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LEAVE UNSET 2021 LEAVE UNSET LEAVE UNSET Machine Learning for Health (ML4H)
2021

# Image Classification with Consistent Supporting Evidence

Peiqi Wang<EMAIL_ADDRESS>
Ruizhi Liao<EMAIL_ADDRESS>
Daniel Moyer<EMAIL_ADDRESS>
Massachusetts Institute of Technology  Cambridge  MA  USA  Seth Berkowitz
<EMAIL_ADDRESS>
Steven Horng<EMAIL_ADDRESS>
Beth Israel Deaconess Medical Center / Harvard Medical School  Boston  MA  USA
Polina Golland<EMAIL_ADDRESS>
Massachusetts Institute of Technology  Cambridge  MA  USA

###### Abstract

Adoption of machine learning models in healthcare requires end users’ trust in
the system. Models that provide additional supportive evidence for their
predictions promise to facilitate adoption. We define consistent evidence to
be both compatible and sufficient with respect to model predictions. We
propose measures of model inconsistency and regularizers that promote more
consistent evidence. We demonstrate our ideas in the context of edema severity
grading from chest radiographs. We demonstrate empirically that consistent
models provide competitive performance while supporting interpretation.

###### keywords:

Interpretability, Medical Image Analysis

##  1 Introduction

Identifying radiological findings and inferring disease stages from medical
images is common in clinical practice. Many models make predictions without
explaining the conclusion. In contrast, human experts often provide specific
explanation based on prior knowledge of human physiology to support their
image-based diagnosis. We aim to build models that are transparent in the
reasoning process, at an appropriate level of understanding consumable by end
users, e.g., clinicians. What additional information should a machine learning
model provide to gain the trust of its end users? We propose a solution
motivated by an example of how radiologists themselves operate.

fig:teaser_consistent_evidence

Figure 1: Our model provides prediction of disease stage $y$ and supporting
evidence. We use $\mathcal{I}(c)$ to denote the set of evidence labels
detected in the image that directly support disease stage $c$. We show
examples of inconsistent evidence highlighted in  red produced by the baseline
model (left column). Our proposed regularizer corrects these mistakes so that
predicted evidence label becomes compatible (top right) and sufficient (bottom
right).

Radiological findings are concepts determined as useful by radiologists.
Findings include image features, pathological states, and observations about
the underlying physiology (Gluecker et al., 1999). The radiologists aggregate
the findings to provide an overall interpretation of the image. They support
the eventual diagnosis by providing an account of the identified findings
based on prior knowledge of relationships between findings and the patient’s
physiological state. We propose and demonstrate an approach that re-
capitulates the reasoning process of domain experts. In addition to primary
predictions, the model provides supporting evidence, i.e., findings, deemed
useful by the end users.

It is critical that predictions and supporting evidence are consistent with
each other. In practice, radiologists cannot draw their conclusions based on
incompatible evidence, nor could they support their conclusions with
insufficient evidence. Similarly, end users will question the credibility of a
model when its predictions and accompanying evidence are incompatible or
insufficient.

In this paper, we build explainable models that supplement their predictions
with consistent supporting evidence, illustrated in Figure
LABEL:fig:teaser_consistent_evidence. We define measures of inconsistency
between the model’s primary output and its supporting evidence and propose
simple regularizers that encourage the classifier to be more consistent. We
demonstrate that we can train consistent models without loss in performance in
the context of pathology grading from a chest radiograph.

##  2 Related Work

###  2.1 Interpretable Machine Learning

While model interpretability is an important topic in machine learning, few
methods take the end users’ needs into account. For example, some method
localize image regions important for a prediction (Zhou et al., 2016;
Selvaraju et al., 2016), but fail to express what properties of the image
region are associated with the model output. Others aim to use a simpler model
(Caruana et al., 2015) or to approximate the behavior of a complex model with
a simpler one (Ribeiro et al., 2016). While effective for handling low
dimensional tabular data where the covariates are physically meaningful
attributes, such methods are less useful for extremely high dimensional
imaging data. Our work provide clinically meaningful supporting evidence
useful to end users of the system, rather than support the developers’
understanding of how the model reaches its decision.

Another approach is to train a classifier whose predictions rely on higher-
level concepts. Unsupervised methods can make for a more interpretable model
for general purpose tasks but cannot take advantage of strong domain knowledge
ubiquitous in healthcare (Alvarez Melis and Jaakkola, 2018). Alternatively,
concept bottleneck models that learn concepts with supervision have been
applied to arthritis grading (Koh et al., 2020), retinal disease
classification (De Fauw et al., 2018), and other applications (Losch et al.,
2019; Bucher et al., 2019). This strategy relies on the appropriate choice of
the concepts to maintain good performance. In contrast, our model predictions
do not solely rely on supporting evidence, thus avoiding the undue influence
the choice of supporting evidence or concepts might have on the accuracy of
the task label prediction.

