\Phi(\mathbf{\hat{y}}_i)$ for the subpopulation of instances where $\mathcal{F}$ output correct predictions (\textit{i.e., } $\mathbf{\hat{y}}_i = \mathbf{\dot{y}}_i$) as well as the subpopulation of instances where $\mathcal{F}$ output incorrect predictions (\textit{i.e., } $\mathbf{\hat{y}}_i \neq \mathbf{\dot{y}}_i$).
A good $\Phi$ should yield similar $\Phi(\mathbf{\hat{y}}_i)$ across both subpopulations.

\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{figures/frame_axioms.png}
\caption{\textbf{\textsc{FRAME}\xspace Axioms.} }
\label{fig:frame_axioms}
\end{figure}

\subsection{\textsc{FRAME}\xspace Implementation}
\label{sec:method:implementation}

\paragraph{Axiom 1: Reference Rationale Upper Bound}
For Axiom 1, we want to check whether each sim variant yields the highest sim scores for $\mathbf{y_i}$.
First, we report sim($\mathbf{y_i}$), the sim score for $\mathbf{y_i}$, where higher scores are better.
Second, given a set of non-reference rationales (\textit{i.e., } $\textbf{r}_i$, $\dot{\textbf{r}}_i$, $\dot{\textbf{y}}_i$) we want to check if the sim variant yields higher scores for the reference rationale than for each of the non-reference rationales.
To account for scaling differences in the sim scores' constituent accuracy values (\textit{i.e., } $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i)$ and $\mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$), we would like to consider the ratio between reference and non-reference sim scores.
However, since sim is computed as $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) - \mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$, sim scores can be negative if $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) < \mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$
This makes the $\text{sim}(\mathbf{\hat{y}}_i) / \text{sim}(\cdot \hspace{0.5mm})$ ratio uninformative, where $\text{sim}(\cdot \hspace{0.5mm})$ is the sim score for the non-reference rationale.
Thus, we instead compute the ratio $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \cdot \hspace{0.5mm})$ for each non-reference rationale, since $\mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$ is constant across all rationales.
We abbreviate $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \cdot \hspace{0.5mm})$ as $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \cdot \hspace{0.5mm})$, where higher $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \cdot \hspace{0.5mm})$ is better.
To aggregate the $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \cdot \hspace{0.5mm})$ ratios, we define the \textbf{M}ean \textbf{O}racle Rationale \textbf{A}ccuracy \textbf{R}atio (\textbf{MOAR}) as the mean $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{r}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \cdot \hspace{0.5mm})$ over all considered non-reference rationales.

\paragraph{Axiom 2: Rationale Perturbation Sensitivity}
For Axiom 2, we want to assess whether each sim variant is appropriately sensitive to semantic perturbation.
Given reference rationale $\mathbf{\hat{y}}_i$ and perturbed reference rationale $\mathbf{\hat{y}}'_i$, we define the \textbf{A}bsolute \textbf{S}imulatability \textbf{D}ifference (\textbf{ASD}) as $\text{ASD}(a, b) = | \text{sim}(a) - \text{sim}(b) |$.
In this case, $a = \mathbf{\hat{y}}_i$, and $b = \mathbf{\hat{y}}'_i$.
If $\mathbf{\hat{y}}'_i$ is obtained via equivalent perturbation, then lower ASD is better, since $\mathbf{\hat{y}}_i$ and $\mathbf{\hat{y}}'_i$ should be similar in meaning.
If $\mathbf{\hat{y}}'_i$ is obtained via contrastive perturbation, then higher ASD is better, since $\mathbf{\hat{y}}_i$ and $\mathbf{\hat{y}}'_i$ should be dissimilar in meaning.

\paragraph{Axiom 3: Robustness to Variation in $\mathcal{F}$}
For Axiom 3, we want to assess whether each sim variant is robust to variation in the number of instances for training $\mathcal{F}$ (100\%, 50\%, 30\%, 10\%), number of noisy-labeled instances for training $\mathcal{F}$ (0\%, 10\%, 30\%, 50\%), number of parameters in $\mathcal{F}$ (T5-Small, T5-Base, T5-Large), and $\mathcal{F}$'s task performance ($y_i = \dot{y}_i$ \textit{vs. } $y_i \neq \dot{y}_i$).
Here, we focus on sim($\mathbf{\hat{y}}_i$), since Axiom 1 tells us that sim($\mathbf{\hat{y}}_i$) should always be high, making sim($\mathbf{\hat{y}}_i$) a more stable score to analyze.
For the first three variation factors, given a sim variant, we compute sim based on each of the variation settings, then obtain the mean and standard deviation sim across all settings.
Since the mean sim can differ greatly across sim variants, it is not effective to directly compare sim variants based on the sim mean and standard deviation.
Thus, we instead report the \textbf{C}oefficient of \textbf{V}ariation of \textbf{S}im (\textbf{CVS}), which is defined as the sim standard deviation divided by the sim mean.
Lower CVS is better, since that means there is little variation in sim across variation settings.
For the fourth variation factor, we compare the sim for $\mathcal{F}$'s correctly predicted test instances to the sim for $\mathcal{F}$'s incorrectly predicted test instances.
We do this by reporting ASD, with $a = \text{sim}_{y_i = \dot{y}_i}(\mathbf{\hat{y}}_i)$, and $b = \text{sim}_{y_i \neq \dot{y}_i}(\mathbf{\hat{y}}_i)$.

\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/sim_variants.png}
\caption{\textbf{Sim Variants.} We consider three baseline sim variants (\textsc{$\mathcal{F}$-Gold, \textsc{$\mathcal{GH}$-Gold}, \textsc{$\mathcal{GH}$-Pred}}) and our proposed \textsc{NP-$\mathcal{GH}$-Pred} sim variant (Sec. \ref{sec:method:sim_variants}). Sim variants denoted with ``$\mathcal{F}$'' use only $\mathcal{F}$ as the simulator. Sim variants denoted with ``$\mathcal{GH}$'' use both $\mathcal{G}$ and $\mathcal{H}$ as simulators. Sim variants denoted with ``Gold'' (yellow background) involve training simulators to predict the gold label. Sim variants denoted with ``Pred'' (blue background) involve training simulators to predict the task LM's predicted label. \textsc{NP-$\mathcal{GH}$-Pred} is the only sim variant that does not pretrain simulators (dotted lines), and we show that simulator pretraining can hinder sim metrics' reliability (Sec. \ref{sec:experiments})}
\label{fig:sim_variants}
\end{figure*}

\subsection{Simulatability Variants}
\label{sec:method:sim_variants}

In this work, we consider four sim variants.
The first three (\textsc{$\mathcal{F}$-Gold, \textsc{$\mathcal{GH}$-Gold}, \textsc{$\mathcal{GH}$-Pred}}) are baselines, while \textsc{NP-$\mathcal{GH}$-Pred} is our proposed sim variant.
These variants differ in how $\mathcal{G}$ and $\mathcal{H}$ are trained.
By default, we use T5-Base as the architecture for $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$.
We describe each sim variant below.

\paragraph{\textsc{$\mathcal{F}$-Gold}}
$\mathcal{F}$-Gold is a baseline sim variant introduced by us.
Recall that $\mathcal{F}$ is first pretrained on external corpora, then finetuned to predict $\mathbf{\dot{y}}_i$.
In \textsc{$\mathcal{F}$-Gold}, $\mathcal{G}$ is replaced by $\mathcal{F}$ with $\mathbf{x}_i$ as input, while $\mathcal{H}$ is replaced by $\mathcal{F}$ with both $\mathbf{x}_i$ and $\mathbf{r}_i$ as input.
As a result, $\mathcal{G}$'s accuracy in predicting $\mathbf{\hat{y}}_i$ will always be 100\%.
Unlike other sim variants, \textsc{$\mathcal{F}$-Gold} does not make the strong assumption that $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ have a sufficiently similiar reasoning process w.r.t. the given task.

\paragraph{\textsc{$\mathcal{GH}$-Gold}}
In \textsc{$\mathcal{GH}$-Gold}, $\mathcal{G}$ and $\mathcal{H}$ are first pretrained on external corpora, then finetuned to predict $\mathbf{\dot{y}}_i$.
\textsc{$\mathcal{GH}$-Gold} is the sim variant used in \citet{wiegreffe2021measuring}.

