$\operatorname{hocolim}(\operatorname{Sing}(T_{i}))\to\operatorname{Sing}(\operatorname{hocolim}(T_{i}))=\operatorname{Sing}(T)$
and $\operatorname{hocolim}(\operatorname{Sing}(T_{i}))\to F$.

######  Lemma 6.16.

Let $F_{*}\to F$ be a dg-Segal hypercover in $\mathbf{dg-Segal}_{\mathcal{O}}$
with $\mathcal{O}=\pi_{0}(F(k))$ constructed as before. Then, for all
$x,y\in\mathcal{O}$, the augmented simplicial complex $T_{*}(x,y)\to F(x,y)$
is a split hypercover in $\mathbf{C(k)}$.

###### Proof.

This follows from the definition of dg-Segal hypercover.
∎

######  Proposition 6.17.

Let $F_{*}\to F$ be a dg-Segal hypercover constructed as in Theorem 6.14. Then
the homotopy colimit of $F_{*}$ is DK-equivalent to $F$ in $\mathbf{dg-
Segal}_{\mathcal{O}}$.

###### Proof.

The proof of this follows very closely the one in Lemma 6.5. Indeed, in order
to prove something is a DK-equivalence, we have two conditions: firstly, that
the morphism

$\left[\operatorname{hocolim}(F_{i})\right]\to\left[F\right]$

is an essentially surjective. But the hypercover has been constructed to have
fixed objects $\pi_{0}(F(k))$, so this condition is verified by construction.

That leaves us with the second condition. We fix $x,y\in\mathcal{O}$. Do we
have that

$(\operatorname{hocolim}F_{i})_{(x,y)}\to F_{(x,y)}$

is a quasi-equivalence? By definition,
$F_{i,(x,y)}=\operatorname{Map}(-,T_{i}(x,y))$ and
$F_{(x,y)}=\operatorname{Map}(-,F(x,y))$, so we can rewrite this condition as
wondering whether

$\operatorname{hocolim}\operatorname{Map}(-,T_{i}(x,y))\to\operatorname{Map}(-,F(x,y))$

is a quasi-equivalence. By Lemma 6.16, we know that $T_{*}(x,y)\to F(x,y)$ is
a split hypercover of complexes, so for all $n\in\mathbb{N}$, $T_{n}(x,y)\to
T_{*}^{\partial\Delta^{n}}(x,y)$ is a split epimorphism, and in consequence
for all $E\in\mathbf{C(k)}$ the morphism

$\pi_{0}(\operatorname{Map}(E,T_{n}(x,y)))\to\pi_{0}(\operatorname{Map}(E,T_{*}(x,y)^{\partial\Delta^{n}}))\simeq\pi_{0}(\operatorname{Map}(E,T_{*}(x,y))^{\partial\Delta^{n}})$

is an epimorphism. That means that
$\operatorname{Map}(E,T_{*}(x,y))\to\operatorname{Map}(E,F(x,y))$ is a
hypercover of simplicial sets, and by Proposition 5.2, we have a quasi-
equivalence

$\operatorname{hocolim}\operatorname{Map}(-,T_{i}(x,y))\simeq\operatorname{Map}(-,F(x,y)).$

So the morphism $\operatorname{hocolim}(F_{i})\to F$ is a DK-equivalence and
we have finished our proof.
∎

Like in the case of the dg-categories, we have proved that the homotopy
colimit is DK-equivalent to $F$, but only with fixed objects. But if we want
to use it, we need to see that it is indeed true in $\mathbf{dg-Segal}$.

######  Proposition 6.18.

Let $F$ be a dg-Segal space and $F_{*}\to F$ a dg-Segal hypercover constructed
as in Theorem 6.14. Then the homotopy colimit of $F_{*}$ is DK-equivalent to
$F$ in $\mathbf{dg-Segal}$.

###### Proof.

This proof is almost identical to the one in Proposition 5.14, and as such we
won’t spend too much time on its details. As was the case there, we have that
the homotopy colimit of the hypercover is DK-equivalent to $F$, but only with
fixed objects. So we’re going to prove that the forgetful functor
$\Phi:\mathbf{dg-Segal}_{\mathcal{O}}\to\mathbf{dg-Segal}$ commutes with those
homotopy colimits. Let us denote the obvious forgetful functor by
$\Xi:\mathbf{dg-Segal}_{\mathcal{O}}\to\coprod\underline{k}/\mathbf{dg-
Segal}$.

For all $G\in\mathbf{dg-Segal}$ we then have a morphism of the form

$\operatorname{Map}(\Phi(F),G)\to\operatorname{Map}(\coprod\underline{k},G)\simeq\prod\operatorname{Map}(\underline{k},G).$

By getting the fiber of this morphism and doing the same with
$\operatorname{hocolim}F_{i}$, we get the following diagram:

${\operatorname{Map}(F,G)}$${\operatorname{Map}(\Xi(F),G)}$${\operatorname{Map}(\Phi(F),G)}$${\prod\operatorname{Map}(\underline{k},G)}$${\operatorname{holim}\operatorname{Map}(F_{i},G)}$${\operatorname{holim}\operatorname{Map}(\Xi(F_{i}),G)}$${\operatorname{holim}\operatorname{Map}(\Phi(F_{i}),G)}$${\prod\operatorname{Map}(\underline{k},G)}$$\scriptstyle{\sim}$$\scriptstyle{f}$$\scriptstyle{=}$

If we can prove that $f$ is a weak equivalence, we have finished. For that,
let us prove that $\Xi$ is fully faithful.