In our work, predictions and evidence are separate outputs of the model. Most
closely related prior method focus on learning a mapping to the product space
of the task label and supporting evidence (Hind et al., 2019; Codella et al.,
2019), with application to text classification (Zaidan et al., 2007; Zhang et
al., 2016). In contrast, we learn a structured output where known
relationships between predictions and evidence are enforced. We inject domain
specific knowledge and require our model to provide supporting evidence that
is clinically feasible under a specific prediction.

A recently demonstrated unsupervised strategy that requires a forward model
that relates supporting evidence to a subset of the input features is also
relevant (Raghu et al., 2021). This method can be difficult to implement in
our radiograph grading task as it assumes knowledge of an accurate forward
model, from supporting evidence to a high dimensional image, which is
infeasible for most medical imaging problems.

###  2.2 Logical Constraints

There are multiple ways to represent symbolic constraints. For example, trees
have been used to express subsumption relationships between attributes, e.g.,
hierarchical annotation of medical images (Dimitrovski et al., 2011).
Hierarchical multi-label learning aims to enforce such constraints (Bi and
Kwok, 2011; Yan et al., 2015; Wehrmann et al., 2018; Giunchiglia and
Lukasiewicz, 2020). Unfortunately, trees are overly restrictive and cannot
express consistency constraints that are important in our application.

Alternatively, boolean statements can be quite expressive in representing
logical constraints. Logical constraints on model outputs can be enforced by
replacing logical operators with their subdifferentiable fuzzy t-norms
(Diligenti et al., 2017; Li et al., 2019) or through the use of specialized
loss functions (Xu et al., 2018). Our approach to representing and enforcing
logical constraints is both easier to interpret and simpler to implement.
Moreover, we enforce logical constraints by making changes to how our model
provides supporting evidence while keeping main task predictions unchanged.

##  3 Method

In this section, we define consistency of evidence and introduce measures of
inconsistency. We provide an example application to ground our definitions. We
construct novel loss functions that encourage consistency and discuss
optimization that arises when training classifiers with consistent supporting
evidence.

###  3.1 Problem Setup

Let $x,y$, and $z=(z_{1},\ldots,z_{K})$ be random variables representing an
image, a $C$-class task label, and $K$ binary evidence labels. Let
$\mathcal{D}_{t}$ be a data set that includes pairs $(x,y)$ and $(x,z_{k})$
for $k=1,\ldots,K$. In this work, we do not assume full tuples
$(x,y,z_{1},\ldots,z_{K})$ are available. Learning joint predictors from
available sub-tuples is an interesting direction that is outside the scope of
this paper. Moreover, we allow the same image $x$ to be included as part of
several different pairs in the data set $\mathcal{D}_{t}$.

We use the training data set $\mathcal{D}_{t}$ to build probabilistic
classifiers $p(y\mid x;\theta)$ and $p(z_{k}\mid x;\theta)$ for
$k=1,\ldots,K$. The maximum a posteriori (MAP) estimates of the task label $y$
and evidence labels $z$ is obtained via

$\displaystyle\hat{y}$
$\displaystyle=\,\,\operatorname*{arg\,max}_{c\in\left[C\right]}\,\,\,p(y=c\mid
x;\theta),$ (1) $\displaystyle\hat{z}_{k}$
$\displaystyle=\operatorname*{arg\,max}_{z\in\left\\{-1,+1\right\\}}p(z_{k}=z\mid
x;\theta),\quad k=1,\ldots,K.$ (2)

We use $\hat{y}(x)$ to express explicit dependence of the predicted label
$\hat{y}$ on the input image $x$.

###  3.2 Consistent Evidence

We assume that domain experts provide domain specific knowledge in the form of
logical constraints between the task label $y$ and the evidence labels $z$. We
identify two major logical constraints that are important in our application,
specifically that supporting evidence should be compatible and sufficient with
the task label.

Let $\mathcal{I}_{1}:\left[C\right]\to\mathcal{P}(\left[K\right])$ be the
indexing function for evidence that is incompatible with a particular value of
task label, where we use $\mathcal{P}(\cdot)$ to denote the power set.
Specifically, if evidence labels $\left\\{z_{i_{1}},\ldots,z_{i_{M}}\right\\}$
are incompatible with task label $y=c$, then
$\mathcal{I}_{1}(c)=\left\\{i_{1},\ldots,i_{M}\right\\}$. Let
$\mathcal{I}_{2}:\left[C\right]\to\mathcal{P}(\left[K\right])$ be the indexing
function for evidence that directly supports a particular value of task label.
We assume that $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are provided by domain
experts.