\paragraph{\textsc{$\mathcal{GH}$-Pred}}
In \textsc{$\mathcal{GH}$-Pred}, $\mathcal{G}$ and $\mathcal{H}$ are first pretrained on external corpora, then finetuned to predict $\mathbf{\hat{y}}_i$.
\textsc{$\mathcal{GH}$-Pred} is very similar in spirit to the sim variant used in \citet{hase2020leakage}.

\paragraph{\textsc{NP-$\mathcal{GH}$-Pred}}
\textsc{NP-$\mathcal{GH}$-Pred} is our proposed sim variant.
Unlike the other variants, the $\mathcal{G}$ and $\mathcal{H}$ are not pretrained on external corpora.
Here, the non-pretrained $\mathcal{G}$ and $\mathcal{H}$ are simply trained to predict $\mathbf{\hat{y}}_i$.
We hypothesize that this will enable us to better isolate the rationale's contributions from the pretraining knowledge obtained by pretrained versions of $\mathcal{G}$ and $\mathcal{H}$.
\section{Experiments}
\label{sec:experiments}

In this section, we present empirical results for \textsc{FRAME}\xspace.
First, we investigate \textsc{FRAME}\xspace Axioms 1-3 in the context of faithfulness evaluation, demonstrating the relative effectiveness of the \textsc{NP-$\mathcal{GH}$-Pred} sim variant in measuring rationale quality (Sec. \ref{sec:exp:faith}).
Second, we conduct a user study about Axiom 1 to show \textsc{FRAME}'s applicability to plausibility evaluation, while further establishing \textsc{NP-$\mathcal{GH}$-Pred}'s effectiveness (Sec. \ref{sec:exp:plaus}).
Third, we use \textsc{FRAME}\xspace to analyze the effects of label leakage on simulatability (Sec. \ref{sec:exp:leakage}).
For all results, we report the mean and standard deviation over three seeds.
In all tables, best performance is highlighted in \colorbox{lgreen}{green}, while second-best performance (if applicable) is highlighted in \colorbox{lblue}{blue}.

\paragraph{Normalized Relative Gain (NRG)}
For each axiom, we would like to summarize all metrics as a single aggregate metric, but this may not be straightforward if different metrics have different scales or different optimal values (\textit{e.g., } higher is better \textit{vs. } lower is better).
Thus, for each axiom, we summarize all metrics using the Normalized Relative Gain (NRG) metric \citep{chan2022unirex}, which normalizes all metrics' scores to values in $[0, 1]$ then outputs the final score by computing the mean of the normalized scores.

\subsection{Datasets}
We consider two types of $m$-class text classification tasks: closed-set and multi-choice.
For closed-set classification (\textit{e.g., } NLI), the same fixed $m$-class label space is used for all instances.
For multi-choice classification (\textit{e.g., } commonsense QA), each instance consists of a different set of $m$ choices.
Following prior works \citep{hase2020leakage, wiegreffe2021measuring}, we experiment with the e-SNLI \citep{camburu2018snli}, CoS-E v1.0 \citep{rajani2019explain, talmor2018commonsenseqa}, and CoS-E v1.1 \citep{rajani2019explain, talmor2018commonsenseqa} datasets.
e-SNLI is a natural language inference (NLI) dataset, while CoS-E v1.0 and CoS-E v1.1 are both commonsense question-answering (QA) datasets.

\subsection{Faithfulness Evaluation}
\label{sec:exp:faith}

\paragraph{Axiom 1: Reference Rationale Upper Bound}

In Table \ref{tab:axiom1}, we report results for Axiom 1.
In Table \ref{tab:axiom1}, we see that \textsc{NP-$\mathcal{GH}$-Pred} performs best overall, achieving the highest $\Phi(\mathbf{\hat{y}}_i)$, MOAR, and NRG on all datasets, besides having the second-highest MOAR on e-SNLI.
Meanwhile, none of the other metrics perform consistently well across datasets.
Based on NRG averaged across all datasets, \textsc{NP-$\mathcal{GH}$-Pred} beats the strongest per-dataset baseline sim variant by 41.7\%.
This shows that \textsc{NP-$\mathcal{GH}$-Pred} effectively satisfies our desired Axiom 1 property of having the reference rationale as a faithfulness upper bound.

\begin{table}[ht!]
\centering
\scalebox{0.6}{
\begin{tabular}{ccccc}
    \toprule
    \multirow{1}{*}{\textbf{Dataset}} & \multirow{1}{*}{\textbf{Sim Variant}} & $\Phi(\mathbf{\hat{y}}_i)$ ($\uparrow$) & MOAR ($\uparrow$) & NRG ($\uparrow$) \\
    \midrule
    \multirow{4}{*}{e-SNLI} & \textsc{$\mathcal{F}$-Gold} & -7.27~($\pm$0.57) & \cellcolor{lgreen}{1.45~($\pm$0.01)} & \cellcolor{lblue}{0.50} \\
    & \textsc{$\mathcal{GH}$-Gold} & -1.70~($\pm$8.25) & 1.01~($\pm$0.07) & 0.04 \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lblue}{9.10~($\pm$0.61)} & 1.01~($\pm$0.01)& 0.13 \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{54.77~($\pm$4.27)} & \cellcolor{lblue}{1.15~($\pm$0.01)} & \cellcolor{lgreen}{0.66} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.0} & \textsc{$\mathcal{F}$-Gold} & -34.39~($\pm$1.83) & 0.85~($\pm$0.01) & 0.00 \\
    & \textsc{$\mathcal{GH}$-Gold} & 17.93~($\pm$1.81) & \cellcolor{lblue}{1.33~($\pm$0.02)} & 0.65 \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lblue}{18.60~($\pm$0.86)} & \cellcolor{lblue}{1.33~($\pm$0.04)} & \cellcolor{lblue}{0.66} \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{23.83~($\pm$2.17)} & \cellcolor{lgreen}{2.04~($\pm$0.09)} & \cellcolor{lgreen}{1.00} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.11} & \textsc{$\mathcal{F}$-Gold} & -35.63~($\pm$1.63) & 0.83~($\pm$0.02) & 0.00 \\
    & \textsc{$\mathcal{GH}$-Gold} & 21.16~($\pm$2.36) & \cellcolor{lblue}{1.47~($\pm$0.06)} & 0.70 \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lblue}{22.17~($\pm$1.04)} & 1.46~($\pm$0.03) & \cellcolor{lblue}{0.71} \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{23.48~($\pm$0.60)} & \cellcolor{lgreen}{2.26~($\pm$0.09)} & \cellcolor{lgreen}{1.00} \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{Axiom 1.} We evaluate sim variants' ability to use reference rationale as a faithfulness upper bound. \textsc{NP-$\mathcal{GH}$-Pred} performs the best overall, while none of the sim variants are consistently good across all three datasets.
}
\label{tab:axiom1}
\end{table}

\paragraph{Axiom 2: Rationale Perturbation Sensitivity}

In Table \ref{tab:axiom2}, we report results for Axiom 2.
In Table \ref{tab:axiom1}, \textsc{NP-$\mathcal{GH}$-Pred} performs best on e-SNLI (w.r.t. Contrastive ASD and NRG) and competitively on CoS-E v1.0 and CoS-E v1.11.
However, \textsc{$\mathcal{GH}$-Gold} performs best overall, yielding the second-highest NRG on e-SNLI and the highest NRG by far on CoS-E v1.0 and CoS-E v1.11.
We find that the equivalent perturbation component of Axiom 2 is easier to satisfy, as all sim variants yield relatively low Equivalent ASD.
However, for contrastive perturbation, \textsc{$\mathcal{GH}$-Pred} and \textsc{$\mathcal{F}$-Gold} are not as strong.
\textsc{$\mathcal{GH}$-Pred} yields relatively low Contrastive ASD on e-SNLI and CoS-E v1.0, while \textsc{$\mathcal{F}$-Gold} yields relatively low Contrastive ASD on e-SNLI.
On the other hand, \textsc{$\mathcal{GH}$-Gold} and \textsc{NP-$\mathcal{GH}$-Pred} consistently satisfy our desired Axiom 2 property of appropriate sensitivity to semantic perturbation.

