# EVIL: Evidential Inference Learning for Trustworthy Semi-supervised Medical
Image Segmentation

###### Abstract

Recently, uncertainty-aware methods have attracted increasing attention in
semi-supervised medical image segmentation. However, current methods usually
suffer from the drawback that it is difficult to balance the computational
cost, estimation accuracy, and theoretical support in a unified framework. To
alleviate this problem, we introduce the Dempster–Shafer Theory of Evidence
(DST) into semi-supervised medical image segmentation, dubbed EVidential
Inference Learning (EVIL). EVIL provides a theoretically guaranteed solution
to infer accurate uncertainty quantification in a single forward pass.
Trustworthy pseudo labels on unlabeled data are generated after uncertainty
estimation. The recently proposed consistency regularization-based training
paradigm is adopted in our framework, which enforces the consistency on the
perturbed predictions to enhance the generalization with few labeled data.
Experimental results show that EVIL achieves competitive performance in
comparison with several state-of-the-art methods on the public dataset.

Index Terms—  Medical Image Segmentation, Semi-Supervised Learning, Evidential
Learning

##  1 Introduction

Medical image segmentation plays an essential role in subsequent clinical or
computer-aided diagnosis and fully-supervised learning has achieved great
success in the field of automatic image segmentation [1]. However, annotating
medical images is laborious and requires rich professional knowledge [2].

Semi-supervised learning (SSL) has shown great potential to alleviate this
problem by leveraging a large set of unlabeled data accompanied with a limited
number of labeled data. These methods can be roughly categorized into two
types: (1) pseudo-label retraining, which incorporates pseudo labels on
unlabeled data for retraining [3, 4, 5]; and (2) consistency regularization,
which enforces the prediction consistency to enhance generalization with
various perturbations, such as input perturbation, feature perturbation, and
network perturbation [6, 7, 8].

However, since these methods rely heavily on the prediction of pseudo label,
false predictions will severely degrade the segmentation performance. To
improve the quality of pseudo labels, some uncertainty-aware methods have been
proposed, including Monte Carlo dropout (MC-dropout)-based [9], Information-
Entropy-based [10], and Prediction Variance-based [11] methods. However, these
methods suffer from some problems: (1) Although MC-dropout is mathematically
guaranteed by Bayesian theory, its training process is costly due to the
multiple sampling operations; (2) Due to the limited sampling times, MC-
dropout can’t obtain accurate uncertainty quantification; (3) Other two
uncertainty estimation methods have advantages in computational cost, but they
lack theoretical support, leading to unstable pseudo label generation.

To handle the above issues, we introduce the Dempster–Shafer Theory of
Evidence (DST) into semi-supervised medical image segmentation, providing a
theoretically guaranteed single-pass solution for uncertainty quantification
inference, dubbed EVidential Inference Learning (EVIL). Following the training
paradigm proposed in [7], EVIL belongs to the consistency regularization
method with network perturbation, which imposes the prediction consistency on
two networks perturbed with different initialization. In particular, the two
networks play different roles. One is a vanilla segmentation network (S-Net)
which directly generates the segmentation result. The other network called
evidential-network (E-Net) is built from the perspective of DST, which is
theoretically guaranteed for reliable predictions. Different from S-Net, the
output of E-Net is regarded as the evidence and parameterized into a Dirichlet
distribution on segmentation probabilities. Subjective Logic theory (SL) [12]
is employed to quantify the predictions and uncertainties of different
categories with the Dirichlet distribution in a single inference, which
significantly reduces the training time. Then, the trustworthy pseudo labels
on unlabeled data are generated. In summary, there are three merits for our
proposed EVIL: lower computation cost due to the single-pass operation,
accurate uncertainty estimation based on SL and theoretical guarantee based on
DST.

The main contributions of this work are summarized as: 1) we introduce DST
into SSL and provide a fast accurate uncertainty estimation with theoretical
guarantee in a unified framework; 2) a novel network perturbation strategy is
proposed, which allows different initialized network optimized with different
objectives; and 3) extensive experiments are conducted to validate the
effectiveness of our proposed EVIL.

Fig. 1: The overview of our EVidential Inference Learning framework (EVIL),
where $\mathcal{M}$ denotes uncertainty map estimated by E-Net and $\odot$
denotes element-wise product. ‘$\rightarrow$’ presents forward operation,
‘$\dashrightarrow$’ presents supervision loss operation and ‘$//$’ on
‘$\rightarrow$’ presents stop-gradient.

##  2 Method

Given a labeled set
$D_{l}=\\{(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\\}^{N_{l}}_{i=1}$ with
${N_{l}}$ samples and an unlabeled set
$D_{u}=\\{\boldsymbol{x}_{i}\\}^{N_{u}}_{i=1}$ with ${N_{u}}$ samples, where
$N_{u}\gg N_{l}$ in semi-supervised task.

As illustrated in Fig. 1, EVIL has two differently initialized networks, E-Net
$\mathcal{F}_{1}$ with parameter set $\theta_{1}$ and S-Net $\mathcal{F}_{2}$
with parameter set $\theta_{2}$. For labeled data, S-Net is optimized with
traditional joint cross-entropy loss and dice loss, while E-Net models a
Dirichlet distribution and is optimized with evidential segmentation loss.
$P_{1}$, $P_{2}$ are the segmentation predictions and $Y_{1}$, $Y_{2}$ are the
corresponding pseudo labels generated by $argmax$ function. For unlabeled
data, E-Net generates pseudo labels $Y_{1}$ and accurate uncertainty
estimations $\mathcal{M}$ simultaneously. Then, the trustworthy pseudo labels
are calculated by $\mathcal{M}\odot Y_{1}$ and used to guide the training of
S-Net. Reversely, the pseudo labels $Y_{2}$ generated by S-Net is leveraged
for E-Net to explore more potential evidence to improve the generation of
pseudo labels from unlabeled data.

###  2.1 Uncertainty Modeling

In this section, we utilize DST to model the segmentation uncertainty and
generate trustworthy prediction. For a $K$-class segmentation task, given an
input $\boldsymbol{x}_{i}$, the evidence vector $\boldsymbol{e}_{i}$ is
obtained with a transform function $g$, which is defined in [13]:

$\boldsymbol{e}_{i}=g(\mathcal{F}_{1}(\boldsymbol{x}_{i}))=exp^{\left(\tanh{\mathcal{F}_{1}(\boldsymbol{x}_{i})/\tau}\right)},$
(1)

where $0<\tau<1$ is a scaling parameter set to $1/K$.
$\mathcal{F}_{1}(\boldsymbol{x}_{i})$ is the output of E-Net with input
$\boldsymbol{x}_{i}$. Subjective Logic [12] computes the belief mass for
category $k$ and uncertainty as:

$b_{i}^{k}=\frac{e_{i}^{k}}{S_{i}}=\frac{\alpha_{i}^{k}-1}{S_{i}}\quad\text{and}\quad
u_{i}=\frac{K}{S_{i}},$ (2)

where $S_{i}=\sum_{k=1}^{K}(e_{i}^{k}+1)$, $u_{i}+\sum_{k=1}^{K}b_{i}^{k}=1$
and $\alpha_{i}^{k}=e_{i}^{k}+1$.
$\boldsymbol{\alpha}_{i}=\left[\alpha_{i}^{1},\dots,\alpha_{i}^{K}\right]$ can
be regarded as the parameters of Dirichlet distribution, which models the
density of segmentation probability and uncertainty [14]. The density function
is defined as:

$D(\boldsymbol{p}_{i}\mid\boldsymbol{\alpha}_{i})=\\{\begin{array}[]{ll}\frac{1}{B(\boldsymbol{\alpha}_{i})}\prod_{k=1}^{K}p_{i}^{\alpha_{i}^{k}-1}&\text{for
}\boldsymbol{p}_{i}\in\mathcal{S}_{K},\\\ 0&\text{otherwise},\end{array}$ (3)

where $\boldsymbol{p}_{i}$ is the segmentation probability,
$B(\boldsymbol{\alpha}_{i})$ is the $K$-dimensional multinomial beta function
for parameter $\boldsymbol{\alpha}_{i}$, and $\mathcal{S}_{K}$ is the
$K$-dimensional simplex.

Fig. 2: Subjective Logic model, where $u_{i}+\sum_{k=1}^{K}b_{i}^{k}=1$.