######  Definition 3.1.

(Consistent Evidence) The task label $y\in\left[C\right]$ and the evidence
label vector $z=(z_{1},\ldots,z_{K})\in\\{-1,+1\\}^{K}$ are consistent if

$\displaystyle\forall k\in\mathcal{I}_{1}(y):\;z_{k}=-1,$ (3)
$\displaystyle\exists k\in\mathcal{I}_{2}(y):\;z_{k}=+1.$ (4)

The first criterion specifies that no evidence is incompatible with the task
label $y$. The second criterion specifies that there should be at least one
direct evidence label present that supports the task label $y$.

In reality, perfectly consistent evidence may not be necessary or possible.
For example, domain experts often specify constraints with a notion of
uncertainty, e.g., $\mathcal{I}_{1}(y)$ is incompatible with $y$ most of the
time except for occasional corner cases. In addition, certain direct evidence
might be so rare that it becomes impossible to include it in
$\mathcal{I}_{2}(y)$. Therefore, it is perfectly sensible that there is no
direct evidence present in some cases, if we have not included the
corresponding evidence label in the construction. This motivates us to
consider these constraints in probabilistic terms.

Definition 3.1 is a specification over the values that random variables can
take. The same definition applies to the true data distribution
$\left\\{y,z\right\\}$ and to the predicted distribution
$\left\\{\hat{y},\hat{z}\right\\}$. In practice, we construct training data
$\left\\{y,z\right\\}$ to be perfectly consistent and demonstrate a training
method that encourages the model outputs $\left\\{\hat{y},\hat{z}\right\\}$ to
be consistent as well.

Moreover, we are not restricted to predicting the findings in
$\cup_{c\in\left[C\right]}\mathcal{I}_{2}(c)$. If some findings provide useful
information but are not directly supportive, they can still be included in the
set of evidence labels.

###  3.3 Example Application

This section illustrates the construction of indexing functions
$\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ for the pulmonary edema grading task
that motivated our work.

Pulmonary edema is defined as an abnormal accumulation of fluid in the lungs.
Higher hydrostatic pressure in the vasculature causes more severe symptoms.
Typically, radiologists grade the severity of edema based on findings that are
typical of the most severe stage of pulmonary edema (Gluecker et al., 1999).

We use a categorization that identifies four edema severity levels, in order
of increasing severity: no edema (0), mild edema (1), moderate edema (2), and
severe edema (3) (Liao et al., 2019; Horng et al., 2021). The edema severity
grading task involves assigning a severity level
$y\in\left\\{0,1,2,3\right\\}$ to a test image. In this task, there are 4
classes, i.e., $C=4$.

tab:severity_evidence  Severity $y$ Findings $\mathcal{I}_{2}(y)$ 0 (none) - 1
(mild) vascular congestion hilar congestion peribronchial cuffing 2 (moderate)
septal lines interstitial abnormality 3 (severe) air bronchograms parenchymal
opacity

Table 1: Findings that directly support a particular severity level.

tab:example_consistent  Severity $y$ Evidence $z$ 1 peribronchial cuffing 2
vascular congestion septal lines interstitial abnormality

Table 2: Examples of consistent evidence.

tab:example_inconsistent  Severity $y$ Evidence $z$ 1 hilar congestion septal
lines 2 vascular congestion interstitial abnormality air bronchograms 1 – 3
septal lines

Table 3: Examples of inconsistent evidence. First two examples are
incompatible. The latter two examples are insufficient.

In our work, we identify $K=7$ supporting evidence labels deemed useful by
clinicians, as shown in Table LABEL:tab:severity_evidence. They are canonical
radiological manifestation of the underlying pathology. End users expect
presence of these findings to be indicative of a specific edema severity
level.

As an example, radiologists grade an image as moderate edema if they observe
septal lines (short parallel lines at the periphery of the lung) or
interstitial abnormality (excess fluids in the supporting tissue within the
lung). Note that presence of evidence from a lower value of edema severity is
not inconsistent. For example, radiologists may at the same time observe
presence of vascular congestion (enlargement of pulmonary veins) and septal
lines in a moderate edema case.

In the severity grading task, we consider an evidence label as incompatible if
its presence directly supports a higher level severity level. Thus define

$\displaystyle\mathcal{I}_{1}(c)=\bigcup_{c^{\prime}>c}\,\mathcal{I}_{2}(c^{\prime}).$
(5)

As an example, a model that grades an image as moderate edema should not use
air bronchograms (opacification of alveoli) as supporting evidence.

We consider evidence as insufficient when no direct evidence for edema
severity grading is present. As an example, a model which grades an image as
severe edema cannot rely on septal lines only to support its prediction.

Tables LABEL:tab:example_consistent and LABEL:tab:example_inconsistent
illustrate further examples of consistent and inconsistent evidence,
respectively.