\begin{table}[ht!]
\centering
\scalebox{0.6}{
\begin{tabular}{ccccc}
    \toprule
    \multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Sim Variant}} & \multicolumn{1}{c}{\textbf{Equivalent}} & \multicolumn{1}{c}{\textbf{Contrastive}} & \textbf{All} \\
    \cmidrule(lr){3-3} \cmidrule(lr){4-4} \cmidrule(lr){5-5}
    & & ASD ($\downarrow$) & ASD ($\uparrow$) & NRG ($\uparrow$) \\
    \midrule
    \multirow{4}{*}{e-SNLI} & \textsc{$\mathcal{F}$-Gold} & 11.80~($\pm$0.60) & 12.50~($\pm$0.44) & 0.10 \\
    & \textsc{$\mathcal{GH}$-Gold} & 6.10~($\pm$7.96) & \cellcolor{lblue}{40.53~($\pm$7.03)} & \cellcolor{lblue}{0.74} \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 5.23~($\pm$1.02) & 0.50 \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lblue}{0.07~($\pm$0.12)} & \cellcolor{lgreen}{40.83~($\pm$2.63)} & \cellcolor{lgreen}{1.00} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.0} & \textsc{$\mathcal{F}$-Gold} & 5.61~($\pm$2.00) & \cellcolor{lblue}{26.91~($\pm$2.37)} & 0.09 \\
    & \textsc{$\mathcal{GH}$-Gold} & \cellcolor{lblue}{0.67~($\pm$0.40)} & \cellcolor{lgreen}{83.09~($\pm$13.16)} & \cellcolor{lgreen}{0.94} \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 15.09~($\pm$4.29)) & 0.50 \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & 0.74~($\pm$0.76) & 26.21~($\pm$2.79) & \cellcolor{lblue}{0.52} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.11} & \textsc{$\mathcal{F}$-Gold} & 4.04~($\pm$0.31) & \cellcolor{lblue}{26.13~($\pm$2.15)} & 0.04 \\
    & \textsc{$\mathcal{GH}$-Gold} & \cellcolor{lblue}{0.96~($\pm$1.65)} & \cellcolor{lgreen}{84.14~($\pm$21.59)} & \cellcolor{lgreen}{0.93} \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 21.57~($\pm$1.04)) & \cellcolor{lblue}{0.50} \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & 1.42~($\pm$0.48) & 25.55~($\pm$0.99) & 0.36 \\

    \bottomrule
\end{tabular}
}
\vspace{0.3cm}
\caption{\small \textbf{Axiom 2.} We evaluate sim variants' sensitivity to semantic perturbation of rationales. We find that \textsc{$\mathcal{GH}$-Gold} performs best overall, while \textsc{NP-$\mathcal{GH}$-Pred} also consistently performs well across all three datasets.
}
\label{tab:axiom2}
\end{table}

\paragraph{Axiom 3: Robustness to Variation in $\mathcal{F}$}

In Table \ref{tab:axiom3} and Fig. \ref{fig:pct_instances}-\ref{fig:perf_vs_sim}, we report results for Axiom 3.
In Table \ref{tab:axiom3}, we see that \textsc{NP-$\mathcal{GH}$-Pred} greatly outperforms other sim variants on all metrics, yielding a perfect NRG of 1.00 on all three datasets.
Meanwhile, none of the other metrics perform consistently well across datasets.
Similarly, in Fig. \ref{fig:pct_instances}-\ref{fig:perf_vs_sim}, we see that \textsc{NP-$\mathcal{GH}$-Pred} yields much flatter curves than other sim variants do, indicating its relative insensitivity to changes in $\mathcal{F}$'s number of train instances (Fig. \ref{fig:pct_instances}), number of noisy train instances (Fig. \ref{fig:pct_noisy}), and task performance (Fig. \ref{fig:perf_vs_sim}).
Based on NRG averaged across all datasets, \textsc{NP-$\mathcal{GH}$-Pred} beats the strongest per-dataset baseline sim variant by 42.9\%.
This shows that \textsc{NP-$\mathcal{GH}$-Pred} effectively satisfies our desired Axiom 3 property of robustness to variation in $\mathcal{F}$'s task performance.

\begin{table}[h]
\centering
\scalebox{0.44}{
\begin{tabular}{ccccccc}
    \toprule
    \multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Sim Variant}} & \multicolumn{1}{c}{\textbf{\% Train}} & \multicolumn{1}{c}{\textbf{\% Noisy Train}} & \multicolumn{1}{c}{\textbf{Capacity}} & \multicolumn{1}{c}{\textbf{Subpop.}} & \textbf{All} \\
    \cmidrule(lr){3-3} \cmidrule(lr){4-4} \cmidrule(lr){5-5} \cmidrule(lr){6-6} \cmidrule(lr){7-7}
    & & CVS ($\downarrow$) & CVS ($\downarrow$) & CVS ($\downarrow$) & ASD ($\downarrow$) & NRG ($\uparrow$) \\
    \midrule
    \multirow{4}{*}{e-SNLI} & \textsc{$\mathcal{F}$-Gold} & 0.27~($\pm$0.06) & 0.34~($\pm$0.13) & 0.81~($\pm$0.18) & \cellcolor{lblue}{6.81~($\pm$1.51)} & 0.27 \\
    & \textsc{$\mathcal{GH}$-Gold} & 0.14~($\pm$0.26) & 0.71~($\pm$0.26) & 0.65~($\pm$0.21) & 20.18~($\pm$4.20) & 0.26 \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lblue}{0.11~($\pm$0.02)} & \cellcolor{lblue}{0.21~($\pm$0.02)} & \cellcolor{lblue}{0.16~($\pm$0.04)} & 32.97~($\pm$3.36) & \cellcolor{lblue}{0.69} \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.02~($\pm$0.01)} & \cellcolor{lgreen}{0.03~($\pm$0.02)} & \cellcolor{lgreen}{0.04~($\pm$0.00)} & \cellcolor{lgreen}{1.85~($\pm$1.12)} & \cellcolor{lgreen}{1.00} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.0} & \textsc{$\mathcal{F}$-Gold} & \cellcolor{lblue}{0.13~($\pm$0.11)} & \cellcolor{lblue}{0.21~($\pm$0.12)} & 0.55~($\pm$0.05) & 23.41~($\pm$1.79) & \cellcolor{lblue}{0.71} \\
    & \textsc{$\mathcal{GH}$-Gold} & 0.25~($\pm$0.13) & 0.40~($\pm$0.22) & 0.43~($\pm$0.18) & 27.56~($\pm$0.87) & 0.15 \\
    & \textsc{$\mathcal{GH}$-Pred} & 0.32~($\pm$0.19) & 0.22~($\pm$0.04) & \cellcolor{lblue}{0.08~($\pm$0.04)} & \cellcolor{lblue}{23.20~($\pm$4.39)} & 0.30 \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.08~($\pm$0.02)} & \cellcolor{lgreen}{0.10~($\pm$0.00)} & \cellcolor{lgreen}{0.05~($\pm$0.02)} & \cellcolor{lgreen}{5.37~($\pm$4.51)} & \cellcolor{lgreen}{1.00} \\

    \midrule

    \multirow{4}{*}{CoS-E v1.11} & \textsc{$\mathcal{F}$-Gold} & \cellcolor{lblue}{0.14~($\pm$0.03)} & \cellcolor{lblue}{0.16~($\pm$0.07)} & 0.63~($\pm$0.08) & 25.48~($\pm$1.77) & 0.57 \\
    & \textsc{$\mathcal{GH}$-Gold} & 0.23~($\pm$0.05) & 0.32~($\pm$0.09) & 0.40~($\pm$0.05) & \cellcolor{lblue}{22.61~($\pm$1.87)} & 0.00 \\
    & \textsc{$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.05~($\pm$0.02)} & 0.22~($\pm$0.02) & \cellcolor{lblue}{0.32~($\pm$0.01)} & 25.47~($\pm$1.22) & \cellcolor{lblue}{0.70} \\