(a) image & GT

(b) MT

(c) UAMT

(d) ICT

(e) CPS

(f) URPC

(g) EVIL (ours)

Fig. 3: Visual comparison of segmentation results with different methods with
10% labeled images.

###  2.2 Evidential Net (E-Net)

We follow [14] and use cross-entropy loss to make the segmentation
probabilities $\boldsymbol{p}_{i}$ approach the ground-truth
$\boldsymbol{y}_{i}$. Notably, the density of $\boldsymbol{p}_{i}$ follows the
Dirichlet distribution parameterized with $\boldsymbol{\alpha}_{i}$. The loss
can be formulated as:

$\displaystyle\mathcal{L}_{dig}$
$\displaystyle=\int\left[\sum_{k=1}^{K}-y_{i}^{k}\log\left(p_{i}^{k}\right)\right]D(\boldsymbol{p}_{i}\mid\boldsymbol{\alpha}_{i})d\boldsymbol{p}_{i}$
(4)
$\displaystyle=\sum_{k=1}^{K}y_{i}^{k}\left(\psi\left({S}_{i}\right)-\psi\left({\alpha}_{i}^{k}\right)\right),$

where $\psi(\cdot)$ is the digamma function. By optimizing
$\mathcal{L}_{dig}$, the evidence of different classes for positive samples is
generated. However, $\mathcal{L}_{dig}$ cannot guarantee that negative samples
generate evidence as close as zero. Therefore, Kullback-Leibler (KL)
divergence is incorporated into our loss function to penalize the divergence
from negative samples, which is defined as:

$\displaystyle\mathcal{L}_{KL}$
$\displaystyle=KL\left[D\left(\boldsymbol{p}_{i}\mid\tilde{\boldsymbol{\alpha}}_{i}\right)\|D\left(\boldsymbol{p}_{i}\mid\boldsymbol{1}\right)\right]$
(5)
$\displaystyle=\log\left(\frac{\Gamma\left(\sum_{k=1}^{K}\widetilde{{\alpha}}_{i}^{k}\right)}{\Gamma(K)\sum_{k=1}^{K}\Gamma\left(\widetilde{\alpha}_{i}^{k}\right)}\right)$
$\displaystyle+\sum_{k=1}^{K}\left(\widetilde{{\alpha}}_{i}^{k}-1\right)\left[\psi\left(\widetilde{{\alpha}}_{i}^{k}\right)-\psi\left(\sum_{k=1}^{K}\widetilde{{\alpha}}_{i}^{k}\right)\right],$

where $\Gamma(\cdot)$ is the gamma function,
$D(\boldsymbol{p}_{i}|\boldsymbol{1})$ is the uniform Dirichlet distribution,
and
$\tilde{\boldsymbol{\alpha}}_{i}=\boldsymbol{y}_{i}+(1-\boldsymbol{y}_{i})\odot\boldsymbol{\alpha}_{i}$.

For segmentation task, the evidence $\boldsymbol{e}_{i}$ is obtained with
$\boldsymbol{x}_{i}$. Then, $\boldsymbol{\alpha}_{i}=\boldsymbol{e}_{i}+1$ is
parameterized into the corresponding Dirichlet distribution and the evidential
loss is:

$\mathcal{L}_{evi}=\mathcal{L}_{dig}+\beta\mathcal{L}_{KL},$ (6)

where $\beta$ is a annealing coefficient and is set to
$\beta(t)=min(1.0,\frac{t}{0.5t_{max}})$. $t$ is the current epoch and
$t_{max}$ is the total number of training epochs.

As shown in Fig. 2, the Subjective Logic model has two parts, the certain part
called belief mass $\boldsymbol{b}_{i}$ and the uncertain part $u_{i}$. The
evidential loss generates evidence to reduce the uncertainty. However, since
the cross-entropy based evidential loss is based on pixel level, which ignores
the relationships between pixels in segmentation task, we use the Dice loss on
the certain part and the certain loss is defined as:

$\displaystyle\mathcal{L}_{certain}=1-\frac{2\sum_{k=1}^{K}y_{i}^{k}\sum_{k=1}^{K}\hat{p}_{i}^{k}}{\sum_{k=1}^{K}y_{i}^{k}+\sum_{k=1}^{K}\hat{p}_{i}^{k}},$
(7)

where $\hat{\boldsymbol{p}}_{i}=softmax(\boldsymbol{b}_{i})$ presents a
simplex transformed from the belief mass $\boldsymbol{b}_{i}$ with a $softmax$
function.

Then, our overall evidential segmentation loss is defined:

$\mathcal{L}_{Eseg}=\mathcal{L}_{evi}+\gamma\mathcal{L}_{certain},$ (8)

where $\mathcal{L}_{evi}$ and $\mathcal{L}_{certain}$ denote the evidential
loss and the certain loss, respectively. $\gamma$ denotes the weighting
parameter, which is set to $1$. By optimizing $\mathcal{L}_{evi}$, E-Net
generates the evidence for positive samples, while reduces the evidence for
negative samples. $\mathcal{L}_{certain}$ is leveraged to constrain the
relationship between different predicted pixels.

###  2.3 EVIL Framework

The total loss $\mathcal{L}$ for our whole framework contains two components:
supervised loss $\mathcal{L}_{sup}$ on labeled data and consistency loss
$\mathcal{L}_{con}$ on unlabeled data:

$\mathcal{L}=\mathcal{L}_{sup}+\lambda\mathcal{L}_{con},$ (9)

where $\lambda$ is the balancing parameter. We use Gaussian ramp-up function
$\lambda(t)=\lambda_{max}*e^{-5\left(1.0-\frac{t}{t_{max}}\right)^{2}}$ and
$\lambda_{max}=0.1$.

The supervision loss is formulated as:

$\displaystyle\mathcal{L}_{sup}=\mathcal{L}_{Eseg}(\mathcal{F}_{1}(\boldsymbol{x}),\boldsymbol{y})+\mathcal{L}_{Sseg}(\mathcal{F}_{2}(\boldsymbol{x}),\boldsymbol{y}),$
(10)

where $\mathcal{L}_{Sseg}=\frac{1}{2}(\mathcal{L}_{ce}+\mathcal{L}_{dice})$
denotes the loss component for S-Net. $\mathcal{L}_{ce}$ and
$\mathcal{L}_{dice}$ are the cross-entropy loss and dice loss, respectively.

The pseudo label can be calculated as $Y_{1}=argmax(\boldsymbol{b}_{i})$ for
E-Net and $Y_{2}=argmax(\mathcal{F}_{2}(\boldsymbol{x}))$ for S-Net. The
consistency loss on the unlabeled data is written as:

$\mathcal{L}_{con}=\mathcal{L}_{evi}(\mathcal{F}_{1}(\boldsymbol{x}),Y_{2})+\mathcal{L}_{ce}(\mathcal{F}_{2}(\boldsymbol{x}),\mathcal{M}\odot
Y_{1}).$ (11)

where $\mathcal{M}={u}<T$ is the mask to filter out high uncertain results
with threshold $T=0.2$. We only use the evidential and cross-entropy losses in
consistency loss term due to the mask operation which preserves only the
reliable pseudo pixel labels. The consistency loss encourages E-Net to
generate potential evidence from S-Net using $\mathcal{L}_{evi}$ and S-Net to
learn the reliable pseudo labels using $\mathcal{L}_{ce}$ from E-Net.