###  3.4 Measuring Inconsistency

We quantify the inconsistency probabilistically based on Definition 3.1.
First, we define a measure of incompatibility as the probability that there is
an incompatible evidence label

$\displaystyle\mathbbm{P}\left[\bigcup_{k\in\mathcal{I}_{1}(y)}\left\\{z_{k}=+1\right\\}\right].$
(6)

To facilitate computation, we upper bound this probability using union bound
by

$\displaystyle\mathcal{R}_{1}(y,z)=\sum_{k\in\mathcal{I}_{1}(y)}\mathbbm{P}\left[z_{k}=+1\right].$
(7)

We provide an estimate of incompatibility over data set $\mathcal{D}$ by
taking expectation over its empirical distribution

$\displaystyle\mathcal{R}_{1}(\mathcal{D})$
$\displaystyle=\mathbb{E}_{(y,z)\sim\mathcal{D}}\left[\sum_{k\in\mathcal{I}_{1}(y)}\mathbbm{1}\left[z_{k}=+1\right]\right],$
(8)

where we have replaced $\mathbbm{P}\left[z_{k}=+1\right]$ with
$\mathbbm{1}\left[z_{k}=+1\right]$ since $z_{k}$ is binary valued.
Intuitively, $\mathcal{R}_{1}(\mathcal{D})$ is the average count of evidence
labels incompatible with the task label.

In this work, we use Equation 8 as a measure of incompatible evidence. We also
note that one could eliminate the dependency on the size of
$\mathcal{I}_{1}(y)$ by defining

$\displaystyle\mathbb{E}_{(y,z)\sim\mathcal{D}}\left[\frac{1}{\left|\mathcal{I}_{1}(y)\right|}\sum_{k\in\mathcal{I}_{1}(y)}\mathbbm{1}\left[z_{k}=+1\right]\right].$
(9)

We prefer the measure of incompatibility defined in Equation 8 for measuring
incompatibility as it arises naturally from bounding the probability of
incompatible evidence.

Similarly, we define a measure of insufficiency as the probability that there
is no sufficient evidence

$\displaystyle\mathbbm{P}\left[\bigcap_{k\in\mathcal{I}_{2}(y)}\left\\{z_{k}=-1\right\\}\right],$
(10)

which leads to an upper bound

$\displaystyle\mathcal{R}_{2}(y,z)$
$\displaystyle=\min_{k\in\mathcal{I}_{2}(y)}\mathbbm{P}\left[z_{k}=-1\right]$
(11)

and its empirical estimate

$\displaystyle\mathcal{R}_{2}(\mathcal{D})$
$\displaystyle=\mathbb{E}_{(y,z)\sim\mathcal{D}}\left[\min_{k\in\mathcal{I}_{2}(y)}\left[1-\mathbbm{1}\left[z_{k}=+1\right]\right]\right]$
(12)
$\displaystyle=1-\mathbb{E}_{(y,z)\sim\mathcal{D}}\left[\max_{k\in\mathcal{I}_{2}(y)}\mathbbm{1}\left[z_{k}=+1\right]\right].$
(13)

Note that $\mathcal{R}_{2}(\mathcal{D})$ is the average count of absence of
direct evidence.

Now we can provide an upper bound on probability of inconsistent evidence
$\mathcal{R}(y,z)=\mathcal{R}_{1}(y,z)+\mathcal{R}_{2}(y,z)$ and its empirical
estimate
$\mathcal{R}(\mathcal{D})=\mathcal{R}_{1}(\mathcal{D})+\mathcal{R}_{2}(\mathcal{D})$.

###  3.5 Consistency Regularization

Models trained naively to predict labels $y$ and $z$ jointly are not
guaranteed to be consistent. Here, we provide regularizers that encourage
supporting evidence to be more consistent.

Observe that Equations 7 and 11 are upper bounds on the true probability of
model being inconsistent. We can simply use these upper bounds, or
modification thereof, as regularizers. We opt to use cross entropy to avoid
inconsistent evidence.

To penalize incompatibility, we define

$\displaystyle\mathcal{R}_{1}(\theta)$
$\displaystyle=-\mathbb{E}_{x\sim\mathcal{D}}\left[\sum_{k\in\mathcal{I}_{1}(\hat{y}(x))}\ln
p(z_{k}=-1\mid x;\theta)\right].$ (14)

Intuitively, $\mathcal{R}_{1}(\theta)$ penalizes evidence probability that is
incompatible with the predicted task label. Including
$\mathcal{R}_{1}(\theta)$ in the loss function is equivalent to supplying
pseudo negative samples for evidence obtained from the predicted task label
$\hat{y}$.