    & \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{0.05~($\pm$0.01)} & \cellcolor{lgreen}{0.07~($\pm$0.01)} & \cellcolor{lgreen}{0.09~($\pm$0.02)} & \cellcolor{lgreen}{6.28~($\pm$2.05)} & \cellcolor{lgreen}{1.00} \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{Axiom 3.} We evaluate sim variants' robustness to four types of variation in $\mathcal{F}$. Here, \textsc{NP-$\mathcal{GH}$-Pred} greatly outperforms other sim variants across all datasets.
}
\label{tab:axiom3}
\end{table}

\begin{figure*}
     \centering
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_instances_esnli.png}
         \caption{e-SNLI}
         \label{fig:pct_instances:esnli}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_instances_cose_v1-0.png}
         \caption{CoS-E v1.0}
         \label{fig:pct_instances:cose_v1-0}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_instances_cose_v1-11.png}
         \caption{CoS-E v1.11}
         \label{fig:pct_instances:cose_v1-11}
     \end{subfigure}
        \caption{\textbf{Axiom 3: Pct. Train Instances for $\mathcal{F}$ \textit{vs. } sim$(\mathbf{\hat{y}}_i)$.} Flatter curves indicate greater robustness to change in $\mathcal{F}$ task performance.}
        \label{fig:pct_instances}
\end{figure*}

\begin{figure*}
     \centering
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_noisy_esnli.png}
         \caption{e-SNLI}
         \label{fig:pct_noisy:esnli}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_noisy_cose_v1-0.png}
         \caption{CoS-E v1.0}
         \label{fig:pct_noisy:cose_v1-0}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/pct_noisy_cose_v1-11.png}
         \caption{CoS-E v1.11}
         \label{fig:pct_noisy:cose_v1-11}
     \end{subfigure}
        \caption{\textbf{Axiom 3: Pct. Noisy Train Instances for $\mathcal{F}$ \textit{vs. } sim$(\mathbf{\hat{y}}_i)$.} Flatter curves indicate greater robustness to change in $\mathcal{F}$ task performance.}
        \label{fig:pct_noisy}
\end{figure*}

\begin{figure*}
     \centering
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/perf_vs_faith_esnli.png}
         \caption{e-SNLI}
         \label{fig:perf_vs_sim:esnli}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/perf_vs_faith_cose_v1-0.png}
         \caption{CoS-E v1.0}
         \label{fig:perf_vs_sim:cose_v1-0}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.32\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/perf_vs_faith_cose_v1-11.png}
         \caption{CoS-E v1.11}
         \label{fig:perf_vs_sim:cose_v1-11}
     \end{subfigure}
        \caption{\textbf{Axiom 3: $\mathcal{F}$ Task Performance \textit{vs. } sim$(\mathbf{\hat{y}}_i)$.} Flatter curves indicate greater robustness to change in $\mathcal{F}$ task performance.}
        \label{fig:perf_vs_sim}
\end{figure*}

\subsection{Plausibility Evaluation}
\label{sec:exp:plaus}

Beyond faithfulness evaluation (Sec. \ref{sec:exp:faith}), we want to show that \textsc{FRAME}\xspace can also be used to evaluate rationale plausibility.
Whereas faithfulness evaluation can be fully automated due to its use of LM simulators, plausibility evaluation is much more expensive due to its reliance on human simulators.
In particular, Axioms 2-3 have more experiment settings than Axiom 1, hence requiring more human annotation effort.
Thus, as a proof of concept for plausibility evaluation, we conduct a user study based only on \textsc{FRAME}\xspace Axiom 1, two of the four sim variants, and the CoS-E v1.0 dataset.

The goal of our user study is to investigate if our previous conclusions for faithfulness evaluation w.r.t. Axiom 1 (Table \ref{tab:axiom1}) still hold for plausibility evaluation.
This means we need to reimplement LM-simulator-based sim variants with human simulators, which requires three assumptions.
First, we assume that our human simulators are already pretrained, since they all possess general language understanding abilities and world knowledge.
Second, we assume that our human simulators already know how to solve the given task (\textit{i.e., } already finetuned to predict gold label), using either $\mathbf{x}_i$ or $(\mathbf{x}_i, \mathbf{r}_i$) as input.
Third, we assume that our human simulators can learn patterns from very few training instances.
Given these assumptions, we compare the \textsc{$\mathcal{GH}$-Gold} and \textsc{NP-$\mathcal{GH}$-Pred} sim variants.
For \textsc{$\mathcal{GH}$-Gold}, given the simulators' assumed pretraining/finetuning, we simply compute $\Phi$ without further training.
However, for \textsc{NP-$\mathcal{GH}$-Pred}, we need to disable pretraining and train the simulator to predict $\mathbf{\hat{y}}_i$.
To simulate the disabling of pretraining, we encrypt every character in the dataset using a Caesar cipher \cite{savarese1999caesar} with right shift 1 (\textit{e.g., } \textit{hello} $\rightarrow$ \textit{ifmmp}).
Each encrypted instance still encodes the same information as before, but the encryption prevents the simulator from effectively leveraging their language priors, since none of the encrypted words will be familiar.
As a result, the simulator must rely on patterns learned while it is newly trained to predict $\mathbf{\hat{y}}_i$.

In our user study, we uniformly sample 50 test instances from CoS-E v1.0, then ask five human annotators to serve as simulators w.r.t. Axiom 1, for both \textsc{$\mathcal{GH}$-Gold} and \textsc{NP-$\mathcal{GH}$-Pred}.
Besides computing $\Phi(\mathbf{\hat{y}}_i)$ and MOAR, as done in Table \ref{tab:axiom1}, we also ask annotators to rate their confidence for each sim accuracy component ($\mathbbm{1}_{\mathcal{G}}(\mathbf{x})$, $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\hat{r}})$, $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\dot{r}})$, $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\dot{y}})$, $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\hat{y}})$).
Confidence is rated on a 4-point Likert scale: 1=``not confident'', 2=``somewhat confident'', 3=``confident'', 4=``very confident''.

Table \ref{tab:plaus} displays the results of the user study.
On $\Phi(\mathbf{\hat{y}}_i)$ and MOAR, we find that \textsc{NP-$\mathcal{GH}$-Pred} greatly outperforms \textsc{$\mathcal{GH}$-Gold}, corroborating the results in Table \ref{tab:axiom1}.
Specifically, across $\Phi(\mathbf{\hat{y}}_i)$ and MOAR, \textsc{NP-$\mathcal{GH}$-Pred} yields a 47.7\% average improvement over \textsc{$\mathcal{GH}$-Gold}.
Furthermore, we find that \textsc{NP-$\mathcal{GH}$-Pred} yields much higher confidence than \textsc{$\mathcal{GH}$-Gold} for $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\hat{y}})$, relative to the other four settings.
In other words, for \textsc{NP-$\mathcal{GH}$-Pred}, annotators not only yield higher sim scores for reference rationales, but also do so more confidently.
These results demonstrate \textsc{FRAME}'s applicability to plausibility evaluation as well as \textsc{NP-$\mathcal{GH}$-Pred}'s effectiveness as a plausibility metric.

\begin{table*}[ht!]
\centering
\scalebox{0.7}{
\begin{tabular}{cccccccccc}
    \toprule
    \multirow{2}{*}{\textbf{Sim Variant}} & \multicolumn{2}{c}{\textbf{Sim Metrics}} & \multicolumn{5}{c}{\textbf{Confidence}} \\
    \cmidrule(lr){2-3} \cmidrule(lr){4-8}

    & $\Phi(\mathbf{\hat{y}}_i)$ ($\uparrow$) & MOAR ($\uparrow$) & $\mathbbm{1}_{\mathcal{G}}(\mathbf{x})$ ($\downarrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\hat{r}})$ ($\downarrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\dot{r}})$ ($\downarrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\dot{y}})$ ($\downarrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}, \mathbf{\hat{y}})$ ($\uparrow$)  \\

    \midrule
    \textsc{$\mathcal{GH}$-Gold} & 51.60~($\pm$28.00) & 2.03~($\pm$0.53) & 2.94~($\pm$0.37) & 3.06~($\pm$0.47) & 3.47~($\pm$0.35) & 3.72~($\pm$0.36) & 3.41~($\pm$0.66) \\

    \textsc{NP-$\mathcal{GH}$-Pred} & \cellcolor{lgreen}{71.60~($\pm$7.65)} & \cellcolor{lgreen}{3.18~($\pm$0.60)} & \cellcolor{lgreen}{1.13~($\pm$0.25)} & \cellcolor{lgreen}{1.82~($\pm$0.66)} & \cellcolor{lgreen}{1.76~($\pm$0.73)} & \cellcolor{lgreen}{2.84~($\pm$1.16)} & \cellcolor{lgreen}{3.44~($\pm$0.86)} \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{Plausibility User Study.} We investigate \textsc{FRAME}'s applicability to plausibility evaluation (\textit{i.e., } using human simulators) w.r.t. Axiom 1. Our results mirror those for faithfulness evaluation, with \textsc{NP-$\mathcal{GH}$-Pred} consistently outperforming \textsc{$\mathcal{GH}$-Gold}.
}
\label{tab:plaus}
\end{table*}

\subsection{Leakage-Adjusted Simulatability}
\label{sec:exp:leakage}

Sometimes, the rationale $\mathbf{r}_i$ can leak the label $\mathbf{\hat{y}}_i$, which means using only $\mathbf{r}_i$ as input is sufficient for predicting $\mathbf{\hat{y}}_i$.
Thus, \citep{hase2020leakage} proposed a sim variant called leakage-adjusted simulatability (LAS), which reports sim as the mean sim across instances with label leakage and instances without label leakage.
Note that LAS' contribution is orthogonal to ours.
In \textsc{FRAME}\xspace Axiom 1, reference rationales are designed to fully leak $\mathbf{\hat{y}}_i$, in order to serve as an analysis tool.
Thus, we investigate LAS' impact on simulatability w.r.t. Axiom 1.