We have an adjunction $\Xi:\operatorname{Ho}(\mathbf{dg-
Segal}_{\mathcal{O}})\rightleftharpoons\operatorname{Ho}(\coprod\underline{k}/\mathbf{dg-
Segal}):\Gamma$ where the right adjoint $\Gamma$ is such that for all
$H\in\coprod\underline{k}/\mathbf{dg-Segal}$ and $L\in
c\mathcal{L}_{\mathbb{S}}$, $\Gamma(H)(L)$ is the following homotopy pullback

${\Gamma(F)(L)}$${F(L)}$${\coprod_{\mathcal{O}_{L}}F(k)}$${\coprod_{\mathcal{O}_{L}}\mathcal{O}}$${\ulcorner^{h}}$

where $\mathcal{O}_{L}=\operatorname{Obj}(L)$ is the set of objects of $L$. It
is trivial that with such a construction,
$\Gamma\Xi=\operatorname{Id}_{\mathbf{dg-Segal}_{\mathcal{O}}}$. We have then
that $\Xi$ is fully faithful and in consequence that
$\operatorname{Map}(\Xi,G)$ is a weak equivalence. By the 2-out-of-3
condition, we have that $f$ is a weak equivalence, and so we have that $\Phi$
commutes with colimits. By Proposition 6.17, we know that
$\operatorname{hocolim}(F_{i})$ is DK-equivalent to $F$ in $\mathbf{dg-
Segal}_{\mathcal{O}}$, and that means that we have a DK-equivalence between
$\operatorname{hocolim}(F_{i})$ and $F$ in $\mathbf{dg-Segal}$ too. We have
finished the proof.
∎

Let us see now about $\operatorname{Sing}(T_{i})\to\operatorname{Sing}(T)$.

######  Lemma 6.19.

Let $F_{*}\to F$ be a dg-Segal hypercover as constructed before, and let
$T_{*}$ be a simplicial object in $\mathbf{dg-cat}_{\mathcal{O}}$ with
$\mathcal{O}=\pi_{0}(F(k))$ such that $\operatorname{Sing}(T_{*})=F_{*}$. We
define $T=\operatorname{hocolim}T_{i}$. Then the augmented simplicial object
$T_{*}\to T$ is a $c\mathcal{L}_{\mathbb{S}}$-hypercover of dg-categories.

###### Proof.

As we have fixed objects on this, by Lemma 5.10 if for all
$x,y\in\mathcal{O}$, the augmented object $T_{*}(x,y)\to T(x,y)$ is a split
hypercover of complexes, then $T_{*}\to T$ is a hypercover of dg-categories
and we have finished.

Let us fix $x,y\in\mathcal{O}$. By Lemma 6.16, $T_{*}(x,y)\to F(x,y)$ is a
split hypercover. But as it is a split hypercover, by Lemma 5.11 we know that
$T(x,y)=\operatorname{hocolim}(T_{i}(x,y))\simeq F(x,y)$. So the augmented
object $T_{*}(x,y)\to T(x,y)$ is a split hypercover on $\mathbf{C(k)}$, and in
consequence $T_{*}\to T$ is a hypercover on $\mathbf{dg-cat}$ and we have
finished our proof.
∎

Now we just need to put everything together.

######  Theorem 6.20.

Let $F$ be a functor that satisfies the dg-Segal conditions. Then there exists
a dg-category $T$ such that the morphism $\operatorname{Sing}(T)\to F$ is a
DK-equivalence.

###### Proof.

Let $\operatorname{Sing}(T_{*})=F_{*}\to F$ be a dg-Segal hypercover
constructed following Theorem 6.14. We have then a diagram of this form:

${\operatorname{Sing}(\operatorname{hocolim}(T_{i}))=\operatorname{Sing}(T)}$${F}$${\operatorname{hocolim}(\operatorname{Sing}(T_{i}))=\operatorname{hocolim}(F_{i})}$$\scriptstyle{\phi}$$\scriptstyle{\zeta}$

We have proven that the corresponding augmented object $T_{*}\to T$ is a
hypercover of dg-categories, and by Lemma 6.5, the morphism $\phi$ is a DK-
equivalence. We have also proven that $\operatorname{hocolim}(F_{i})\to F$ is
also a DK-equivalence, so by Remark 4.33, that means that our morphism
$\operatorname{Sing}(T)\to F$ is also a DK-equivalence and we have finished.
∎

And we are done.

######  Theorem 6.21.

Assume Hypothesis 4.35 is true. Then, if we take the complete dg-Segal model
structure on
$\operatorname{Fun}^{\mathbb{S}}(c\mathcal{L}_{\mathbb{S}}^{op},\mathbf{sSet})$,
the functor $\operatorname{Sing}:\mathbf{dg-cat}\to\mathbf{dg-Segal_{c}}$ is
essentially surjective. Hence there exists an equivalence of categories of the
form

$\operatorname{Ho}(\mathbf{dg-cat})\to\operatorname{Ho}(\mathbf{dg-
Segal_{c}}).$

##  7 Epilogue

###  7.1 Dealing with the hypothesis

We have thus seen that we can define complete dg-Segal spaces, give their
ambient functor category a model structure such that the complete dg-Segal
spaces are its fibrant objects, and define a class of morphisms that would be
the weak equivalences for such a model structure. We have also proven that
assuming the DK-equivalences to actually be those weak equivalences, we have
that complete dg-Segal spaces are a model for dg-categories.