Instead of penalizing incompatibility with respect to MAP estimate of the task
label $\hat{y}(x)$, we can penalize incompatibility for each value of task
label weighted by the posterior probability, i.e.,

$\displaystyle\widetilde{\mathcal{R}}_{1}(\theta)=-\mathbb{E}_{x\sim\mathcal{D}}\Bigg{[}\Bigg{.}\sum_{c\in[C]}\sum_{k\in\mathcal{I}_{1}(c)}$
$\displaystyle p(y=c\mid x;\theta)\,\cdot$ $\displaystyle\ln p(z_{k}=-1\mid
x;\theta)\Bigg{.}\Bigg{]}.$ (15)

In contrast to Equation 14 where gradients cannot flow through $\hat{y}(x)$
due to the $\operatorname*{arg\,max}$ operator, Equation 15 provides a softer
regularizer that affects the predictions of both the task and the evidence
labels.

Similarly, we define

$\displaystyle\mathcal{R}_{2}(\theta)$
$\displaystyle=-\mathbb{E}_{x\sim\mathcal{D}}\left[\ln\underset{k\in\mathcal{I}_{2}(\hat{y}(x))}{\max}p(z_{k}=+1\mid
x;\theta))\right].$ (16)

Intuitively, $\mathcal{R}_{2}(\theta)$ encourages presence of some evidence to
support predicted task label. Including $\mathcal{R}_{2}(\theta)$ in the loss
function is equivalent to supplying pseudo positive samples obtained from the
predicted task label $\hat{y}$.

Similar to Equation 15, we can penalize insufficiency using posterior
probability as weights,

$\displaystyle\widetilde{\mathcal{R}}_{2}(\theta)=-\mathbb{E}_{x\sim\mathcal{D}}\Bigg{[}\Bigg{.}\sum_{c\in[C]}$
$\displaystyle p(y=c\mid x;\theta)\,\cdot$
$\displaystyle\ln\underset{k\in\mathcal{I}_{2}(c)}{\max}p(z_{k}=+1\mid
x;\theta))\Bigg{.}\Bigg{]}.$ (17)

In our work, we focus on regularizers $\mathcal{R}_{1}(\theta)$ and
$\mathcal{R}_{2}(\theta)$. In Appendix B, we provide preliminary comparison of
regularizers $\mathcal{R}_{1}(\theta)$ and $\mathcal{R}_{2}(\theta)$ with the
soft regularizers $\widetilde{\mathcal{R}}_{1}(\theta)$ and
$\widetilde{\mathcal{R}}_{2}(\theta)$. We leave further investigation of soft
regularizers to future work.

###  3.6 Optimization

We apply deep multitask learning for joint predictions of
$y,z_{1},\cdots,z_{K}$. In particular, we parameterize $p(y\mid x;\theta)$ and
$p(z_{k}\mid x;\theta)$ for $k=1,\ldots,K$ with neural network $f(x;\theta)$
and assume function $f$ outputs logits over $K+1$ marginals.

Given a classification loss function $L(\cdot,\cdot)$, the objective is simply
the empirical risk,

$\displaystyle\mathcal{L}(\theta)$
$\displaystyle=\mathbb{E}_{(x,y)\sim\mathcal{D}_{t}}\left[L(y,f(x;\theta))\right]$
(18)
$\displaystyle+\frac{1}{K}\sum_{k=1}^{K}\mathbb{E}_{(x,z_{k})\sim\mathcal{D}_{t}}\left[L(z_{k},f(x;\theta))\right].$
(19)

We add consistency regularization to multitask classification loss, which
yields a regularized empirical risk minimization problem

$\displaystyle\min_{\theta}\,\mathcal{L}(\theta)+\omega_{1}\mathcal{R}_{1}(\theta)+\omega_{2}\mathcal{R}_{2}(\theta),$
(20)

where $\omega_{1},\omega_{2}\in\mathbb{R}^{+}$ are coefficients that control
the degree of regularization.

fig:results_ic_v3_measure_inconsistency

Figure 2: Model inconsistency across different values of $\hat{y}$ evaluated
on the test set $\hat{\mathcal{D}}$ for a naively trained model, i.e.,
$\omega_{1}=\omega_{2}=0$. The majority of inconsistent evidence comes from
incompatible evidence associated with small values of the predicted task label
$\hat{y}$.

fig:consistentreg_vary_regularization_strength [Vary $\omega_{1}$, fix
$\omega_{2}=0$] [Fix $\omega_{1}=0$, vary $\omega_{2}$] [Vary
$\omega_{1}=\omega_{2}$ together]

Figure 3: The effect of varying strength of regularizations during training on
model consistency. Here $\mathcal{R}_{1},\mathcal{R}_{2}$ is short hand for
$\mathcal{R}_{1}(\hat{\mathcal{D}}),\mathcal{R}_{2}(\hat{\mathcal{D}})$,
respectively. The proposed regularizers
$\mathcal{R}_{1}(\theta),\mathcal{R}_{2}(\theta)$ encourage the model to
provide more compatible evidence (left) or more sufficient evidence (middle),
respectively. The application of regularizers at the same time encourages the
model to provide more consistent evidence (right).