In Table \ref{tab:las_axiom1}, we compare \textsc{$\mathcal{GH}$-Gold} and \textsc{$\mathcal{GH}$-Gold} + LAS, which applies LAS to \textsc{$\mathcal{GH}$-Gold}, w.r.t. Axiom 1.
We find that \textsc{$\mathcal{GH}$-Gold} + LAS achieves better $\Phi(\mathbf{\hat{y}}_i)$ and MOAR on e-SNLI, while \textsc{$\mathcal{GH}$-Gold} achieves better $\Phi(\mathbf{\hat{y}}_i)$ and MOAR on CoS-E v1.0 and CoS-E v1.11.
For $\Phi(\mathbf{\hat{y}}_i)$, the difference is quite significant.
This shows that LAS' effect on Axiom 1 performance is dataset-dependent and that label leakage is not strongly correlated with reference rationale simulatability.

In Table \ref{tab:las_leakage}, we report $\Phi(\mathbf{\hat{y}}_i)$, $\mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$ ($\uparrow$), and $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{\hat{y}}_i)$ for \textsc{$\mathcal{GH}$-Gold} + LAS, decomposing the default `Mean'' result into ``Leak'' (instances with label leakage) and ``No Leak'' (instances without label leakage) subpopulations.
The goal of this experiment is to further explore label leakage's relationship with reference rationale simulatability and its constituent accuracy components.
As expected, we find that instances with label leakage generally yield significantly higher simulatability and accuracy.
This effect should be even greater when using reference rationales.

\begin{table}[ht!]
\centering
\scalebox{0.6}{
\begin{tabular}{ccccc}
    \toprule
    \multirow{1}{*}{\textbf{Dataset}} & \multirow{1}{*}{\textbf{Sim Variant}} & $\Phi(\mathbf{\hat{y}}_i)$ ($\uparrow$) & MOAR ($\uparrow$) \\
    \midrule
    \multirow{2}{*}{e-SNLI} & \textsc{$\mathcal{GH}$-Gold} & -1.70~($\pm$8.25) & 1.01~($\pm$0.07) \\
    & \textsc{$\mathcal{GH}$-Gold} + LAS & \cellcolor{lgreen}{13.46~($\pm$5.32)} & \cellcolor{lgreen}{1.02~($\pm$0.03)} \\

    \midrule

    \multirow{2}{*}{CoS-E v1.0} & \textsc{$\mathcal{GH}$-Gold} & \cellcolor{lgreen}{17.93~($\pm$1.81)} & \cellcolor{lgreen}{1.33~($\pm$0.02)} \\
    & \textsc{$\mathcal{GH}$-Gold} + LAS & -1.66~($\pm$7.76) & 1.18~($\pm$0.16) \\

    \midrule

    \multirow{2}{*}{CoS-E v1.11} & \textsc{$\mathcal{GH}$-Gold} & \cellcolor{lgreen}{21.16~($\pm$2.36)} & \cellcolor{lgreen}{1.47~($\pm$0.06)} \\
    & \textsc{$\mathcal{GH}$-Gold} + LAS & 3.76~($\pm$6.26) & 1.13~($\pm$0.18) \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{LAS: Axiom 1.}
}
\label{tab:las_axiom1}
\end{table}

\begin{table}[ht!]
\centering
\scalebox{0.57}{
\begin{tabular}{ccccc}
    \toprule
    \multirow{1}{*}{\textbf{Dataset}} & \multirow{1}{*}{\textbf{Leakage}} & $\Phi(\mathbf{\hat{y}}_i)$ ($\uparrow$) & $\mathbbm{1}_{\mathcal{G}}(\mathbf{\hat{y}}_i | \mathbf{x}_i)$ ($\uparrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{\hat{y}}_i | \mathbf{x}_i, \mathbf{\hat{y}}_i)$ ($\uparrow$) \\
    \midrule
    \multirow{3}{*}{e-SNLI} & Leak & 12.84~($\pm$5.34) & \cellcolor{lgreen}{77.16~($\pm$9.30)} & \cellcolor{lgreen}{90.00~($\pm$11.29)} \\
    & No Leak & \cellcolor{lgreen}{14.07~($\pm$5.51)} & 71.40~($\pm$2.44) & 85.47~($\pm$7.57) \\
    \cmidrule(lr){2-5}
    & Mean & 13.46~($\pm$5.32) & 74.28~($\pm$4.62) & 87.74~($\pm$8.78) \\

    \midrule

    \multirow{3}{*}{CoS-E v1.0} & Leak & \cellcolor{lgreen}{26.76~($\pm$2.19)} & \cellcolor{lgreen}{73.24~($\pm$2.19)} & \cellcolor{lgreen}{100.00~($\pm$0.00)} \\
    & No Leak & -30.08~($\pm$13.36) & 30.08~($\pm$13.36) & 0.00~($\pm$0.00) \\
    \cmidrule(lr){2-5}
    & Mean & -1.66~($\pm$7.76) & 51.66~($\pm$7.76) & 50.00~($\pm$0.00) \\

    \midrule

    \multirow{3}{*}{CoS-E v1.11} & Leak & \cellcolor{lgreen}{33.54~($\pm$1.16)} & \cellcolor{lgreen}{66.46~($\pm$1.16)} & \cellcolor{lgreen}{100.00~($\pm$0.00)} \\
    & No Leak & -26.01~($\pm$13.51) & 26.01~($\pm$13.51) & 0.00~($\pm$0.00) \\
    \cmidrule(lr){2-5}
    & Mean & 3.76~($\pm$6.26) & 46.24~($\pm$6.26) & 50.00~($\pm$0.00) \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{LAS: Leak \textit{vs. } No Leak.}
}
\label{tab:las_leakage}
\end{table}

\section{Related Work}
\label{sec:related_work}

\subsection{Rationale Generation}
Free-text rationales are designed to explain LMs' decisions more flexibly via natural language \citep{narang2020wt5, lakhotia2020fid, kumar2020nile, rajani2019explain, camburu2018snli, park2018multimodal, hendricks2016generating}.
Compared to extractive rationales, free-text rationales are designed to be more flexible and intuitive \citep{narang2020wt5, kumar2020nile, camburu2018snli} as well as more applicable when the LM's reasoning process requires latent knowledge (\textit{i.e., } encoded in the LM's parameters) beyond the task input --- \textit{e.g., } in commonsense reasoning tasks \citep{rajani2019explain, wiegreffe2021measuring}.
Although a number of works generate the task and explanation outputs separately (\textit{i.e., } as a pipeline of separate models) \citep{camburu2018snli, kumar2020nile}, most works follow the self-rationalizing LM paradigm, which generates the task and explanation outputs jointly \citep{narang2020wt5, lakhotia2020fid, tang2020exploring, do2020snli, rajani2019explain, wu2018faithful, liu2018towards, ehsan2018rationalization, camburu2018snli, hancock2018training, kim2018textual, hendricks2016generating}.