Now, the proof of that hypothesis is proving to be trickier than we expected.
In [Rez01], Rezk gets around this issue by defining a completion functor that
takes Segal spaces to complete Segal spaces, and such that the morphism from a
Segal space to its completion is always a Dwyer-Kan equivalence. Then, the
problem is reduced to proving that a Dwyer-Kan equivalence between two
complete Segal spaces is a weak equivalence for the complete Segal model
structure, and a relatively straightforward argument gives us the proof.

Unfortunately for us, the completion functor from Rezk’s case doesn’t work for
us. Indeed, one integral part of its definition is the use of exponentials of
simplicial sets, for which he uses the cartesian product. We cannot do that,
as the lack of compatibility between monoidal and model structure in dg-
categories means that the exponential is not well-defined in our case.

There are several possible ways to solve this issue. The best option, of
course, would be to define a monoidal structure on the functor category that
is compatible with the model structure. Indeed, doing this would not only
solve our hypothesis, but also open the door to writing linear versions to
classical category theory theorems that cannot be expressed yet.
Unfortunately, the obvious answer, the convolution product, wouldn’t work
here. Indeed, although we can write our desired

$\operatorname{Sing}(T)\otimes\operatorname{Sing}(T^{\prime})=\operatorname{Sing}(T\otimes
T^{\prime}),$

it is easy to see that $T\otimes T^{\prime}$ is not a free dg-category. This
is, of course, not the only way to define a monoidal structure on a category,
and other avenues are being explored. Another less ambitious option would be
to abandon the idea of finding a monoidal model structure for the moment, and
concentrate on defining a simplicial Hom.

###  7.2 Refining the definition of complete dg-Segal spaces

Although free dg-categories of finite type are enough for our definitions,
they are probably not the minimal structure necessary in order to define
complete dg-Segal spaces. Indeed, during our time working on this we have
considered another, much smaller category, that we have called the linear
simplicial category and denoted $\Delta_{k}$. This category $\Delta_{k}$ would
be the full subcategory of $\mathbf{dg-cat}$ of dg-categories with a finite
number of objects and such that for every two "consecutive" objects $i$ and
$i+1$ the complex between them is concentrated on a single degree, where it is
$k^{n}$ for a certain $n\in\mathbb{N}$. Such a category has several advantages
over $c\mathcal{L}$: for one, all its objects are cofibrant for Tabuada’s
model structure. On top of that, we have that we can find an explicit
description of the Hom from one object in $\Delta_{k}$ to a general dg-
category, and also a description of weak equivalences in dg-categories via
$\Delta_{k}$. Lastly, it seems to be a good candidate for our complete dg-
Segal spaces: the image of the linearisation functor is entirely contained in
$\Delta_{k}$.

Lastly, although this is still a work in progress, the author is convinced
that managing to write complete dg-Segal spaces in terms of $\Delta_{k}$ would
give us a relatively easy way to compare complete dg-Segal spaces to Mertens’
$S^{+}\mathbf{Mod}(k)-cat$, their own model for dg-categories.

###  7.3 Automorphisms of dg-categories

As we said in the introduction, one of the potential uses of a construction
such as the one of complete dg-Segal spaces would be to compute functors from
dg-categories to other categories, or, in particular, the automorphisms of dg-
categories. In his 2005 article, Toën proves that the group of automorphisms
of $\infty$-categories is $\mathbb{Z}/2\mathbb{Z}$, consisting only of the
identity and the dualisation functor. We expect the automorphism group of dg-
categories to contain $\mathbb{Z}/2\mathbb{Z}$, of course, but not
exclusively: at the very least, the group of automorphisms of the ring $k$
will have to be there, and also probably the Brauer group.

It would also be interesting to take a look at higher homotopy groups of
$Aut(\mathbf{dg-cat})$: we expect the Hochschild homology to appear at some
point.

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aainstitutetext: University of Tennessee, Knoxville, TN, USAbbinstitutetext:
Harvard University, Cambridge, MA, USA

# Strategies for Beam-Induced Background Reduction at Muon Colliders

D. Ally a L. Carpenter a T. Holmes a,b L. Lee a P. Wagenknecht
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Future collider detectors at muon colliders will be bombarded by Beam-Induced
Backgrounds (BIB) due to the in-flight muon decays from the beam line. These
backgrounds can inhibit the ability of the detector and subsequent data
analysis to successfully reconstruct collision products. We explore methods
for geometrically reducing these effects for use in the readout, triggering,
and data analysis of future experiments. Studies are performed for a collision
energy of 1.5 TeV, and a detector with a tungsten nozzle designed to block the
majority of the BIB. In this context, detector strategies are explored to
further reduce the BIB, with a focus on the innermost layers of the tracker
where its density is highest. In addition, a conceptual design of a
calorimeter built to reject BIB is presented.

##  1 Introduction

Muon colliders are a promising option for future colliders: they are arguably
the most cost-, space-, and energy-efficient way to explore the energy
frontier Long:2020wfp (1). From the perspective of detector performance, the
largest blocker for a successful muon collider is the Beam-Induced Background
(BIB) muc-perf-2020 (2, 3, 4). The metastable muon decays in flight, producing
a halo of secondary particles that travel along with the beam. As in previous
works Bartosik:2019a9 (5), a detector design employing a tungsten nozzle to
block this BIB is used.

Even with this shield in place, the remaining BIB drowns out the signals of
particles from the collision. While the BIB extends throughout the tracker and
into the calorimeter, we chose to first address this problem where it is the
most intense, in the detector elements closest to the beam. With the exception
of Section 4, this paper explores strategies to reduce BIB at the hit level in
the innermost layers of the tracker, to reduce this overwhelming background
before it becomes a combinatorial blocker for tracking algorithms.