##  4 Implementation Details

We use residual networks to parameterize our probabilistic classifiers (He et
al., 2016). The network is modified to output a $(C+K)$-dimensional vector
representing the posterior marginal probabilities for $y,z_{1},\ldots,z_{K}$.

We use weighted cross entropy loss as $L(\cdot,\cdot)$ to handle class
imbalances. We employ the Adam optimizer with a constant learning rate of
$2\cdot 10^{-4}$ with mini-batch size of 32 for stochastic optimization of
network parameters (Kingma and Ba, 2015). Each gradient update involves random
sampling of a label (task or evidence), assembling a mini-batch of data
corresponding to the sampled label, computing the objective function, and
updating parameters with backpropagated gradients. This approach enables us to
learn even if some labels are missing for some images.

We normalize images to zero mean and unit variance, and resize them to 224x224
pixels. We apply random image augmentations to images, e.g., crop, horizontal
flip, brightness and contrast variations, to alleviate model overfitting.

We implement Equation 16 by substituting the $\max$ operator with a soft
maximum operator, i.e., $\text{LSE}_{i\in[n]}(x_{i})=\log\sum_{i}\exp(x_{i})$.
This way, we enforce sufficiency of evidence by upscaling probabilities of
direct evidence that are larger to begin with.

We use exactly the same network architecture, data augmentation, and
optimization parameters to isolate the impact of the proposed regularization
on consistency and performance.

We compute mean and standard deviation statistics for inconsistency and test
prediction from 3 runs with different random seed.

##  5 Experiments

###  5.1 Data

We use a subset of 238,086 frontal-view chest X-ray from the MIMIC-CXR data
set (Johnson et al., 2019). We split the data set into training (217,016),
validation (10,445), and test (10,625) sets randomly. The performance of
predicted evidence is computed over this test set. There is no patient overlap
between training, validation and test sets.

Edema severity labels are extracted from associated reports by searching for
keywords that are indicative of a specific disease stage. The 7,802 labeled
image/report pairs are split into training (6,656), validation (648), and test
(498) set. The test set was corrected for keyword matching errors by an expert
radiologist, as detailed in prior work (Chauhan et al., 2020). We use
$\hat{\mathcal{D}}$ to denote this test set that includes images and predicted
labels $(x,\hat{y},\hat{z})$. All subsequent evaluations of model consistency
and performance is computed using $\hat{\mathcal{D}}$.

###  5.2 Model Inconsistency

We examine model inconsistency overall and over partitions of data with
respect to values of predicted label $\hat{y}$. The sum of model inconsistency
over the partitions gives the quantities $\mathcal{R}_{1}(\hat{\mathcal{D}})$
in Equation 8 and $\mathcal{R}_{2}(\hat{\mathcal{D}})$ in Equation 13.

Figure LABEL:fig:results_ic_v3_measure_inconsistency reports model
inconsistency over partitions of $\hat{y}$ for a model that is trained without
consistency regularization, i.e., $\omega_{1}=\omega_{2}=0$. We observe that
$\mathcal{R}_{1}(\hat{\mathcal{D}})$ is typically larger than
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ due to the fact that
$\mathcal{R}_{1}(\hat{\mathcal{D}})$ is essentially an average count of
potentially many incorrect evidence labels, while
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ is an average count of missing evidence,
and therefore is upper bounded by 1. We also observe a downward trend in
values of $\mathcal{R}_{1}(\hat{\mathcal{D}})$ with increasing values for
$\hat{y}$. This is reasonable as there are many ways to make mistake with a
small $\hat{y}$, while no way to provide conflicting evidence when $\hat{y}=3$
in our framework.

fig:test_image_with_evidence

Figure 4: Correctly and incorrectly classified test images with supporting
evidence given by a consistent model ($\omega_{1}=\omega_{2}=10.$). We use
$\mathcal{I}(c)$ to denote the set of evidence labels detected in the image
that directly support disease stage $c$.

###  5.3 Consistency Regularization

To demonstrate that proposed regularization promotes model consistency, we
vary values of $\omega_{1},\omega_{2}$ in the objective function and train
multiple models. We select the most accurate model on the validation set and
compute inconsistency on the test set $\hat{\mathcal{D}}$.

Figure LABEL:fig:consistentreg_vary_regularization_strength demonstrates the
effects of regularization on model consistency. We observe that the
regularizers $\mathcal{R}_{1}(\theta)$ and $\mathcal{R}_{2}(\theta)$ are
effective in reducing the respective intended model inconsistency, indicated
by a reduction of $\mathcal{R}_{1}(\hat{\mathcal{D}})$ in Figure 3 and
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ in Figure 3 respectively. Additionally,
we observe that penalizing $\mathcal{R}_{1}(\hat{\mathcal{D}})$ inadvertently
makes $\mathcal{R}_{2}(\hat{\mathcal{D}})$ larger and vice versa. This makes
intuitive sense, since a model that is more likely to predicts absence of
evidence will (i) less likely to provide incompatible evidence and (ii) less
likely to provide some direct evidence. We observe that we can reduce both
types of inconsistency by regularizing with both loss terms, as shown in
Figure 3.