Table 1: The comparison of different methods on ACDC dataset on different semi-supervised labeled data ratio settings. Method | 10% | 20% | 30%
---|---|---|---
$DSC$ $\uparrow$ |  ${HD}_{95}$ $\downarrow$ |  $ASD$ $\downarrow$ |  $DSC$ $\uparrow$ |  ${HD}_{95}$ $\downarrow$ |  $ASD$ $\downarrow$ |  $DSC$ $\uparrow$ |  ${HD}_{95}$ $\downarrow$ |  $ASD$ $\downarrow$
Unet | 80.05 | 7.41 | 2.38 | 84.90 | 8.94 | 2.52 | 87.07 | 6.61 | 1.95
E-Net (ours) | 81.05 | 11.17 | 3.26 | 85.68 | 7.39 | 2.12 | 87.45 | 8.12 | 2.23
MT | 81.06 | 10.17 | 2.64 | 86.01 | 8.13 | 2.40 | 87.37 | 4.81 | 1.49
UA-MT | 80.81 | 11.73 | 3.52 | 85.38 | 7.77 | 2.70 | 87.53 | 6.32 | 2.05
ICT | 83.54 | 8.42 | 2.46 | 85.28 | 5.65 | 1.64 | 87.49 | 8.25 | 2.23
CPS | 84.70 | 8.25 | 2.35 | 87.47 | 5.98 | 1.74 | 88.21 | 6.49 | 1.90
URPC | 82.07 | 5.62 | 1.88 | 85.13 | 5.71 | 1.75 | 86.99 | 4.43 | 1.31
EVIL (ours) | 85.91 | 3.91 | 1.36 | 88.22 | 4.01 | 1.21 | 89.43 | 3.84 | 1.07

##  3 Experiment

###  3.1 Experiment Setup

We evaluate our method on the Automated Cardiac Diagnosis Challenge (ACDC)
[15] dataset which contains 200 annotated short-axis cardiac MR-cine images
from 100 patients. We leverage 70 patients (140 scans) for training, 10
patients (20 scans) for validation and 20 patients (40 scans) for testing. All
short-axis slices within 3D scans are resized to 256 $\times$ 256 as 2D
images. See SSL4MIS 111https://github.com/HiLab-git/SSL4MIS for more details.
For semi-supervised experiments, images from 7 patients, 14 patients and 21
patients are set as labeled ratio 10%, 20% and 30% in the training set,
respectively. Standard data augmentation, including random cropping, random
rotating, and random flipping, is used to enlarge the training set. Three
widely used metrics, Dice Coefficient ($DSC$), Hausdorff Distance 95
(${HD}_{95}$) and Average Surface Distance ($ASD$) are employed to evaluate
the performance of our method.

For the sake of fairness, Unet [1] is chosen as the backbone in all methods,
and SGD is adopted as the optimizer. The initial learning rate is set to 0.01,
and polynomial scheduler strategy is employed to update the learning rate. We
implement the proposed framework with PyTorch, using a single NVIDIA GTX
1080Ti GPU. The batch size is set to 24, where 12 images are labeled. All
methods perform 30000 iterations during training.

###  3.2 Experimental Results

Several recently proposed semi-supervised segmentation methods are compared,
including: Mean-Teacher (MT) [6], Uncertainty-Aware Mean Teacher (UA-MT) [9],
Interpolation Consistency Training (ICT) [16], Cross Pseudo Supervision (CPS)
[7], and Uncertainty Rectified Pyramid Consistency (URPC) [17]. For all
competing methods, official parameter settings are adopted.

Tab. 1 illustrates the quantitative results on ACDC. The first and second rows
list the quantitative results of supervised Unet and E-Net. In different
labeled data ratio settings, EVIL outperforms all the other methods. When only
10% of data are labeled, our method improves $DSC$ by more than 3% compared
with other SOTA uncertainty-aware methods (UAMT and URPC). Moreover, we
achieve 4 points improvement in ${HD}_{95}$ and 1 point in $ASD$ compared with
CPS. Especially, we can see that the performance of EVIL using 20% labeled
data has surpassed all compared methods using 30% labeled data.

Table 2: The comparison of training time. Method | Num | Uncertainty | Time | Cost
---|---|---|---|---
Unet | 1 | $\times$ | 0.076 s | -
ICT | 1 | $\times$ | 0.090 s | \+ 18.42 %
URPC | 1 | $\surd$ | 0.089 s | \+ 17.11 %
E-Net (ours) | 1 | $\surd$ | 0.085 s | \+ 11.84 %
MT | 2 | $\times$ | 0.101 s | -
CPS | 2 | $\times$ | 0.137 s | \+ 35.64 %
UA-MT | 2 | $\surd$ | 0.337 s | \+ 233.66%
EVIL (ours) | 2 | $\surd$ | 0.148 s | \+ 46.53 %

Fig. 3 visualizes the segmentation results of two cases using different
methods with 10% labeled data. It is easy to see that the compared methods
mis-classify many pixels while EVIL obtains more accurate prediction. As shown
in Fig. 4, sampling times affect the uncertainty estimation quality of MC-
dropout and our E-Net has best accurate estimation.

Tab. 2 shows the training time with fixed batch size = 24, where ‘Num’,
‘Uncertainty’, ‘Time’, ‘Cost’ denotes the network number, uncertainty-based or
not, time consuming, and the additional time consuming cost respectively. We
treat Unet as the upper bound of single network method and MT as the baseline
of the multi-network framework since it is the fastest method compared to
others. Specially, we can see that the proposed method improve significantly
without introducing too much computation overhead.

(a) Input

(b) Label

(c) Ours

(d) EVIL

(e) S=2

(f) S=8

(g) S=64

(h) UA-MT

Fig. 4: Visualization of uncertainty estimation. ‘S’ denotes the MC-dropout
sampling times.

##  4 Conclusion

In this paper, we propose a novel uncertainty-aware semi-supervised medical
image segmentation framework. The proposed EVIL introduces DST into the
consistency regularization training paradigm and achieves fast accurate
uncertainty estimation with solid theoretical guarantee. Extensive experiments
demonstrate that EVIL achieves state-of-the-art performance on the widely used
ACDC dataset.

##  5 CONFLICTS OF INTEREST

The authors declare that they have no conflicts of interest.

##  6 COMPLIANCE WITH ETHICAL STANDARDS

This research study was conducted retrospectively using real clinical exams
acquired at the University Hospital of Dijon. Ethical approval was not
required as confirmed by the license attached with the open access data.

##  7 ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of
China under Grant 62271335; in part by the Sichuan Science and Technology
Program under Grant 2021JDJQ0024; and in part by the Sichuan University “From
0 to 1” Innovative Research Program under Grant 2022SCUH0016.

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# Inference to the Best Explanation in Large Language Models

Dhairya Dalal1 , Marco Valentino2, André Freitas2,3,4, Paul Buitelaar1,5
1 SFI Centre for Research and Training in Artificial Intelligence, University
of Galway, Ireland
2 Idiap Research Institute, Switzerland
3 Department of Computer Science, University of Manchester, UK
4 National Biomarker Centre, CRUK-MI, University of Manchester, UK
5 Insight SFI Research Centre for Data Analytics, University of Galway,
Ireland
Email<EMAIL_ADDRESS>
###### Abstract

While Large Language Models (LLMs) have found success in real-world
applications, their underlying explanatory process is still poorly understood.
This paper proposes IBE-Eval, a framework inspired by philosophical accounts
on _Inference to the Best Explanation (IBE)_ to advance the interpretation and
evaluation of LLMs’ explanations. IBE-Eval estimates the plausibility of
natural language explanations through a combination of explicit logical and
linguistic features including: _consistency_ , _parsimony_ , _coherence_ , and
_uncertainty_. Extensive experiments are conducted on _Causal Question
Answering (CQA)_ , where IBE-Eval is tasked to select the most plausible
causal explanation amongst competing ones generated by LLMs (i.e., GPT 3.5 and
Llama 2). The experiments reveal that IBE-Eval can successfully identify the
best explanation with up to 77% accuracy ($\approx 27\%$ above random),
improving upon a GPT 3.5-as-a-Judge baseline ($\approx+17\%$) while being
intrinsically more efficient and interpretable. Additional analyses suggest
that, despite model-specific variances, LLM-generated explanations tend to
conform to IBE criteria and that IBE-Eval is significantly correlated with
human judgment, opening up opportunities for future development of automated
explanation verification tools.

Figure 1: The IBE framework qualifies LLM-generated explanations with a set of
logical and linguistic selection criteria to identify the most plausible
hypothesis.

##  1 Introduction

Large Language Models (LLMs) such as OpenAI’s GPT Brown et al. (2020) and
Llama 2 Touvron et al. (2023) have been highly effective across a diverse
range of language understanding and reasoning tasks Liang et al. (2023). While
LLM performances have been thoroughly investigated across various benchmarks
Wang et al. (2019); Srivastava et al. (2023); Gao et al. (2023); Touvron et
al. (2023), the principles and properties behind their step-wise reasoning
process are still poorly understood Valentino et al. (2021). LLMs are
notoriously black-box models and can be difficult to interpret Chakraborty et
al. (2017); Danilevsky et al. (2020). Moreover, the commercialization of LLMs
has led to strategic secrecy around model architectures and training details
Xiang (2023); Knight (2023). Finally, neural models are susceptible to
hallucinations and adversarial perturbations Geirhos et al. (2020); Camburu et
al. (2020), often producing plausible but factually incorrect answers Ji et
al. (2023); Huang et al. (2023). As the size and complexity of LLM
architectures increase, the systematic study of generated explanations becomes
crucial to better interpret and validate the LLM’s internal inference and
reasoning processes Wei et al. (2022b); Lampinen et al. (2022); Huang and
Chang (2022).