##  2 Simulated Samples

Simulated samples were provided by the International Muon Collider
Collaboration (IMCC) imcc (6) or produced using its software as made available
for the Snowmass process. The BIB background overlay was provided by the
larger community (originally produced in MARS15 by N. Mokhov osti_1282121
(7)). Di-Higgs production is used as an example signal process in these
studies, and all Higgs bosons are assigned Standard Model couplings. These
events were produced privately, generated in MadGraph5_aMC@NLO Alwall:2014hca
(8) interfaced with Pythia8 pythia2015 (9), and the provided Marlin framework
Gaede:2003ip (10). A beamspot spread of 1 cm was used. The results presented
here were produced using energy depositions from particle interactions as
simulated by Geant4 AGOSTINELLI2003250 (11). Each simulated energy deposition
will be referred to as a _sim hit_.

The detector geometry assumed in the simulation is a modified CLIC detector
design, CLIC_o3_v14_mod4, and the majority of these results focus on the
vertex detector, the innermost silicon tracking system, with a tungsten nozzle
added on either side of the detector to block incoming BIB Bartosik:2019a9
(5).

##  3 BIB Discrimination Strategy in Trackers

The primary source of BIB in the detector is not direct muon decay products,
but the low-energy particles resulting from those decay products interacting
with the nozzle. To travel from the nozzle to the vertex detector, these
particles must have a shallow angle with respect to the beam line, and they
can arrive in a large time window.

In this study, the target is to retain sensitivity to collision product
particles originating at the interaction point (IP) which could feasibly be
tracked, here defined as those with momentum greater than 1 GeV. In the
simulated collision events, any vertex detector hit associated with a charged
particle matching these requirements is classified as signal. All hits from
BIB contributions are considered background, regardless of particle momentum.
A fiducial region in time is defined such that only sim hits within the range
$[-250,+300]$ ps with respect to the expected time of arrival assuming
$\beta=1$ are included.

Even after this fiducial cut on the BIB, it is overwhelmingly large compared
to signal. In the samples used here, a typical event contained approximately
600,000 hits from BIB in the vertex detector, but only about 250 signal hits.

These two populations have strikingly different features at the track level,
but discrimination is needed before this stage to make tracking feasible. In
this study, we explore two strategies for hit-level BIB reduction: one using
precision timing, and one using double tracker layers to make angular
measurements from hit pairs.

###  3.1 Timing Measurements

Figure 1: [Top] The correlation between the hit time $t$ and the expected hit
time $t_{\rm exp}$ is shown. The red contribution from the collision products
shows a strong correlation, while the color scale shows the structure due to
BIB contributions. The measured time $t$ [Bottom Left] and relative time
$t_{\Delta}$ [Bottom Right] as a function of longitudinal position $z$ are
also shown.

Precise timing measurements can serve to reject energy depositions due to BIB,
particularly in the tracker. Each detector element’s position has a particular
expected time of arrival $t_{exp}$ for particles with relativistic $\beta=1$.
As shown in Figure 1, for particles from the collision, the actual time of
arrival $t$ is highly correlated with $t_{exp}$. The energy depositions due to
BIB do not show such a correlation and often arrive to the sensor earlier than
expected (and earlier than allowed by the speed of light) if originating at
the IP. The corrected time

$\displaystyle t_{\Delta}=t-t_{exp}$ (1)

should strongly peak at zero for particles from the IP, while the BIB has a
characteristic timing distribution similar to the distribution seen for BIB at
the LHC. These behaviors can be seen in Figure 1. Selecting for small values
of $t_{\Delta}$ will retain the interactions due to collision products while
removing a large amount of the BIB.

While strict requirements on $t_{\Delta}$ may reduce BIB contributions, they
may also reduce the acceptance for low-$\beta$ particles and may negatively
affect searches for heavy BSM long-lived particles either produced with a
$\beta<1$ or that have non-trivial flight paths from intermediate displaced
decays Lee_2019 (12).

###  3.2 Angular Measurements

Figure 2: [Top] The correlation between the incident polar angle $\theta$ of
the particle creating a hit and the expected angle $\theta_{\rm exp}$ is
shown. The contribution from the collision products (red) shows a strong
correlation, while the color scale shows the structure due to BIB
contributions. The measured angle $\theta$ [Bottom Left] and relative angle
$\theta_{\Delta}$ [Bottom Right] as a function of longitudinal position $z$
are also shown.

As in the case for timing, particles emerging from the IP have an expected
incident angle $\theta$, while interactions from BIB may arrive at the sensors
at other angles. Figure 2 shows that particles from the IP have a $\theta$
value well correlated with the expected $\theta_{exp}$ that will vary around
the volume of the detector. BIB interactions are not well correlated in this
plane, but in fact show a distinctive structure, with each beam producing BIB
hits with incidence angles biased towards the original beam direction.

Once again, the correlation for particles from the IP can be exploited by a
corrected $\theta$ variable,

$\displaystyle\theta_{\Delta}=\theta-\theta_{exp},$ (2)

for which the collision products have a value close to zero, while BIB
contributions are found at other preferred angles that vary across the
detector.

In order to take advantage of this feature, the tracker would need to provide
a high-resolution $\theta$ measurement for incoming particles. This could be
achieved via doublet layers of silicon giving a stub-based $\theta$
measurement, or via some other technology. In the case of doublet layers, the
angular resolution depends on the pitch, the distance between layers, and the
incident angle. For a 50 $\mu$m pitch with a 2 mm distance between layers, the
maximum angular resolution is about 0.025 radians.