\subsection{Rationale Evaluation}

\subsubsection{Faithfulness Evaluation}
For both extractive rationales \citep{deyoung2019eraser, shrikumar2017learning, sanyal2021discretized, hase2020evaluating} and free-text rationales \citep{hase2020leakage, wiegreffe2021measuring}, existing works have generally defined rationale faithfulness as the association between the rationale and the LM task output being explained by rationale.
Importantly, rationale faithfulness cannot be measured based on human-annotated gold rationales, which would entangle plausibility \citep{jacovi2020towards, jacovi2021aligning}.

\paragraph{Extractive Rationales}
Given that extractive rationales are more common than free-text rationales \citep{denil2014extraction, sundararajan2017axiomatic, jain2020learning, chan2022unirex, lundberg2017unified, li2016understanding, luo2021local}, most existing rationale faithfulness metrics are designed for extractive rationales.
Some metrics are heuristic functions and do not require any model training \citep{deyoung2019eraser, shrikumar2017learning, sanyal2021discretized, hase2020evaluating}.
In particular, \citet{deyoung2019eraser} proposed the popular comprehensiveness and sufficiency metrics, as part of the ERASER benchmark.
Also, a number of metrics are computed by retraining the model by using the rationale to modify the input \citep{hooker2018benchmark} or as supervision signal \citep{pruthi2020evaluating}.
For example, \citet{pruthi2020evaluating} measures the faithfulness of a teacher LM's rationales as how well a student LM can predict the teacher's outputs when regularized using rationales explaining the teacher's outputs.

Our proposed \textsc{NP-$\mathcal{GH}$-Pred} sim variant is similar to \citet{pruthi2020evaluating} in that a student LM is used to predict a teacher LM's outputs, although we focus on free-text rationales and simply use the rationale as additional input context when training the student.
Furthermore, recall that \textsc{FRAME}\xspace is based on the axiom that good metrics should score reference (\textit{i.e., } faithful-by-construction) rationales highly.
However, we do not apply this axiom to evaluating extractive rationale faithfulness, since it is not straightforward to construct extractive reference rationales.
Meanwhile, this is more feasible for free-text rationales, since we can derive the reference rationale from the task LM's output token sequence.

\paragraph{Free-Text Rationales}
Meanwhile, relatively few works focus on measuring the faithfulness of free-text rationales \citep{hase2020leakage, wiegreffe2021measuring}.
All of these works are based on the concept of simulatability \cite{doshi2017towards}, which measures a rationale's faithfulness as how predictive the rationale is of the task LM's output.
Note that simulatability is just one aspect of faithfulness and should not be seen as a comprehensive evaluation of whether a rationale can fully explain the task LM's reasoning process.
Simulatability can be automatically computed as how much the rationale helps a simulator LM predict the task LM's output \citep{hase2020evaluating, hase2020leakage}.
\citet{hase2020leakage} proposes LAS, a simulatability-based metric that aims to control for label leakage.
\citet{wiegreffe2021measuring} proposes two simulatability-based metrics: RE aims to measure the label-rationale association w.r.t. similarity in robustness to various noise levels, while FIA aims to measure label-rationale association w.r.t. similarity in agreement with feature importance scores.

Crucially, all of these simulatability-based metrics are built upon a implementation of simulatability where the simulator LM is pretrained on external corpora, which makes it difficult to isolate the rationale's contribution from the simulator LM's pretraining knowledge.
Using \textsc{FRAME}, we show that this causes existing metrics to be produce a muddled faithfulness signal and lack robustness to irrelevant changes in how the task LM generates its predicted label.
Plus, we propose a new \textsc{NP-$\mathcal{GH}$-Pred} sim variant where the simulator LM is not pretrained, then show that \textsc{NP-$\mathcal{GH}$-Pred} performs better on our \textsc{FRAME}\xspace meta-metrics.
Note that the contributions of our proposed simulatability implementations are orthogonal to those in LAS, RE, and FIA, our new simulatability implementations can be applied to all of these metrics.

\subsection{Plausibility Evaluation}
In the explainability literature, there exist two general approaches for evaluating rationale plausibility.
The first approach is by using gold rationale similarity \cite{deyoung2019eraser, chan2022unirex, narang2020wt5}.
Given a gold rationale (annotated w.r.t. gold label), a rationale's plausibility is measured as its similarity to the gold rationale.
However, evaluating plausibility via gold rationale similarity is noisy because gold rationales are defined w.r.t. the gold label, not the task LM's predicted label \cite{chan2022unirex}.

The second approach is by asking human annotators to rate whether the rationale makes sense as an explanation for the task LM's predicted label \citep{doshi2017towards, hase2020evaluating, strout2019human}.
Within this category, we focus on simulatability \citep{doshi2017towards, hase2020evaluating}, which defines plausibility as how much the rationale helps a human simulator predict the task LM's predicted label.

\section{Limitations}
\label{sec:limitations}

\section{Appendix}
\label{sec:appendix}

\subsection{\textsc{FRAME}\xspace Implementation}
\label{sec:app:method:implementation}

\subsubsection{Axiom 2: Rationale Perturbation Sensitivity}
\label{sec:app:method:implementation}

Here, we describe how equivalent and contrastive perturbations can be applied to a given rationale $\mathbf{r}_i$.

\paragraph{Equivalent Perturbation}
Let $\Psi_e$ be an equivalent perturbation function which modifies $\mathbf{r}_i$'s surface form without changing $\mathbf{r}_i$'s semantic content.
That is, $\mathbf{r}_i$ and $\Psi_e(\mathbf{r}_i)$ have different surface forms but share the same meaning.
In this paper, we consider the closed-set (e-SNLI) and multi-choice (CoS-E v1.0, CoS-E v1.11) text classification tasks.
Each task requires a different form of $\Psi_e$.

For closed-set classification, $\mathbf{r}_i$ is generally a single-word label (\textit{e.g., } ``entailment''), hence yielding a single-word reference rationale.
Such single-word rationales are not representative of free-text rationales, which are motivated as being accessible natural language sentences.
Here, we design $\Psi_e$ to construct $\Psi_e(\mathbf{r}_i)$ by paraphrasing $\mathbf{r}_i$ as a natural language sentence.
For each class $c_j$ in the $m$-class label space $C = \{c_j\}_{j=1}^{m}$, we manually write a set of $k$ paraphrase sentences $S_j$, based only on $c_j$ (\textit{i.e., } not on $\mathbf{x}_i$).
\aaron{Refer to appendix for list of actual paraphrase sentences.}
Given task instance $i$ and $\mathcal{F}$'s predicted label $y_i$, suppose $y_i = c_j$ for some $j$.
After setting $\mathbf{r}_i = y_i$, we obtain $\Psi_e(\mathbf{r}_i)$ by randomly sampling a paraphrase sentence $s$ from $S_j$ then setting $\Psi_e(\mathbf{r}_i) = s$.
For example, if $\mathbf{r}_i = $ ``\texttt{contradiction}'', and $s = $ ``\texttt{The hypothesis conflicts with the premise.}'', then the resulting $\Psi_e(\mathbf{r}_i)$ would be ``\texttt{The hypothesis conflicts with the premise.}''

For multi-choice classification, $\mathbf{r}_i$ is generally a multi-word choice (\textit{e.g., } ``have fun in the park''), hence yielding a multi-word reference rationale $\mathbf{r}_i$.
Since each instance has a different set of choices, it is not feasible to manually write paraphrase sentences specifically for every choice across all instances in the dataset.
Instead, we design $\Psi_e$ to perturb $\mathbf{r}_i$ by inserting a generic supporting sentence in front of $\mathbf{r}_i$.
First, we manually write a set, $S$, of $k$ sentences, each generically expressing affirmation for a subsequent rationale.
Note that these sentences are written independently of any particular task input, label, or rationale.
Then, for each instance $i$, we randomly sample one sentence $s$ from $S$ to insert before $\mathbf{r}_i$.
For example, if $\mathbf{r}_i = $ ``\texttt{I am a rationale.}'', and $s = $ ``\texttt{The following rationale is faithful: }'', then the resulting $\Psi_e(\mathbf{r}_i)$ would be ``\texttt{The following rationale is faithful: I am a rationale.}''

For both tasks, we can see that $\Psi_e(\mathbf{r}_i)$ is constructed in a way that does not change the meaning of $\mathbf{r}_i$.
Therefore, in both cases, a good $\Phi$ should yield a low value of $|\Phi(\mathbf{r}_i) - \Phi(\Psi_e(\mathbf{r}_i))|$.