The automatic evaluation of natural language explanations presents several
challenges Atanasova et al. (2023); Camburu et al. (2020). Without resource-
intensive annotation Wiegreffe and Marasovic (2021); Thayaparan et al. (2020);
Dalvi et al. (2021); Camburu et al. (2018), explanation quality methods tend
to rely on either weak supervision, where the identification of the correct
answer is taken as evidence of explanation quality, or require the injection
of domain-specific knowledge Quan et al. (2024). In this paper, we seek to
better understand the LLM explanatory process through the investigation of
explicit linguistic and logical properties. While explanations are hard to
formalize due to their open-ended nature, we hypothesize that they can be
analyzed as linguistic objects, with measurable features that can serve to
define criteria for assessing their quality.

Specifically, this paper investigates the following overarching research
question: _“Can linguistic and logical properties associated with LLMs’
generated explanations be used to qualify the models’ reasoning process?”_. To
this end, we propose an interpretable framework inspired by philosophical
accounts of abductive inference, also known as _Inference to the Best
Explanation (IBE)_ \- i.e. the process of selecting among competing
explanatory theories Lipton (2017). In particular, we aim to measure the
extent to which LLM-generated explanations conform to IBE expectations when
attempting to identify the most plausible explanation. To this end, we present
IBE-Eval, a framework designed to estimate the plausibility of natural
language explanations through a set of explicit logical and linguistic
features, namely: _logical consistency_ , _parsimony_ , _coherence_ , and
_linguistic uncertainty_.

To evaluate the efficacy of IBE-Eval, we conduct extensive experiments in the
multiple-choice Causal Question Answering (CQA) setting. The overall results
and contributions of the paper can be summarized as follows:

  1. 1.

To the best of our knowledge, we are the first to propose an interpretable
framework inspired by philosophical accounts on Inference to the Best
Explanation (IBE) to automatically assess the quality of natural language
explanations.

  2. 2.

We propose IBE-Eval, a framework that can be instantiated with external tools
for the automatic evaluation of LLM-generated explanations and the
identification of the best explanation in the multiple-choice CQA settings.

  3. 3.

We provide empirical evidence that LLM-generated explanations tend to conform
to IBE expectations with varying levels of statistical significance correlated
to the LLM’s size.

  4. 4.

We additionally find that uncertainty, parsimony, and coherence are the best
predictors of plausibility and explanation quality across all LLMs. However,
we also find that the LLMs tend to be strong rationalizers and can produce
consistent explanations even for less plausible candidates, making the
consistency metric less effective in practice.

  5. 5.

IBE-Eval can successfully identify the best explanation supporting the correct
answers with up to 77% accuracy (+$\approx 27\%$ above random and +$\approx
17\%$ over GPT 3.5-as-a-Judge baselines)

  6. 6.

IBE-Eval method is significantly correlated with human judgment, outperforming
a GPT3.5-as-a-Judge baseline in terms of alignment with human preferences.

For reproducibility, the entire experimental code and analysed explanations
are available online111anonymous_link to encourage future research in the
field.

##  2 Inference to the Best Explanation (IBE)

Explanatory reasoning is a distinctive feature of human rationality
underpinning problem-solving and knowledge creation in both science and
everyday scenarios Lombrozo (2012); Deutsch (2011). Accepted epistemological
accounts characterize the creation of an explanation as composed of two
distinct phases: conjecturing and criticism Popper (2014). The explanatory
process always involves a conflict between plausible explanations, which is
typically resolved through the criticism phase via a selection process, where
competing explanations are assessed according to a set of criteria such as
parsimony, coherence, unification power, and hardness to variation Lipton
(2017); Harman (1965); Mackonis (2013); Thagard (1978, 1989); Kitcher (1989);
Valentino and Freitas (2022).

As LLMs become interfaces for natural language explanations, epistemological
frameworks offer an opportunity for developing criticism mechanisms to
understand the explanatory process underlying state-of-the-art models. To this
end, this paper considers an LLM as a conjecture device producing linguistic
objects that can be subject to criticism. In particular, we focus on a subset
of criteria that can be computed on explicit linguistic and logical features,
namely: consistency, parsimony, coherence, and uncertainty.

To assess the LLM’s alignment to such criteria, we focus on the task of
selecting among competing explanations in a multiple-choice CQA setting
(Figure 1). Specifically, given a set of competing hypotheses (i.e. the
multiple-choice options), $H=\\{h_{1},h_{2},\ldots,h_{n}\\}$, we prompt the
LLM to generate plausible explanations supporting each hypothesis (Section 3).
Subsequently, we adopt the proposed IBE selection criteria to assess the
quality of the generated explanations (Section 4). IBE-Eval computes an
explanation score derived from the linear combination of the computed
selection criteria. The explanation with the highest score is selected as the
predicted answer and additionally assessed as to the extent to which the
observable IBE features are correlated with QA accuracy. We hypothesize that
IBE-Eval will produce higher scores for the explanation associated with the
correct answer and that IBE criteria should meaningfully differentiate between
competing explanations.

##  3 Explanation Generation

For the first stage, the LLM is prompted to generate competing explanations
for the hypotheses using a modified Chain-of-Thought (CoT) prompt Wei et al.
(2022a). Specifically, the COT prompt is modified to instruct the LLM to
produce an explanation for each competing hypothesis (see Figure 1). We adopt
a methodology similar to Valentino et al. (2021), where the generated
explanation is constrained into an entailment form for the downstream IBE
evaluation. In particular, we posit that a valid explanation should
demonstrate an entailment relationship between the premise and conclusion
which are derived from the question-answer pair.

To elicit logical connections between explanation steps and facilitate
subsequent analysis, the LLM is constrained to use weak syllogisms expressed
as If-Then statements. Additionally, the LLM is instructed to produce the
associated causal or commonsense assumption underlying each explanation step.
This output is then post-processed to extract the explanation steps and
supporting knowledge for evaluation via the IBE selection criteria. Additional
details and examples of prompts are reported in Appendix A.2.

##  4 Linguistic & Inference Criteria

To perform IBE, we investigate a set of criteria that can be automatically
computed on explicit logical and linguistic features, namely: consistency,
parsimony, coherence, and uncertainty.

##### Consistency.

Consistency aims to verify whether the explanation is logically valid. Given a
hypothesis, composed of a premise $p_{i}$, a conclusion $c_{i}$, and an
explanation consisting of a set of statements $E={s_{1},...,s_{i}}$, we define
$E$ to be logically consistent if $p_{i}\cup E\vDash c_{i}$. Specifically, an
explanation is logically consistent if it is possible to build a deductive
proof linking premise and conclusion. To evaluate logical consistency, we
leverage external symbolic solvers along with autoformalization - i.e., the
translation of natural language into a formal language Wu et al. (2022).
Specifically, hypotheses and explanations are formalized into a Prolog program
which will attempt to generate a deductive proof via backward chaining Weber
et al. (2019). To perform autoformalization, we leverage the translation
capabilities of GPT 3.5. Specifically, we instruct GPT 3.5 to convert each IF-
Then implication from the generated explanation into an implication rule and
the premise statement into grounding atoms. On the other end, the entailment
condition and the conclusion are used to create a Prolog query. Further
details about the autoformalization process can be found in Appendix A.3.
After autoformalization, following recent work on neuro-symbolic integration
for LLM explanations Quan et al. (2024), we adopt an external Prolog solver
for entailment verification222https://github.com/neuro-symbolic-
ai/explanation_based_ethical_reasoning. The explanation is considered
consistent if the Prolog solver can satisfy the query and successfully build a
deductive proof. Technical details can be found in Appendix A.6.