As in the case of the timing measurements above, strict requirements on
$\theta_{\Delta}$ may also reduce the acceptance for BSM long-lived particles
producing non-trivial flight paths from intermediate displaced decays Lee_2019
(12).

###  3.3 Combined Discrimination

Figure 3: The two-dimensional distribution of $t_{\Delta}$ vs.
$\theta_{\Delta}$ is shown. The contribution from the collision products (red)
is largely localized at the origin with a tail extending to positive
$t_{\Delta}$, while the color scale shows the structure due to BIB
contributions.

For mitigating BIB in the inner tracking detector, we explore simultaneous
requirements on $t_{\Delta}$ and $\theta_{\Delta}$. This space, shown in
Figure 3, has contributions from collision products localized at (0 rad, 0 ns)
with a tail extending up to positive $t_{\Delta}$ values. The BIB
contributions, while structured, extend to large values of both $t_{\Delta}$
and $\theta_{\Delta}$. Selecting increasingly small rectangles in this space
can serve to reject the BIB contributions while retaining particles emerging
from the IP.

Figure 4: ROC surfaces are shown for a few example resolution values
$\sigma(t)$ and $\sigma(\theta)$. Each resolution configuration is shown in a
different hue. The darkness of the color represents the $t_{\Delta}$
requirement, while each $\theta_{\Delta}$ requirement (shown in the same hue
and darkness) forms a traditional, one-dimensional ROC curve. Different hue
ribbons represent different detector capabilities, while the surface of the
ribbon represents the potential performance of the discriminators described in
the text. Efficiencies are shown after a fiducial $t_{\Delta}$ selection of
[-250, +300] ps.

Performance of the two-dimensional discriminator for rejecting BIB in the
tracker can be found in Figure 4 for various measurement resolutions as
defined by a gaussian smearing of the measured $t$ and $\theta$ values. Since
there are two rectangular cuts, a ROC _ribbon_ , comprising a collection of
individual ROC curves, can be calculated. A single point along one of these
ribbons corresponds to the performance of a single set of cuts and resolution
values. For a set of resolutions $(\sigma(t),\sigma(\theta))=(20\mbox{
ps},0.1\mbox{ rad})$, collision particle interactions can be selected with an
efficiency of $99\%$ while rejecting $81.5\%$ of the fiducial contribution
from BIB. Poorer resolutions at $(\sigma(t),\sigma(\theta))=(100\mbox{
ps},0.1\mbox{ rad})$ result in only $60.5\%$ of the fiducial BIB contribution
being rejected. The spatial distribution of vertex detector sim hits is shown
in Figure 5 before and after the application of a requirement that
$|t_{\Delta}|<50$ ps and $|\theta_{\Delta}|<0.2$ rad for
$(\sigma(t),\sigma(\theta))=(20\mbox{ ps},0.1\mbox{ rad})$.

Figure 5: The spatial distribution of vertex detector sim hits is shown before
[Left] and after [Right] a requirement of $|t_{\Delta}|<50$ ps and
$|\theta_{\Delta}|<0.2$ rad is applied for
$(\sigma(t),\sigma(\theta))=(20\mbox{ ps},0.1\mbox{ rad})$. On the left, BIB
contributions on the first tracking layer from the forward regions closest to
the nozzles is apparent. After placing these requirements to reject BIB, the
only remaining components are those that mirror the distributions of collision
products, emerging approximately from the IP. The spatial distribution is
shown in transverse radius $r$ and longitudinal distance $z$.

This performance is highly dependent on the geometry of the vertex detector
and the size of the beamspot. Because the vertex detector barrel is close to
the beamline and extends only to 6.5 cm in $z$, the 1 cm beamspot spread means
that a large range of angles are consistent with collision products. A study
performed with a 1 mm beamspot showed that the BIB could be reduced an
additional order of magnitude, so long as a detector angular resolution of
0.01 rad could be achieved to take advantage of it. For detectors at larger
distances from the IP, this dependence would be smaller, making this strategy
interesting for larger radius trackers and the calorimeter.

##  4 Angular Discrimination in Calorimeters

Conceptual Designfor a Triscopic Calorimeter

Conceptual Designfor a Triscopic Calorimeter

Conceptual Designfor a Triscopic Calorimeter

Figure 6: [Top] A three-quarters cut-away view of the proposed triscopic
calorimeter is shown. IP slices are shown in purple while the BIB slices are
shown in blue. [Bottom Left] A half view of the IP slices is shown. Elements
are aligned such that they are projective from the IP. [Bottom Right] The BIB
slices are shown, inter-weaved such that half of the layers are angled to be
consistent with BIB from each beam.

In calorimeter systems, precise timing information may be difficult to
achieve. Angular information can be used for the rejection of BIB, but the
geometry of the detector elements will greatly affect this discrimination
power. At the LHC, calorimeter elements are generally designed with the IP in
mind, prioritizing energy deposition in tower volumes that are projective from
the IP. For example, the ATLAS LAr calorimeter comprises square tower units in
sampling layer 2 that have a volume of roughly
$\Delta\eta\approx\Delta\phi\approx 0.025$ with a radial length of roughly
$450$ mm. Integrating energy deposition over this entire volume would be very
sensitive to the high-BIB environment of a muon collider.

Conceptual Design for a Triscopic Calorimeter Figure 7: An example BIB slice
is shown, targeting BIB due to the beam entering the detector from the right.
The optimal angle to accept BIB varies across the longitudinal length of the
detector from $\theta=97.40^{\circ}$ to $127.41^{\circ}$ as pictured. The
opposing BIB slice mirrors this structure. The linear dimensions are arbitrary
and for drawing purposes only.