\paragraph{Contrastive Perturbation}
Let $\Psi_c$ be an contrastive perturbation function which modifies $\mathbf{r}_i$'s surface form while also significantly changing $\mathbf{r}_i$'s semantic content.
That is, $\mathbf{r}_i$ and $\Psi_e(\mathbf{r}_i)$ should have dissimilar surface forms and meaning.

For both closed-set and multi-choice classification, we can construct $\Psi_c(\mathbf{r}_i)$ in the same way.
For each instance $i$, $y_i$ is one of $m$ classes, so we obtain $\Psi_c(\mathbf{r}_i)$ by randomly sampling one of the other $m-1$ classes to replace $\mathbf{r}_i$ with.
We can see that $\Psi_c(\mathbf{r}_i)$ is constructed in a way that fundamentally changes the meaning of $\mathbf{r}_i$.
Therefore, a good $\Phi$ should yield a high value of $|\Phi(\mathbf{r}_i) - \Phi(\Psi_c(\mathbf{r}_i))|$.

\subsection{Related Work}

\subsubsection{Rationale Generation}
Free-text rationales are designed to explain LMs' decisions more flexibly via natural language \citep{narang2020wt5, lakhotia2020fid, kumar2020nile, rajani2019explain, camburu2018snli, park2018multimodal, hendricks2016generating}.
Compared to extractive rationales, free-text rationales are designed to be more understandable to humans \citep{narang2020wt5, kumar2020nile, camburu2018snli} and more applicable when the LM's reasoning process requires latent knowledge (\textit{i.e., } encoded in the LM's parameters) beyond the task input --- \textit{e.g., } in commonsense reasoning tasks \citep{rajani2019explain, wiegreffe2021measuring}.
Although a number of works generate the task and explanation outputs separately (\textit{i.e., } as a pipeline of separate models) \citep{camburu2018snli, kumar2020nile}, most works follow the self-rationalizing LM paradigm, which generates the task and explanation outputs jointly \citep{narang2020wt5, lakhotia2020fid, tang2020exploring, do2020snli, rajani2019explain, wu2018faithful, liu2018towards, ehsan2018rationalization, camburu2018snli, hancock2018training, kim2018textual, hendricks2016generating}.

Self-rationalizing LMs are popular due to their high task performance, parameter efficiency, and rationale plausibility \citep{wiegreffe2021measuring}.
In particular, text-to-text transformers (\textit{e.g., } T5 \citep{raffel2019exploring}), which use a vanilla encoder-decoder transformer \citep{vaswani2017attention} to output the predicted label and rationale as a single token sequence, have shown promise as self-rationalizing LMs \citep{narang2020wt5, lakhotia2020fid}.
Prior works on free-text rationale generation tend to focus on producing plausible rationales and are often supervised with human-annotated gold rationales \citep{narang2020wt5, lakhotia2020fid, camburu2018snli, rajani2019explain}.
Meanwhile, few works on free-text rationale generation are optimized or specially designed for faithfulness \citep{hase2020leakage, kumar2020nile, wu2018faithful}.
Following recent works on evaluating free-text rationales \citep{hase2020leakage, wiegreffe2021measuring}, we focus on T5-based self-rationalizing LMs in this paper.

\subsection{Additional Empirical Results}
Tables \ref{tab:axiom1_full}-\ref{tab:axiom3_full} show additional results for Axioms 1-3.

\begin{table*}[h]
\centering
\scalebox{0.58}{
\begin{tabular}{ccccccc}
    \toprule
    \multirow{1}{*}{\textbf{Dataset}} & \multirow{1}{*}{\textbf{Sim Variant}} & $\text{sim}(\mathbf{y}_i)$ ($\uparrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{y}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{r}_i)$ ($\uparrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{y}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \dot{\mathbf{r}}_i)$ ($\uparrow$) & $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{y}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \dot{\mathbf{y}}_i)$ ($\uparrow$) & Mean $\mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \mathbf{y}_i) / \mathbbm{1}_{\mathcal{H}}(\mathbf{x}_i, \cdot\hspace{0.5mm})$ ($\uparrow$) \\
    \midrule
    \multirow{3}{*}{e-SNLI} & Vanilla & -1.70~($\pm$8.25) & 0.91~($\pm$0.08) & 1.03~($\pm$0.09) & 1.08~($\pm$0.03) & 1.01~($\pm$0.07)\\
    & Pretrain+Retrain & \cellcolor{lblue}{9.10~($\pm$0.61)} & 1.01~($\pm$0.01) & 1.13~($\pm$0.01) & \cellcolor{lblue}{1.10~($\pm$0.01)} & 1.08~($\pm$0.01) \\
    & $\mathcal{F}$-Sim & -7.27~($\pm$0.57) & \cellcolor{lgreen}{1.45~($\pm$0.00)} & \cellcolor{lgreen}{1.85~($\pm$0.03)} & 1.04~($\pm$0.01) & \cellcolor{lgreen}{1.45~($\pm$0.01)} \\

    & Retrain & \cellcolor{lgreen}{54.77~($\pm$4.27)} & \cellcolor{lblue}{1.03~($\pm$0.01)} & \cellcolor{lblue}{1.25~($\pm$0.01)} & \cellcolor{lgreen}{1.18~($\pm$0.01)} & \cellcolor{lblue}{1.15~($\pm$0.01)} \\

    \midrule

    \multirow{3}{*}{CoS-E v1.0} & Vanilla & 17.93~($\pm$1.81) & \cellcolor{lblue}{1.10~($\pm$0.01)} & \cellcolor{lblue}{1.45~($\pm$0.01)} & \cellcolor{lblue}{1.46~($\pm$0.03)} & \cellcolor{lblue}{1.33~($\pm$0.02)} \\
    & Pretrain+Retrain & \cellcolor{lblue}{18.60~($\pm$0.86)} & \cellcolor{lblue}{1.10~($\pm$0.01)} & 1.43~($\pm$0.05) & 1.45~($\pm$0.05) & \cellcolor{lblue}{1.33~($\pm$0.04)} \\
    & $\mathcal{F}$-Sim & -34.39~($\pm$1.83) & 0.87~($\pm$0.01) & 0.85~($\pm$0.01) & 0.84~($\pm$0.02) & 0.85~($\pm$0.01) \\

    & Retrain & \cellcolor{lgreen}{23.83~($\pm$2.17)} & \cellcolor{lgreen}{1.75~($\pm$0.05)} & \cellcolor{lgreen}{2.85~($\pm$0.18)} & \cellcolor{lgreen}{1.52~($\pm$0.03)} & \cellcolor{lgreen}{2.04~($\pm$0.09)} \\

    \midrule

    \multirow{3}{*}{CoS-E v1.11} & Vanilla & 21.16~($\pm$2.36) & 1.09~($\pm$0.03) & \cellcolor{lblue}{1.62~($\pm$0.06)} & \cellcolor{lgreen}{1.70~($\pm$0.09)} & \cellcolor{lblue}{1.47~($\pm$0.06)} \\
    & Pretrain+Retrain & \cellcolor{lblue}{22.17~($\pm$1.04)} & \cellcolor{lblue}{1.10~($\pm$0.02)} & 1.59~($\pm$0.03) & \cellcolor{lblue}{1.69~($\pm$0.03)} & 1.46~($\pm$0.03) \\
    & $\mathcal{F}$-Sim & -35.63~($\pm$1.63) & 0.84~($\pm$0.02) & 0.83~($\pm$0.03) & 0.80~($\pm$0.01) & 0.83~($\pm$0.02) \\