##### Parsimony.

The parsimony principle, also known as Ockham’s razor, favors the selection of
the simplest explanation consisting of the fewest elements and assumptions
Sober (1981). Epistemological accounts posit that an explanation with fewer
assumptions tends to leave fewer statements unexplained, improving specificity
and alleviating the infinite regress Thagard (1978). Further, parsimony is an
essential feature of causal interpretability, as only parsimonious solutions
are guaranteed to reflect causation in comparative analysis Baumgartner
(2015). In this paper, we adopt two metrics as a proxy of parsimony, namely
_proof depth_ , and _concept drift_. Proof depth, denoted as $Depth$, is
defined as the cardinality of the set of rules, $R$, required by the Prolog
solver to connect the conclusion to the premise via backward chaining. Let $h$
be a hypothesis candidate composed of a premise $p$ and a conclusion $c$, and
let $E$ be a formalized explanation represented as a set of rules
$R^{\prime}$. The proof depth is the number of rules $|R|$, with $R\subseteq
R^{\prime}$, traversed during backward chaining to connect the conclusion $c$
to the premise $p$:

$Depth(h)=|R|$

Concept drift, denoted as $Drift$, is defined as the number of additional
concepts and entities, outside the ones appearing in the hypothesis (i.e.,
premise and conclusion), that are introduced by the LLM to support the
entailment. For simplicity, we consider nouns as concepts. Let
$N=\\{Noun_{p},Noun_{c},Noun_{E}\\}$ be the unique nouns found in the premise,
conclusion, and explanation steps. Concept drift is the cardinality of the set
difference between the nouns found in the explanation and the nouns in the
hypothesis:

$Drift(h)=|Noun_{E}-(Noun_{p}\cup Noun_{c})|$

Intuitively, the parsimony principle would predict the most plausible
hypothesis as the one supported by an explanation with the smallest observed
proof depth and concept drift. Implementation details can be found in Appendix
A.7.

##### Coherence.

Coherence attempts to measure the logical relations within individual
explanatory statements and implications. An explanation can be formally
consistent on the surface while still including implausible or ungrounded
intermediate assumptions. Coherence evaluates the quality of each intermediate
If-Then implication by measuring the entailment strength between the If and
Then clauses. To this end, we employ a fine-tuned natural language inference
(NLI) model. Formally, let $S$ be a set of explanation steps, where each step
$s$ consists of an If-Then statement, $s=(If_{s},Then_{s})$. For a given step
$s_{i}$, let $ES(s_{i})$ denote the entailment score obtained via the NLI
model between $If_{s}$ and $Then_{s}$ clauses. The step-wise entailment score
$SWE(S)$ is then calculated as the averaged sum of the entailment scores
across all explanation steps $|S|$:

$\text{SWE}(S)=\frac{1}{|S|}\sum_{i=1}^{|S|}\text{ES}(s_{i})$

We hypothesize that the LLM should generate a higher coherence score for more
plausible hypotheses, as such explanations should exhibit stronger step-wise
entailment. Additional details can be found in Appendix A.8.

##### Uncertainty.

Finally, we consider the linguistic certainty expressed in the generated
explanation as a proxy for plausibility. Hedging words such as probably, might
be, could be, etc typically signal ambiguity and are often used when the truth
condition of a statement is unknown or improbable. Pei and Jurgens (2021)
found that the strength of scientific claims in research papers is strongly
correlated with the use of direct language. In contrast, they found that the
use of hedging language suggested that the veracity of the claim was weaker or
highly contextualized. To measure the linguistic certainty ($LC$) of an
explanation, we consider the explanation’s underlying assumptions ($A_{i}$)
and the overall explanation summary ($S$). The linguistic certainty score is
extracted using the fine-tuned sentence-level RoBERTa model from Pei and
Jurgens (2021). The overall linguistic certainty score ($LC_{\text{overall}}$)
is the sum of the assumption and explanation summary scores:

$LC_{\text{overall}}=LC(A)+LC(S)$

Where $LC(A)$ is the sum of the linguistic certainty scores ($LC(A)$) across
all the assumptions $|A|$ associated with each explanation step $i$:

$LC(A)=\sum_{i=1}^{|A|}LC(a_{i})$

and linguistic certainty of the explanation summary $LC(S)$. We hypothesize
that the LLM will use more hedging language when explaining the weaker
hypothesis resulting in a higher uncertainty score. Further details can be
found in Appendix A.9.

###  4.1 Inference to Best Explanation

After the IBE criteria are computed for each competing hypothesis, they are
used to generate the final explanation score. We define a simple linear
regression model $\theta(\cdot)$, which was fitted on a small set of training
examples consisting of extracted IBE features to predict the probability that
an explanation $E_{i}$ corresponds to the correct answer. Specifically, we
employ IBE-Eval to score each generated explanation independently and then
select the final answer $a$ via argmax:

$a=\operatorname*{argmax}_{i}[\theta(E_{1}),\ldots,\theta(E_{n})]$

Additional details can be found in Appendix A.10.

Figure 2:  Regression analysis measuring the correlation between IBE criteria and question accuracy. All LLMs tend to conform to IBE expectations with GPT 3.5 exhibiting the most consistent and significant alignment. Linguistic uncertainty is the strongest IBE predictor for explanation quality, where higher uncertainty is negatively correlated with question accuracy. Statistical significance is noted as: ‘***’ p < 0.001, ‘**’ p < 0.01 ‘*’ p < 0.05. | COPA | E-CARE
---|---|---
| GPT 3.5 | LlaMA 2 13B | LlaMA 2 7B | GPT 3.5 | LlaMA 2 13B | LlaMA 2 7B
Baselines |  |  |  |  |  |
GPT3.5 Judge | .59 | .47 | .63 | .43 | .61 | .52
Human | .95 | 1.0 | .91 | .90 | .91 | .92
IBE Features |  |  |  |  |  |
Consistency | .51 | .52 | .55 | .54 | .54 | .54
Depth (Parsimony) | .67 | .53 | .63 | .66 | .56 | .54
Drift (Parsimony) | .67 | .63 | .58 | .66 | .57 | .57
Coherence | .66 | .66 | .56 | .56 | .57 | .59
Linguistic Uncertainty | .70 | .65 | .61 | .59 | .56 | .60
Composed Model |  |  |  |  |  |
Random | .50 | .50 | .50 | .50 | .50 | .50
\+ Consistency | .51 | .52 | .55 | .54 | .54 | .54
\+ Depth | .67 | .53 | .63 | .66 | .56 | .56
\+ Drift | .70 | .65 | .65 | .72 | .66 | .65
\+ Coherence | .73 | .71 | .69 | .73 | .68 | .69
\+ Linguistic Uncertainty | .77 | .74 | .70 | .74 | .70 | .73
Table 1:  Ablation study and evaluation of the IBE criteria and the composed
IBE-Eval model. IBE-Eval outperforms the GPT 3.5 Judge baseline with an
average by +17.5% across all all models and tasks.

##  5 Experimental Setting

Causal Question-Answering (CQA) requires reasoning about the causes and
effects given an event description. We specifically consider the task of cause
and effect prediction in the multiple-choice setting, where given a question
and two candidate answers, the LLM must decide which is the most plausible
cause or effect. Causal reasoning is a challenging task as the model must both
possess commonsense knowledge about causal relationships and consider the
event context which would make one option more plausible than the other. For
our experiments, we use the Choice of Plausible Alternatives (COPA) Gordon et
al. (2012) and E-CARE Du et al. (2022) datasets.

##### COPA.

COPA is a multiple-choice commonsense causal QA dataset consisting of 500
train and test examples that were manually generated. Each multiple-choice
example consists of a question premise and a set of answer candidates which
are potential causes or effects of the premise. COPA is a well-established
causal reasoning benchmark that is both a part of SuperGlue Wang et al. (2019)
and the CALM-Bench Dalal et al. (2023).

##### E-CARE.

E-CARE is a large-scale multiple-choice causal QA dataset consisting of crowd-
sourced 15K train and 2k test examples. Similar to COPA, the task requires the
selection of the most likely cause or effect provided an event description. We
randomly sample 500 examples from the E-CARE test set for our experiments.

##### LLMs.

We consider GPT-Turbo-3.5, LLaMA 2 13B and LLaMA 2 7B for all experiments. GPT
3.5 is a proprietary model Brown et al. (2020) and is highly effective across
a wide range of natural language reasoning tasks Laskar et al. (2023). We
additionally evaluate the open-source Llama 2 model Touvron et al. (2023). We
consider both the 13B and 7B variants of Llama 2 as both are seen as viable
commodity GPT alternatives and have been widely adopted by the research
community for LLM benchmarking and evaluation.