Using the distributions shown in Figure 2, assuming the angles of incidence
are representative of what the calorimeter would see, we propose a toy
calorimeter geometry designed to give offline analysis as much information as
possible to distinguish BIB from the collision products. This geometry
contains $\phi$-segmentation of three alternating types. A particular slice of
the calorimeter in $\phi$ will be a) an “IP” slice designed with IP-projective
towers, b) a “BIB-1” slice designed for particles consistent with BIB from
Beam 1, or c) a “BIB-2” slice corresponding to BIB from Beam 2. The detector
$\theta-\phi$ segmentation can be fine enough such that a typical hadronic jet
deposits energy in all of these types of layers, giving a _triscopic_
calorimeter measurement. The triscopic angles vary across the detector to
account for the changing incidence angles along the beam-line. An example
geometry demonstrating this structure can be seen in Figure 6. The angular
structure of the BIB layers, as derived from Figure 2, can be seen in Figure
7.

A jet that has been reconstructed in the calorimeter can be characterized by
the particular pattern of triscopic calorimeter signals that form the jet. For
example, defining a width measure in the longitudinal location $z$ as

$\displaystyle\sigma_{z}=\sum_{i}\frac{E_{i}}{E_{\rm{total}}}\Delta z_{i},$
(3)

where the sum runs over the calorimeter measurements that compose the jet, and
$\Delta z_{i}$ is the longitudinal distance between the measurement and the
longitudinal location of the center of the jet.

Measuring $\sigma_{z}$ exclusively in each type of $\phi$-layer, allows for
comparisons between layers. For each jet, we can construct a quantity such as

$\displaystyle R_{\rm BIB}=\frac{\min{\sigma_{z,\rm BIB}}}{\sigma_{z,\rm
IP}}\cdot\frac{\min\sigma_{z,\rm BIB}}{\max\sigma_{z,\rm
BIB}}=\frac{\min\sigma_{z,\rm BIB}^{2}}{\sigma_{z,\rm IP}\ \max\sigma_{z,\rm
BIB}},$ (4)

where $\min$ and $\max\sigma_{z,\rm BIB}$ represent the smaller and larger of
the two values $\sigma_{z,\rm BIB-1}$ and $\sigma_{z,\rm BIB-2}$,
respectively. This variable can discriminate jets originating from BIB and
those originating from the collision – BIB-induced jets from either beam will
result in a low value of $R_{\rm BIB}$, while collision jets will have a
relatively large value. In general, the triscopic calorimetry is designed to
give as much information to offline analysis as possible, prioritizing the
identification and rejection of BIB. For triggered readout systems, this
information can also easily be included into the calorimeter trigger systems
to reduce the rate of the jet triggers due to BIB.

##  5 Conclusions and Future Works

In the vertex detector, the combination of timing and angular information can
be used to build a discriminator that strongly rejects BIB. Even after a tight
fiducial timing window, nearly an order of magnitude of additional BIB
rejection can be achieved with realistic timing detectors and angular
measurements from paired layers. For a detector with 20 ps timing resolution
and 0.1 rad angular resolution, 90% BIB reduction can be achieved while
retaining 95% of signal hits. This performance is highly dependent on the size
of the beamspot; the reduction of the beamspot by a factor of 10 increased
rejection power by approximately an order of magnitude.

In the calorimeter, this kind of precision is less feasible, but dedicated
detector geometries can be explored to help lessen the impact of BIB. However,
in this comparatively high-$r$ and high-$z$ detector, the impact of the
beamspot size is much smaller. More studies are needed to determine the
quantitative impact of these designs.

These rejection techniques are all predicated on the assumption that signal
processes originate at the IP, so tight cuts necessarily reduce sensitivity to
the decays of long-lived particles. Further studies are required to fully
explore this trade-off.

###### Acknowledgements.

We thank the University of Wisconsin-Madison experimental HEP group for useful
discussions and computing, with particular thanks to S. Dasu and C. Vuosalo.
We also thank the larger community of Muon Collider enthusiasts for useful
feedback, especially the International Muon Collider Collaboration, S.
Jindariani, and S. Pagan Griso. This work was supported by U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier
Research Centers program under Award Number DE-SC0020267 (T.H., D.A., L.C.),
and Award Number DE-SC0007881 (L.L.).

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11institutetext: KU Leuven, Leuven, Belgium
11email<EMAIL_ADDRESS>22institutetext: Inria Centre, Lille
22email<EMAIL_ADDRESS>
# Generating Explanations in Medical Question-Answering by Expectation
Maximization Inference over Evidence

Wei Sun 11 Mingxiao Li 11 Damien Sileo 22 Jesse Davis 11 Marie-Francine Moens
11

###### Abstract

Medical Question Answering (medical QA) systems play an essential role in
assisting healthcare workers in finding answers to their questions. However,
it is not sufficient to merely provide answers by medical QA systems because
users might want explanations, that is, more analytic statements in natural
language that describe the elements and context that support the answer. To do
so, we propose a novel approach for generating natural language explanations
for answers predicted by medical QA systems. As high-quality medical
explanations require additional medical knowledge, so that our system extract
knowledge from medical textbooks to enhance the quality of explanations during
the explanation generation process. Concretely, we designed an expectation-
maximization approach that makes inferences about the evidence found in these
texts, offering an efficient way to focus attention on lengthy evidence
passages. Experimental results, conducted on two datasets MQAE-diag and MQAE,
demonstrate the effectiveness of our framework for reasoning with textual
evidence. Our approach outperforms state-of-the-art models, achieving a
significant improvement of 6.86 and 9.43 percentage points on the Rouge-1
score; 8.23 and 7.82 percentage points on the Bleu-4 score on the respective
datasets.