It important to note that even though $\mathcal{R}_{2}(\hat{\mathcal{D}})$ is
relatively small in models trained with $\omega_{1}=0$, regularizing with
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ is necessary as we want to avoid
situations in Figure 3 where $\mathcal{R}_{2}(\hat{\mathcal{D}})$ becomes
intolerably large.

###  5.4 Interpretability

fig:perf_consist_tradeoff

Figure 5: The effect of regularization on model inconsistency (left 2) and
performance of predicting task label $y$ (middle 2) and evidence labels $z$
(right 2). Here $\mathcal{R}_{1},\mathcal{R}_{2}$ is short hand for
$\mathcal{R}_{1}(\hat{\mathcal{D}}),\mathcal{R}_{2}(\hat{\mathcal{D}})$,
respectively. When regularizing for both $\mathcal{R}_{1},\mathcal{R}_{2}$,
e.g., down the diagonals of the matrix, we notice dramatic decrease in model
inconsistency and competitive performance for the regularized model.

Figure LABEL:fig:test_image_with_evidence illustrates how a consistent model
(trained with $\omega_{1}=\omega_{2}=10$) provides supporting evidence for
randomly sampled test images. We provide correctly and incorrectly classified
test images for each severity level. We observe that the regularized model
provides consistent evidence in all 8 examples, even in cases where model
prediction of task label is not correct.

How does providing consistent supporting evidence build trust in the model ?
We note that supporting findings are already described in radiological reports
and can be easily mined for training and verified in an image. When the
supporting evidence is clearly correct, it builds additional trust in the
predicted task label. When the consistent but wrong evidence is presented, it
is easy to see in the image and helps the end users understand why the main
task label is wrong. Our method avoid confusions that arise from model
providing inconsistent evidence.

Crucially, evidence labels should not only be consistent, but also correct. To
this point, we reported performance of evidence detection in Section 5.5. Our
proposed regularizers offers a complementary tool to help end users understand
why the model erred. Our method can be integrated with technologies, e.g.,
RCNN, GradCAM (Girshick et al., 2014; Selvaraju et al., 2016), that provide
localization, i.e., confirmation that the model is focusing on the correct
regions in the image.

###  5.5 Performance-Consistency Tradeoff

Next, we show that we can achieve good model consistency without compromising
predictive performance. We vary $\omega_{1},\omega_{2}$ together in the
objective function and train multiple models. We select for the most accurate
model on the validation set for subsequent evaluations.

Figure LABEL:fig:perf_consist_tradeoff demonstrates that we can ensure
satisfactory model consistency. At the same time, the regularized model
achieves similar performance on the severity grading task. The improvement in
performance can be attributed empirically to fact that heavily regularized
models over-fit less. However, we do observe a drop in the average performance
of the model for predicting evidence. The drop in predicted evidence
performance is tolerable if we consider that the model rarely provides
inconsistent supporting evidence.

Figure LABEL:fig:perf_consist_tradeoff reinforces previous observation that
penalizing $\mathcal{R}_{1}(\hat{\mathcal{D}})$ makes
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ higher and vice versa, when $\omega_{i}$
for $i=1,2$ being held constant is different from 0. Figure
LABEL:fig:consistentreg_vary_regularization_strength shows the first column,
the first row, and the diagonal slices of the grid in left 2 sub-figures in
Figure LABEL:fig:perf_consist_tradeoff. We refer the reader to Table
LABEL:tab:consistency_performance_tradeoff in Appendix A for detailed numbers
of inconsistency and performance along the diagonal slice of the grid.

##  6 Conclusion

We argue for supplementing model predictions with supporting evidence that is
deemed useful by end users. We defined a notion of consistent evidence via
incorporating domain specific constraints. Then, we proposed ways to measure
and enforce such constraints during model training. We evaluated our method on
the pulmonary edema severity grading task, which provides a grounding for our
consistent evidence framework. We demonstrated that consistent models remain
competitive on the main task.

This work was supported in part by NIH NIBIB NAC P41EB015902 grant, MIT
Lincoln Laboratory, MIT JClinic, MIT Deshpande Center, and Philips.

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##  Appendix A Table for Figure 5.