##### Baselines.

We employ LLM-as-a-Judge Zheng et al. (2023) and human evaluators as baseline
methods for the selection of the best explanation in the CQA setting. (Zheng
et al., 2023) found LLMs can align with human judgment and be utilized for
automated evaluation and judgment. We specifically uses GPT 3.5 as the LLM
judge. For each CQA example, we present the judges with two competing
explanations generated by the target LLM. The judge is asked to identify the
best and most plausible explanation. Additional details about the baselines
can be found in Appendix A.4 and A.5.

##  6 Preliminary Analysis

We present a preliminary sanity-check measuring the extent to which LLMs
generate _self-evident or tautological_ explanations - i.e., explanations that
simply restate the premises and conclusions. Tautological explanations
represent a risk for IBE-Eval as the metrics would be theoretically
uninformative if the LLM adopts the tested causal relation as the explanation
itself (e.g. A → B) without providing additional supporting statements.

We consider the _parsimony_ metric to compute the percentage of explanations
with _proof depth_ equal to 1 (i.e, explanations containing only one inference
step) and _concept drift_ equal to 0 (i.e. no additional concepts other than
the ones stated in premises and conclusions appear in the explanation). In
such cases, the LLM is effectively generating a self-evident or tautological
explanation.

We found that about 2% of the cases consist of self-evident explanations. For
GPT 3.5, Llama 2 13B and 7B, 2% of the generated explanations exhibit a
concept drift of 0, and on average 1.5% of the explanations have a proof depth
of 1. Next, we conduct an error analysis to evaluate the cases where IBE-Eval
selected a self-evident explanation as the best one. Across all LLMs, less
than 0.1% of the errors were caused by the selection of such explanations. Our
analysis suggests that the impact of self-evident explanations is not
significant and that the IBE framework can be robustly applied to identify
such cases.

##  7 Results

To assess LLMs’ alignment with the proposed IBE framework and evaluate the
efficacy of IBE-Eval, we run a regression analysis and conduct a set of
ablation studies to evaluate the relationship between IBE and question
accuracy. The main results are presented in Figure 2 and Table 1.

Our regression analysis finds that IBE criteria are generally consistent
across all LLMs as demonstrated by similar correlation patterns found across
both the board. GPT 3.5 exhibits the strongest alignment to IBE expectations
with nearly all criteria being statistically significant. Thus our proposed
IBE criteria can serve as promising build blocks for future work on automated
explanation evaluation.

In Table 1 we evaluate the accuracy of IBE criteria and IBE-Eval in selecting
the most plausible explanation in CQA setting. We find that independently the
IBE criteria are generally limited in their ability to identify the more
plausible explanation - though still outperform the GPT 3.5 judge. IBE-Eval,
which considers all IBE criteria, improves the ability to select the best
explanation by 17% over both the GPT 3.5 and random baselines. We can achieve
up 77% accuracy utilizing just the extracted IBE criteria demonstrating IBE’s
potential value for automatic explanation evaluation.

Next, we explore each explanation feature in further detail to better
understand the variances across criteria and LLMs.

##### Consistency.

We found that LLMs are surprisingly strong conjecture models, being able to
generate logically consistent explanations for all hypotheses as observed by
similar consistency scores for correct and incorrect (Figure 3). Moreover, we
observe that consistency tends to be a statistically insignificant predictor
for the Llama models. Therefore, we conclude that evidence of logical
consistency provides a limited signal for plausibility and is better
understood in the context of other IBE criteria. For the incorrect candidate
explanations, we find that LLMs over-rationalize and introduce additional
premises to demonstrate entailment in their explanations. We further explore
this phenomenon in Section 4.

Figure 3: Evaluation of explanation consistency. LLMs are strong rationalizers
and can generate logically consistent explanations at equal rates for both
correct and incorrect answers.

##### Parsimony.

Figure 4: Explanation parsimony is evaluated using proof depth and concept
drift. Both metrics are consistently lower for explanations supporting the
correct answers suggesting that LLMs are able efficient explanations for the
more plausible hypothesis.

The results suggest that parsimony has a more consistent effect and is a
better predictor of explanation quality. We observe a negative correlation
between proof depth, concept drift, and question-answering accuracy,
suggesting that LLMs tend to introduce more concepts and explanation steps
when explaining less plausible hypotheses. On average, we found the depth and
drift to be 6% and 10% greater for the incorrect option across all LLMs
(Figure 4). Moreover, the results suggest that as the LLM size grows, the
ability to over-rationalize tends to grow linearly. This is attested by the
fact that the average difference in depth and drift is the greatest in GPT
3.5, suggesting that the model tends to find the most efficient explanations
for stronger hypotheses and articulates explanations for weaker candidates.
Finally, we found that the Llama models tend to generate more complex
explanations overall, with Llama 2 13B exhibiting the largest concept drift
for less plausible hypotheses. The parsimony criterion supports the IBE
predictive power with an average of 14% improvement over consistency.

##### Coherence.

Figure 5: The average coherence score is consistently higher for explanations
corresponding to the correct hypotheses.

Similarly to parsimony, we found coherence to be a better indicator of
explanation quality being statistically significant for both GPT 3.5 and Llama
2 13B on COPA and both Llama 2 models on E-Care. We found that the average
coherence score is consistently greater for the stronger hypothesis across all
LLMs and datasets (see Figure 5). Both GPT and Llama 2 13B exhibit a higher
relative difference between the correct and incorrect hypotheses in contrast
to Llama 2 7B.

##### Uncertainty.

Figure 6: LLMs tend to use more hedging language in explanations supporting
less plausible hypotheses. This language is mostly classified as _epistemic_.

The results reveal that linguistic uncertainty is the strongest predictor of
explanation quality and is statistically significant for all LLMs. This
suggests that LLMs use more qualifying language when explaining weaker
hypotheses (see Figure 6). We found that uncertainty can improve accuracy by
13pp on COPA and 4pp on E-CARE. We also examine the uncertainty cues expressed
by LLMs by analyzing both the frequency of hedge words and the types of hedge
cues employed in incorrect explanations. We find the distribution of hedge
cues across LLMs tends to be similar, with only minor differences between LLMs
(Figure 6). Epistemic cues were most frequently used by all three models, with
Llama 2 7B being more likely to use conditional cues. See Appendix A.9 for
further details..

###  7.1 Correlation with Human Judgement.

Finally, We calculate the Spearman’s rank correlation between GPT 3.5 Judge,
IBE-Eval, and human judgment. We found that GPT 3.5 Judge exhibits a weak and
statistically insignificant correlation with human judgment (0.31). In
contrast, we find that the IBE-Eval is significantly aligned with human
preferences (with a Spearman’s correlation of 0.64 with p < 0.01) further
suggesting the IBE’s potential for automatic explanation evaluation.

##  8 Related Work

Explorations of LLM reasoning capabilities across various domains (e.g.
arithmetic, commonsense, planning, symbolic, etc) are an emerging area of
interest Xu et al. (2023); Huang and Chang (2023). Prompt-based methods Wei et
al. (2022b); Zhou et al. (2023); Wang et al. (2023), such as CoT, investigate
strategies to elicit specific types of reasoning behavior through direct LLM
interaction. Olausson et al. (2023) investigate automatic proof generation and
propose a neurosymbolic framework with an LLM semantic parser and external
solver. Creswell et al. (2022) propose an inference framework where the LLM
acts as both a selection and inference module to produce explanations
consisting of causal reasoning steps in entailment tasks. This paper primarily
draws inspiration from recent work on the evaluation of natural language
explanations Quan et al. (2024); Valentino et al. (2021); Wiegreffe and
Marasovic (2021); Thayaparan et al. (2020); Dalvi et al. (2021); Camburu et
al. (2018). However, differently from previous methods that require extensive
human annotations or specific domain knowledge, we are the first to propose a
set of criteria that can be automatically computed on explicit linguistic and
logical features.