###### Keywords:

Medical Explanation Generation Expectation-Maximization.

##  1 Introduction

Medical Question Answering (Medical QA) is an essential component of modern
healthcare. By automating the process of answering medical queries, it spares
users the effort of manual searches. Despite the development of advanced
medical QA systems [1, 16], the ideal medical QA systems should not only
provide accurate answers but also explanations that describe the elements and
context that support a choice [6]. Explanations serve as a linchpin for
promoting the reliability of answers and facilitating fact-checking [6, 8], so
that users, i.e., patients and clinicians, can trust medical QA systems by
addressing their concerns. An alternative explanation generation approach is
to generate concise, self-contained natural language explanations that are
more comprehensible and accessible to users [3]. This approach can empower
clinicians and patients to better understand the medical QA system’s behavior,
facilitating informed decisions and improved patient care [9].

In this paper, we will explore what we call Question-Answering Explanation
(QAE), where the goal is to provide a natural language explanation of the
answer or decision in a medical QA system. QAE provides explanations for the
returned answer given a question, a patient’s case description, and possibly
external data. Prior works [2, 5] demonstrate that evidence retrieved from an
external knowledge base helps to improve the quality and effectiveness of NLP
systems. For medical explanation generation, a key challenge to improve the
quality of explanations lies in how to fuse medical knowledge from multiple
lengthy documents This is difficult because it is often unclear which evidence
pieces are relevant and each document is quite long. Classical multi-head
attention mechanisms struggle to deal with multiple lengthy documents. It is
expensive to conduct full attention across concatenated evidence because their
time complexity increases exponentially as the number of input tokens
increases. To overcome this challenge, we propose Expectation-Maximization
Inference over evideNces (EMIN), implemented as an encoder-decoder model with
a dual encoder for encoding question and answer, and evidence texts. This
approach iteratively optimizes both the relevance weights of each piece of
evidence and the text generation model’s parameters. This permits attending
over separated evidence rather than performing full attention across
concatenated evidence.

Our main contributions are the following:

  * •

We introduce a novel task known as Question-Answering Explanation (QAE) in
medical domain. We present two new datasets, MQAE-diag and MQAE, curated from
publicly available medical resources.

  * •

We propose a novel framework, Expectation-Maximization Inference over
evidences (EMIN), designed for generating contextualized explanations. EMIN
enables inference of the relevance of each piece of evidence and leverages
attention in a scalable way.

  * •

Experimental results show that our model outperforms state-of-the-art
baselines by a large margin on the MQAE and MQAE-diag datasets across
different evaluation scores. Notably, our model achieved improvements by 6.86
and 9.43 percentage points on the Rouge-1 score; 8.23 and 7.82 percentage
points on the Bleu-4 score on the MQAE-diag and MQAE datasets.

This paper is organized as follows: Section 2 details the EMIN model. Section
3 we presents results on two medical QAE datasets and an error analysis.
Finally, section 4 describes related works and section 5 summarizes this
paper.

Figure 1: The overall architecture of the EMIN model (training stage). The
evidence query is the question $Q_{i}$’s answer. Model parameters consist of
the parameters of the question, evidence encoders and dual cross-attention
decoder. We use of ground-truth answers during the training phase, while
relying on the predictions generated by a trained QA system during the
inference phase.

##  2 Methods

The goal of Question-Answering Explanation (QAE) is to generate a natural
language explanation of the predicted answer to a given question. We utilize
the ground-truth answer for training purposes and rely on the predictions
generated by a trained QA system during the inference phase. We will approach
this as a text-to-text generation task. More formally, for training we assume
access to a dataset ((${Q}_{1}$, ${A}_{1}$,
$E_{1}$),${Exp}_{1}$),$\cdots$,((${Q}_{N}$, ${A}_{N}$, $E_{N}$),${Exp}_{N}$),
where ${Q}_{i}$ is a question, ${A}_{i}$ is the answer to this question,
$E_{i}$ refers to the set of relevant evidence paragraphs and ${Exp}_{i}$ is
the corresponding explanation. The goal is to learn a question-answering
system $\mathbf{f}^{\prime}:{Q}_{i}\rightarrow{A}_{i}$ and an explanation
generator $\mathbf{f}:\\{{Q}_{i},{A}_{i},{E}_{i}\\}\rightarrow{Exp}_{i}$ where
the $Exp_{i}$ is a text composed of natural language sentences.

Figure 1 shows the overall architecture of our approach, which works as
follows. First, we automatically identify a limited set of potential pieces of
evidence in a document collection, typically represented as relevant
paragraphs corresponding to each question. Second, we employ a text generation
model based on a BART encoding-decoding neural architecture [10]. We fine-tune
this model using an expectation-maximization (EM) approach. This iterative
process refines not only the model’s parameters, including attention
parameters but also updates the importance weights associated with each
potential evidence piece. During inference, we employ an analogous EM-based
approach to generate explanations for question-answer pairs that have not been
previously encountered. In this phase, we rely on a trained question-answering
system to predict the answers. The following subsections describe each
component of the approach in detail.