    & Retrain & \cellcolor{lgreen}{23.48~($\pm$0.60)} & \cellcolor{lgreen}{1.86~($\pm$0.11)} & \cellcolor{lgreen}{3.28~($\pm$0.11)} & 1.64~($\pm$0.05) & \cellcolor{lgreen}{2.26~($\pm$0.09)} \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{Axiom 1 Results.} Reference rationale as simulatability upper bound.
}
\label{tab:axiom1_full}
\end{table*}

\begin{table}[t]
\centering
\scalebox{0.55}{
\begin{tabular}{cccc}
    \toprule
    \multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Sim Variant}} & \multicolumn{1}{c}{\textbf{Equivalent}} & \multicolumn{1}{c}{\textbf{Contrastive}} \\
    \cmidrule(lr){3-3} \cmidrule(lr){4-4}
    & & $| \text{sim}(\mathbf{y}_i) - \text{sim}(\mathbf{y}'_i) |$ ($\downarrow$) & $| \text{sim}(\mathbf{y}_i) - \text{sim}(\mathbf{y}'_i) |$ ($\uparrow$) \\
    \midrule
    \multirow{3}{*}{e-SNLI} & Vanilla & 6.10~($\pm$7.96) & \cellcolor{lblue}{40.53~($\pm$7.03)} \\
    & Pretrain+Retrain & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 5.23~($\pm$1.02) \\
    & $\mathcal{F}$-Sim & 11.80~($\pm$0.60) & 12.50~($\pm$0.44) \\

    & Retrain & \cellcolor{lblue}{0.07~($\pm$0.12)} & \cellcolor{lgreen}{40.83~($\pm$2.63)} \\

    \midrule

    \multirow{3}{*}{CoS-E v1.0} & Vanilla & \cellcolor{lblue}{0.67~($\pm$0.40)} & \cellcolor{lgreen}{83.09~($\pm$13.16)} \\
    & Pretrain+Retrain & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 15.09~($\pm$4.29)) \\
    & $\mathcal{F}$-Sim & 5.61~($\pm$2.00) & \cellcolor{lblue}{26.91~($\pm$2.37)} \\

    & Retrain & 0.74~($\pm$0.76) & 26.21~($\pm$2.79) \\

    \midrule

    \multirow{3}{*}{CoS-E v1.11} & Vanilla & \cellcolor{lblue}{0.96~($\pm$1.65)} & \cellcolor{lgreen}{84.14~($\pm$21.59)} \\
    & Pretrain+Retrain & \cellcolor{lgreen}{0.00~($\pm$0.00)} & 21.57~($\pm$1.04)) \\
    & $\mathcal{F}$-Sim & 4.04~($\pm$0.31) & \cellcolor{lblue}{26.13~($\pm$2.15)} \\

    & Retrain & 1.42~($\pm$0.48) & 25.55~($\pm$0.99) \\

    \bottomrule
\end{tabular}
}
\vspace{0.3cm}
\caption{\small \textbf{Axiom 2 Results.} Sensitivity to semantic perturbation of rationales.
}
\label{tab:axiom2_full}
\end{table}

\begin{table*}[h]
\centering
\scalebox{0.6}{
\begin{tabular}{cccccccc}
    \toprule
    \multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Sim Variant}} & \multicolumn{1}{c}{\textbf{\% Train Instances}} & \multicolumn{1}{c}{\textbf{\% Noisy Train Instances}} & \multicolumn{1}{c}{\textbf{Model Size}} & \multicolumn{3}{c}{\textbf{Subpopulation}} \\
    \cmidrule(lr){3-3} \cmidrule(lr){4-4} \cmidrule(lr){5-5} \cmidrule(lr){6-8}
    & & $\text{CV}(\text{sim}(\mathbf{y}_i))$ ($\downarrow$) & $\text{CV}(\text{sim}(\mathbf{y}_i))$ ($\downarrow$) & $\text{CV}(\text{sim}(\mathbf{y}_i))$ ($\downarrow$) & $\text{sim}_{y_i = \dot{y}_i}(\mathbf{y}_i)$ ($\uparrow$) & $\text{sim}_{y_i \neq \dot{y}_i}(\mathbf{y}_i)$ ($\uparrow$) & $| \text{sim}_{y_i = \dot{y}_i}(\mathbf{y}_i) - \text{sim}_{y_i \neq \dot{y}_i}(\mathbf{y}_i) |$ ($\downarrow$) \\
    \midrule
    \multirow{3}{*}{e-SNLI} & Vanilla & 0.14~($\pm$0.26) & 0.71~($\pm$0.26) & 0.65~($\pm$0.21) & -4.79~($\pm$7.86) & 15.39~($\pm$11.22) & 20.18~($\pm$4.20) \\
    & Pretrain+Retrain & \cellcolor{lblue}{0.11~($\pm$0.02)} & \cellcolor{lblue}{0.21~($\pm$0.02)} & \cellcolor{lblue}{0.16~($\pm$0.04)} & \cellcolor{lblue}{4.05~($\pm$0.22)} & \cellcolor{lblue}{37.03~($\pm$3.58)} & 32.97~($\pm$3.36) \\
    & $\mathcal{F}$-Sim & 0.27~($\pm$0.06) & 0.34~($\pm$0.13) & 0.81~($\pm$0.18) & -6.22~($\pm$0.76) & -13.04~($\pm$0.84) & \cellcolor{lblue}{6.81~($\pm$1.51)} \\

    & Retrain & \cellcolor{lgreen}{0.02~($\pm$0.01)} & \cellcolor{lgreen}{0.03~($\pm$0.02)} & \cellcolor{lgreen}{0.04~($\pm$0.00)} & \cellcolor{lgreen}{54.74~($\pm$4.31)} & \cellcolor{lgreen}{54.88~($\pm$4.73)} & \cellcolor{lgreen}{1.85~($\pm$1.12)} \\

    \midrule

    \multirow{3}{*}{CoS-E v1.0} & Vanilla & 0.25~($\pm$0.13) & 0.40~($\pm$0.22) & 0.43~($\pm$0.18) & 8.81~($\pm$1.83) & \cellcolor{lblue}{36.37~($\pm$1.76)} & 27.56~($\pm$0.87) \\
    & Pretrain+Retrain & 0.32~($\pm$0.19) & 0.22~($\pm$0.04) & \cellcolor{lblue}{0.08~($\pm$0.04)} & \cellcolor{lblue}{14.78~($\pm$9.09)} & \cellcolor{lgreen}{37.97~($\pm$4.82)} & \cellcolor{lblue}{23.20~($\pm$4.39)} \\
    & $\mathcal{F}$-Sim & \cellcolor{lblue}{0.13~($\pm$0.11)} & \cellcolor{lblue}{0.21~($\pm$0.12)} & 0.55~($\pm$0.05) & -26.64~($\pm$1.47) & -50.05~($\pm$2.81) & 23.41~($\pm$1.79) \\

    & Retrain & \cellcolor{lgreen}{0.08~($\pm$0.02)} & \cellcolor{lgreen}{0.10~($\pm$0.00)} & \cellcolor{lgreen}{0.05~($\pm$0.02)} & \cellcolor{lgreen}{21.29~($\pm$2.37)} & 26.29~($\pm$2.72) & \cellcolor{lgreen}{5.37~($\pm$4.51)} \\

    \midrule

    \multirow{3}{*}{CoS-E v1.11} & Vanilla & 0.23~($\pm$0.05) & 0.32~($\pm$0.09) & 0.40~($\pm$0.05) & \cellcolor{lblue}{11.34~($\pm$1.86)} & \cellcolor{lblue}{33.94~($\pm$3.26)} & \cellcolor{lblue}{22.61~($\pm$1.87)} \\
    & Pretrain+Retrain & \cellcolor{lgreen}{0.05~($\pm$0.02)} & 0.22~($\pm$0.02) & \cellcolor{lblue}{0.32~($\pm$0.01)} & 11.10~($\pm$0.68) & \cellcolor{lgreen}{36.57~($\pm$1.51)} & 25.47~($\pm$1.22) \\
    & $\mathcal{F}$-Sim & \cellcolor{lblue}{0.14~($\pm$0.03)} & \cellcolor{lblue}{0.16~($\pm$0.07)} & 0.63~($\pm$0.08) & -24.55~($\pm$2.48) & -50.04~($\pm$0.86) & 25.48~($\pm$1.77) \\

    & Retrain & \cellcolor{lgreen}{0.05~($\pm$0.01)} & \cellcolor{lgreen}{0.07~($\pm$0.01)} & \cellcolor{lgreen}{0.09~($\pm$0.02)} & \cellcolor{lgreen}{20.76~($\pm$0.53)} & 27.03~($\pm$1.68) & \cellcolor{lgreen}{6.28~($\pm$2.05)} \\

    \bottomrule
\end{tabular}
}
\caption{\small \textbf{Axiom 3 Results.} Robustness to variation in $\mathcal{F}$.
}
\label{tab:axiom3_full}
\end{table*}

redpajama_set_name: RedPajamaArXiv