##  9 Conclusion

This paper proposed IBE-Eval, an interpretable framework for LLM explanation
evaluation inspired by philosophical accounts of Inference to the Best
Explanation (IBE). IBE-Eval can identify the best explanation supporting the
correct answer with up to 77% accuracy in CQA scenarios, improving upon a GPT
3.5 Judge baselines by +17%. Our regression study suggests that LLM
explanations tend to conform to IBE expectations and that IBE-Eval is strongly
correlated with human judgment. Linguistic uncertainty is the stronger IBE
predictor for explanation quality closely followed by parsimony and coherence.
However, we also found that LLMs tend to be strong conjecture models able to
generate logically consistent explanations for less plausible hypotheses,
suggesting limited applicability for the logical consistency criterion in
isolation. We believe our findings can open new lines of research on external
evaluation methods for LLMs as well as interpretability tools for
understanding the LLM’s underlying explanatory process.

##  10 Limitations

IBE offers an interpretable explanation evaluation framework utilizing logical
and linguistic features. Our current instantiation of the framework is
primarily limited in that it does not consider grounded truth for factuality.
We observe that the model can generate factually incorrect but logically
consistent explanations. In some cases, the coherence metric can identify
those factual errors when the step-wise entailment score is comparatively
lower. However, our reliance on aggregated metrics can hide weaker entailment
especially when the explanation is longer or the entailment strength of the
surrounding steps is stronger. Future work can introduce metrics to evaluate
grounded knowledge or perform more granular evaluations of explanations to
better weight factual inaccuracies.

Finally, the list of criteria considered in this work is not exhaustive and
can be extended in future work. However, additional criteria for IBE might not
be straightforward to implement (e.g., unification power, hardness to
variation) and would probably require further progress in both epistemological
accounts and existing NLP technology.

##  11 Ethics Statement

The human annotators for computing the human judgment baseline are all authors
of the papers and as such were not further compensated for the annotation
task.

## Acknowledgements

This work was partially funded by the Swiss National Science Foundation (SNSF)
project NeuMath (200021_204617), by the EPSRC grant EP/T026995/1 entitled
“EnnCore: End-to-End Conceptual Guarding of Neural Architectures” under
Security for all in an AI enabled society, by the CRUK National Biomarker
Centre, and supported by the Manchester Experimental Cancer Medicine Centre,
the Science Foundation Ireland under grants SFI/18/CRT/6223 (Centre for
Research Training in Artificial Intelligence), SFI/12/RC/2289_P2 (Insight),
co-funded by the European Regional Development Fund, and the NIHR Manchester
Biomedical Research Centre.

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##  Appendix A Appendix

###  A.1 Reproducibility

All experimental code is available online333TBA to encourage future research
in the field. We additionally summarize all the model implementations and
technical resources used for the computation of the proposed IBE criteria
below:

  * •

We adopt the Prolog solver for neuro-symbolic integration from Quan et al.
(2024).

  * •

We use Spacy Honnibal and Montani (2017) to tokenize and extract part-of-
speech (POS) tags.

  * •

To compute coherence, we employ a RoBERTa-based NLI model Nie et al. (2020)
that has been finetuned on a range of NLI and fact verification datasets
consisting of SNLI Bowman et al. (2015), aNLI Nie et al. (2020), multilingual
NLI Williams et al. (2018)), and FEVER-NLI Nie et al. (2019).

  * •

To measure sentence-level uncertainty, we employ a finetuned RoBERTa model
provided by Pei and Jurgens (2021).

  * •

We use a fine-tuned BERT-based token classification model to classify all the
words in the generated explanation with uncertainty categories introduced in
the 2010 CoNLL shared task on Hedge Detection Farkas et al. (2010).

###  A.2 Explanation Prompting

Figure 7: To perform IBE we convert the CQA context and answer candidates into
an entailment form (i.e., EEV) Valentino et al. (2021).  Figure 8: An example
of the modified CoT prompt template for explanation generation.

A modified CoT prompt is used to instruct the LLM to generate explanations.
The prompt includes a set of instructions for explanation generation and an
in-context example. Appended to the end of the prompt are the CQA context,
causal question, and answer candidates. The LLM is instructed to first convert
the options into the EEV format consisting of a premise and conclusion. The
EEV format will differ depending on the directionality of the causal question
(see Figure 7). Cause prediction questions will treat the answer candidate as
the premise and the context as the conclusion. In contrast, effect prediction
reverses the relationship treating the context as the premise and the answer
options as the conclusion. After the EEV conversion, the model is instructed
to generate a step-by-step explanation consisting of IF-THEN statements and
the associated causal or commonsense assumptions. For ease of post-processing,
the LLM is instructed to use headers and enumerate steps using the Step #
format. A full example of the prompt template is exhibited in Figure 8.

Figure 9: An example of the autoformalization prompt.

###  A.3 Autoformalisation

Autoformalisation is the process of translating natural language descriptions
into formal specifications Wu et al. (2022). We adopt the translational
capabilities of GPT-3.5-Turbo to convert the explanation into a formal
entailment hypothesis. The IF-THEN explanation steps are converted into a set
of Prolog rules, the entailment description is used to generate Prolog atoms,
and the conclusion statement is translated into a Prolog query. We provide an
example of the autoformalization prompt in Figure 9 and an example of the
formalized output in Figure 11. After autoformalization, we deploy a post-
processing script to extract the formalized rules, atoms, and query and
generate a Prolog program for entailment verification.

###  A.4 Human Evaluation Baseline

We sample 100 generated explanation pairs across both COPA and E-CARE and all
three language models and provide them to two human evaluators. The human
evaluator is provided with only the explanations and instructed to select
which explanation is more plausible among the two. No additional information
about the original question nor the correct answer is provided. We find the
human evaluators on average were able to identify the explanation associated
with the correct answer 96% and 91% for the COPA and E-CARE datasets.
Additionally, we compute the inter-evaluator agreement score between two human
evaluators and find that there is Cohen’s Kappa score of .68 suggesting there
is a strong agreement between the two evaluators.

###  A.5 LLM-as-a-Judge Baseline

Figure 10: An example of prompt used by the LLM-as-a-Judge model for
evaluating competing explanations.

GPT 3.5 is used as the LLM for the LLM-as-a-Judge baseline. Similar to the
human evaluators, GPT is presented with both generated explanations and asked
to identify which explanation is more plausible.

###  A.6 Logical Consistency

Figure 11: An example of the autoformalization prompt.

An explanation hypothesis is considered logically consistent if the external
solver can build a deductive proof connecting the conclusion to the premise.
We use NLProlog Weber et al. (2019), a neuro-symbolic Prolog solver
integrating backward chaining with word embedding models via a weak
unification mechanism. NLProlog allows for a level of flexibility and
robustness that is necessary for NLP use cases (e.g. unification applied to
synonyms). We provide the autoformalized query, atoms, and rules to NLProlog.
If NLProlog can satisfy the entailment query, it will return the proof
consisting of the set of rules traversed, the weak unification score, and the
proof depth. For simplicity, we assign a score of one if the entailment query
is satisfied and zero if it is not. The proof depth score is evaluated as part
of the parsimony analysis. An end-to-end example of consistency evaluation can
be found in Figure 11.

1

Input : Symbolic KB $kb$, Goal $goal$, Glove embedding model $e(\cdot)$

Output : proof chain $chain$, proof depth $depth$

2

3threshold $\leftarrow$ 0.13;

4

5$depth$ $\leftarrow$ 1 ;

6 $chain$ $\leftarrow$ $emptylist$ ;

7

8foreach  _step $t$ in backward_chaining($kb$, $goal$)_ do

9 foreach  _$max\\_unification(q,q_{t})$_ do

10 $unification\\_score$ $\leftarrow$
$CosineSimilarity$($e(q,m_{s}),e(q_{t},m_{s})$);

11 $depth$ $\leftarrow$ $depth$ $\times$ $unification\\_score$ ;

12

13 end foreach

14 $chain$ $\leftarrow$ backward_chaining($kb$, $goal$);

15

16 end foreach

17

18if  _$chain$ is not empty and $depth$ $>$ threshold_ then

19 $chain\leftarrow$ current_proof_chain$[0]$;

20

21 end if

22else

23 $depth$ $\leftarrow$ 0 ;

24 end if

25

26return _$chain$ , $depth$ _;

Algorithm 1 Neuro-symbolic Solver

###  A.7 Parsimony

Parsimony measures the complexity of an explanation and is represented by the
proof depth and concept drift metrics. Proof depth is automatically calculated
by NLProlog and reflects the number of rules traversed by the solver to
satisfy the entailment query. If the hypothesis is not logically consistent,
depth is set to zero. The concept drift metric measures the entropy of novel
concepts introduced to bridge the premise and conclusion. To compute the drift
of an explanation, we consider the nouns found in the premise, conclusion, and
explanation steps. We use Spacy Honnibal and Montani (2017) to tokenize and
extract part-of-speech (POS) tags. All tokens with the ’NOUN’ POS tag
extracted. For normalization purposes, we consider the lemma of the tokens.
Concept drift then is calculated as the set difference between the unique
nouns found across all explanation steps and those found in the premise and
conclusion.