Table LABEL:tab:consistency_performance_tradeoff provides detailed numbers on
inconsistency and performance.

tab:consistency_performance_tradeoff  $\omega_{1},\omega_{2}$ * 0.0,0.0
3.0,3.0 10.0,6.0 30.0,10.0 $\mathcal{R}_{1}(\hat{\mathcal{D}})$ -
$0.507\pm\text{\footnotesize 0.084}$ $0.277\pm\text{\footnotesize 0.103}$
$0.218\pm\text{\footnotesize 0.205}$ $0.112\pm\text{\footnotesize 0.049}$
$\mathcal{R}_{2}(\hat{\mathcal{D}})$ - $0.048\pm\text{\footnotesize 0.011}$
$0.023\pm\text{\footnotesize 0.019}$ $0.030\pm\text{\footnotesize 0.023}$
$0.019\pm\text{\footnotesize 0.007}$ acc ($y$) $0.524\pm\text{\footnotesize
0.017}$ $0.524\pm\text{\footnotesize 0.005}$ $0.523\pm\text{\footnotesize
0.008}$ $0.512\pm\text{\footnotesize 0.014}$ $0.499\pm\text{\footnotesize
0.022}$ auc ($y$) $0.836\pm\text{\footnotesize 0.008}$
$0.852\pm\text{\footnotesize 0.004}$ $0.865\pm\text{\footnotesize 0.004}$
$0.862\pm\text{\footnotesize 0.002}$ $0.846\pm\text{\footnotesize 0.006}$ acc
(vascular congestion) - $0.795\pm\text{\footnotesize 0.018}$
$0.793\pm\text{\footnotesize 0.026}$ $0.777\pm\text{\footnotesize 0.005}$
$0.763\pm\text{\footnotesize 0.015}$ acc (hilar congestion) -
$0.771\pm\text{\footnotesize 0.047}$ $0.801\pm\text{\footnotesize 0.031}$
$0.759\pm\text{\footnotesize 0.016}$ $0.679\pm\text{\footnotesize 0.040}$ acc
(peribronchial cuffing) - $0.804\pm\text{\footnotesize 0.041}$
$0.802\pm\text{\footnotesize 0.032}$ $0.795\pm\text{\footnotesize 0.026}$
$0.789\pm\text{\footnotesize 0.040}$ acc (septal lines) -
$0.869\pm\text{\footnotesize 0.010}$ $0.859\pm\text{\footnotesize 0.043}$
$0.796\pm\text{\footnotesize 0.037}$ $0.845\pm\text{\footnotesize 0.023}$ acc
(interstitial abnormality) - $0.649\pm\text{\footnotesize 0.011}$
$0.647\pm\text{\footnotesize 0.002}$ $0.630\pm\text{\footnotesize 0.014}$
$0.606\pm\text{\footnotesize 0.006}$ acc (air bronchograms) -
$0.863\pm\text{\footnotesize 0.019}$ $0.881\pm\text{\footnotesize 0.030}$
$0.888\pm\text{\footnotesize 0.015}$ $0.860\pm\text{\footnotesize 0.018}$ acc
(parenchymal opacity) - $0.706\pm\text{\footnotesize 0.018}$
$0.764\pm\text{\footnotesize 0.020}$ $0.756\pm\text{\footnotesize 0.009}$
$0.738\pm\text{\footnotesize 0.009}$

Table 4: The effect of regularization on model inconsistency and performance
of predicting task label $y$ and evidence label $z$. $*$ stands for baseline
model that is trained to predict task label $y$. Standard deviation is shown
after $\pm$. We notice dramatic decrease in model inconsistency and
competitive performance for the regularized model.

##  Appendix B Training with Soft Regularizer

We compare behavior of model trained using hard regularizers
$\mathcal{R}_{1}(\theta),\mathcal{R}_{2}(\theta)$ versus the soft regularizers
$\tilde{\mathcal{R}}_{1}(\theta),\tilde{\mathcal{R}}_{2}(\theta)$.

In Figure LABEL:fig:perf_consist_tradeoff_compare_hard_soft, we note that the
soft regularizers reduce model inconsistency while maintain model consistency
in a similar manner to hard regularizers. Unlike those trained with hard
regularizers, models trained with soft regularizers seem to avoid the decrease
in performance for evidence detection.

fig:perf_consist_tradeoff_compare_hard_soft

Figure 6: The effect of regularization on model inconsistency (left 2) and
performance of predicting task label $y$ (middle 2) and evidence labels $z$
(right 2), when model is trained using the hard regularizers
$\mathcal{R}_{1}(\theta),\mathcal{R}_{2}(\theta)$ (top) versus the soft
regularizers $\tilde{\mathcal{R}}_{1}(\theta),\tilde{\mathcal{R}}_{2}(\theta)$
(bottom). Here $\mathcal{R}_{1},\mathcal{R}_{2}$ is short hand for
$\mathcal{R}_{1}(\hat{\mathcal{D}}),\mathcal{R}_{2}(\hat{\mathcal{D}})$,
respectively.