1

2

Input : Premise, Conclusion, Explanation, Spacy model $spacy$($\cdot$)

Output : Drift Score $drift$

3

4$Noun_{p}\leftarrow$ $spacy$($Premise$);

5 $Noun_{c}\leftarrow$ $spacy$($Conclusion$);

6 $Noun_{E}\leftarrow$ $spacy$($Explanation$);

7 $N\leftarrow\\{Noun_{p},Noun_{c},Noun_{E}\\}$;

8 $drift\leftarrow length(set(Noun_{E})-set(Noun_{p}\cup Noun_{c})$);

9 return _$drift$_ ;

Algorithm 2 Concept Drift

###  A.8 Coherence

Coherence evaluates the plausibility of the intermediate explanation. We
propose stepwise entailment as a metric to measure the entailment strength of
the If-then implications. We employ a RoBERTa-based NLI model Nie et al.
(2020) that has been finetuned on a range of NLI and fact verification
datasets consisting of SNLI Bowman et al. (2015), aNLI Nie et al. (2020),
multilingual NLI Williams et al. (2018)), and FEVER-NLI Nie et al. (2019). To
compute the stepwise entailment score, we first measure the entailment
strength between the If and Then propositions. For example, to calculate the
score of the statement _“IF a balloon is pricked, THEN the balloon may
deflate”_ we consider _“a balloon is pricked”_ and _“the balloon may deflate”_
as input sentences for the NLI model. The NLI will produce independent scores
for the entailment and contradiction labels. We compute the entailment
strength by subtracting the contraction label score from the entailment label
score. An entailment strength of one indicates the If-then implication is
strongly plausible whereas a score of zero suggests that it is likely
implausible. The overall stepwise entailment score is the average of
entailment strength measures across all explanation steps.

1

2

Input : Explanation $E$, NLI Model $nli$($\cdot$)

Output : Average Entailment Strength $strength$

3

4$EntailmentStrengthScores\leftarrow$ empty list;

5

6foreach  _Step $(If_{s},Then_{s})$ in $E$_ do

7 $EntailmentScore\leftarrow$ $nli$($If_{s}$, $Then_{s}$);

8 $ContradictionScore\leftarrow$ $nli$($If_{s}$, $Then_{s}$);

9 $EntailmentStrength\leftarrow EntailmentScore-ContradictionScore$;

10 Append $EntailmentStrength$ to $EntailmentStrengthScores$;

11

12 end foreach

13

14$strength\leftarrow$ $Avg$($EntailmentStrengthScores$);

15 return _$strength$_ ;

Algorithm 3 Stepwise Entailment

###  A.9 Linguistic Uncertainty

Linguistic uncertainty measures the confidence of a statement where hedging
cues and indirect language suggest ambiguity around the proposition. To
measure sentence-level uncertainty, we employ a finetuned RoBERTa model
provided by Pei and Jurgens (2021). The model was trained on a sentence-level
dataset consisting of findings and statements extracted from new articles and
scientific publications and human annotated evaluation of sentence certainty.
A scale from one to six was used to annotate sentences where one corresponds
to the lowest degree of certainty and six is the highest expressed by the
sentence. We invert the scale to retrieve the uncertainty scores. To compute
the overall linguistic uncertainty of an explanation, we first compute the
uncertainty for each assumption and the explanation summary and then average
all the scores.

We use a fine-tuned BERT-based token classification model to classify all the
words in the generated explanation with uncertainty categories introduced in
the 2010 CoNLL shared task on Hedge Detection Farkas et al. (2010). Farkas et
al. (2010) classify hedge cues into three categories: epistemic, doxatic, and
conditional. Epistemic cues refer to hedging scenarios where the truth value
of a proposition can be determined but is unknown in the present (e.g. the
blocks may fall). Doxatic cues refer to beliefs and hypotheses that can be
held to be true or false by others (e.g. the child believed  the blocks would
fall). Finally, conditional cues refer to propositions whose truth value is
dependent on another proposition’s truth value (e.g. if the balloon is pricked
it may deflate).

1

2

Input : Assumptions, Explanation Summary, Uncertainty Estimator Model
$uc(\cdot)$

Output : Overall Uncertainty

3

4$AssumptionUncertaintyList\leftarrow$ empty list;

5

6foreach  _Assumption in Assumptions_ do

7 $UncertaintyScore\leftarrow$ $uc$($UncertaintyModel$, $Assumption$);

8 Append $UncertaintyScore$ to $AssumptionUncertaintyList$;

9

10 end foreach

11

12$AverageAssumptionUncertainty\leftarrow$ $Avg$($AssumptionUncertaintyList$);

13

14$ExplanationUncertainty\leftarrow$ $uc$($UncertaintyModel$,
$ExplanationSummary$);

15

16$OverallExplanationUncertainty\leftarrow
AverageAssumptionUncertainty+ExplanationUncertainty$;

17

18return _$OverallExplanationUncertainty$_ ;

Algorithm 4 Linguistic Uncertainty

###  A.10 Inference to Best Explanation

To perform IBE, we first fit a linear regression model over the extracted
explanation features from the COPA train set and 500 random sample train
examples from the E-CARE train set. We consider all explanations independently
and annotate each explanation with a 1 if it corresponds to a correct answer
or 0 if corresponds to an incorrect answer. After the linear model is fitted,
we evaluate the COPA and E-CARE test sets. For each example, we use the
trained linear model to score each answer candidate explanation and then
select a candidate with the highest score. We use the linear regression
implementation from scikit-learn Buitinck et al. (2013) for the IBE model. We
additionally use the R stats package R Core Team (2013) for conducting our
regression analysis.

###  A.11 E-CARE Results

####  A.11.1 E-CARE Consistency

See Figure 13.

####  A.11.2  E-CARE Proof Depth

See Figure 14.

####  A.11.3  E-CARE Concept Drift

See Figure 14.

####  A.11.4  E-CARE Coherence

See Figure 16.

####  A.11.5  E-CARE Uncertainty

See Figure 17.

####  A.11.6  E-CARE Hedge Ratio

See Figure 18.

####  A.11.7  E-CARE Hedge Distribution

See Figure 19.

###  A.12 Causal Directionality

Figure 12: Accuracy in predicting the most plausible causes vs effects on
COPA.

When considering the causal directionality (i.e. cause vs effect), we observed
that accuracy tended to differ between LLMs on COPA. In particular, we found
both GPT and Llama 2 7B to be more accurate in predicting the effects in
causal scenarios (see Figure 12). We hypothesize that LLMs may suffer the
challenge of causal sufficiency as the space of potential causal explanations
can be far greater than the range of effects once an event has been observed.
This hypothesis is partly supported by the fact that GPT and Llama 2 7B
express greater linguistic uncertainty and produce more complex explanations
when predicting causes rather than effects.

###  A.13 Dataset Details

COPA is released under a BSD-2 license and made available for broad research
usage with copyright notification restrictions 444people.ict.usc.edu/
gordon/copa.html. We do not modify or use COPA outside of its intended use
which is primarily open-domain commonsense causal reasoning. E-CARE is
released under the MIT license and can be used for broad purposes with
copyright notification restrictions 555github.com/Waste-
Wood/e-CARE?tab=MIT-1-ov-file#readme. We do not modify or use E-CARE outside
of its intended use which is causal reasoning evaluation of language models.

Figure 13: Average consistency comparison between correct and incorrect
options for the E-CARE dataset. Figure 14: Comparison of average proof depth
between correct and incorrect options. Figure 15: Comparison of average
concept drift between correct and incorrect options. Figure 16: Comparison of
average coherence scores between correct and incorrect options. Figure 17:
Comparison of average uncertainty scores between correct and incorrect
options. Figure 18: Comparison of the average ratio of hedge cues between
correct and incorrect options. Figure 19: Distribution of hedge cues across
incorrect explanations.