###  2.1 Evidence Selection

The initial step involves selecting a set of evidence paragraphs from a
document collection that are likely to be relevant for generating the
explanation. To do so, we employ the Dense Passage Retrieval (DPR) model [7],
which retrieves documents related to the answer from a medical document
collection (in our case medical textbooks). The retrieval query is the answer
${A}_{i}$. Before initiating the retrieval process, both documents and
retrieval queries undergo preprocessing. This involves tokenization into
words, removal of stop words, and conversion of all remaining words to
lowercase. The DPR algorithm subsequently returns a set of evidence documents
based on their relevance scores. Following the retrieval of documents, the
next step involves breaking them down into evidence paragraphs. These evidence
paragraphs are then ranked using the DICE similarity score concerning the
textual retrieval query. The top-$k$ ranked textual evidence paragraphs are
selected, taking into account normalized word overlap.111This approach
resulted in better explanation generation results than first splitting the
documents into evidence paragraphs and then ranking them with DPR.

###  2.2 Neural Backbone

The dual encoder-decoder model is based on BART [10]. It has two encoders: one
encodes the question-answer and the other encodes the evidence paragraphs. We
initialize the dual encoder-decoder model using the pretrained parameters of
the BART base model.

####  2.2.1 Encoder

The BART encoder has a transformer architecture characterized by applying
self-attention to the representations at a specific layer $l$ of the
transformer encoder. Let $X_{i}$ be the concatenation of the question string
$Q_{i}$ and answer string $A_{i}$. We pass $X_{i}$ to the encoder and use
$\mathbf{X}_{i}^{l}\in\mathbb{R}^{L_{x}\times d}$ to represent its output at a
specific layer $l$ where $L_{x}$ is the sequence length of $X_{i}$ and $d$ is
the dimension of the hidden states. Evidence features ${\mathbf{E}}_{i}$ are
obtained by feeding each selected evidence paragraph
$\mathbf{E}_{i,j}\in\mathbb{R}^{L_{e}\times d}$ into the evidence encoder.
${\mathbf{E}}_{i}^{l}$ refers to the encoded evidence paragraphs at a specific
layer $l$ of the transformer encoder. $L_{e}$ represents the evidence’s
sequence length.

####  2.2.2 Dual Cross Attention Decoder

##### Vanilla BART decoder

The BART decoder has a transformer architecture characterized by self-
attention and cross-attention over the tokens of the encoder. The decoder is
composed of different layers. Given the hidden state of the $l$th layer,
$\mathbf{H}^{l}\in\mathbb{R}^{L_{d}\times d}$, which represents the hidden
states of the decoder sequence. $L_{d}$ is the length of the decoder sequence
in terms of tokens. The hidden state of the next layer $\mathbf{H}^{l+1}$ is
computed as:

$\displaystyle\mathbf{SA_{i}^{l}}$ $\displaystyle=\operatorname{Self-
Attn}\Bigl{(}\mathbf{H}^{l}_{i},\mathbf{H}^{l}_{i}\Bigl{)},$ (1)
$\displaystyle\mathbf{CA}_{i}^{l}$ $\displaystyle=\operatorname{Cross-
Attn}\Bigl{(}\mathbf{SA}_{i}^{l},\mathbf{X}_{i}^{l}\Bigl{)},$ (2)

where $\operatorname{Self-Attn}$ and $\operatorname{Cross-Attn}$ operations
compute the scaled dot-product attention [15]. The decoder autoregressively
generates explanations. That is, at each timestep of the decoding process, the
input of the decoder is the embedding of the previous token (embedding of the
ground truth token during training and embedding of the generated token during
inference).

##### EMIN decoder

In EMIN, we explicitly model evidence weights as
$\mathbf{z}_{i}\in\mathbb{R}^{k}$ that capture the relevance of each selected
evidence paragraph for instance $i$. As we do not have ground truth evidence
weights, we infer them using an expectation-maximization algorithm. The
inferred evidence weights make the attention of many texts or on a long text
split in paragraphs scalable for realistic applications.

The EMIN decoder follows the general architecture of the vanilla BART decoder.
This entails that the hidden state of each layer is updated with the context
vector that contains the encoded information of the evidence tokens weighted
by the cross-attention weights obtained by $\operatorname{Cross-
Attn}\Bigl{(}\mathbf{CA}_{i}^{l},\mathbf{E}_{i}^{l}\Bigl{)}$. In EMIN, each
evidence paragraph $\mathbf{E}_{i,j}$ is also weighted with its evidence
weight $\mathbf{z}_{i,j}$ by element-wise multiplication of their respective
vectors.

The intuition is that every transformer layer of the decoder cross-attends to
relevant feature extracted by the corresponding encoder layer builds a
representation to be decoded that captures different linguistic information.
The representation of each layer can be used to update the evidence weight
$\mathbf{z}_{i,j}$. Equation ($3$) becomes:

$\displaystyle\mathbf{H}_{i}^{l+1}$
$\displaystyle=\mathbf{z}_{i}\odot\operatorname{Cross-
Attn}\Bigl{(}\mathbf{CA}_{i}^{l},\mathbf{E}_{i}^{l}\Bigl{)}.$ (3)

###  2.3 Expectation-Maximization Inference over Evidence

The evidence weights $\mathbf{z}_{i}\in\mathbb{R}^{k}$ capture the importance
of the evidence paragraphs ${\mathbf{E}}_{i}$. The evidence weights are the
hidden variables in the EM algorithm with
$\operatorname{sum}(\mathbf{z}_{i})=1$. During training for each question-
answer input $X_{i}$, the weights of the evidence paragraphs $\mathbf{z}_{i}$
and parameters $\mathbf{\Theta}$ of the dual encoder-decoder model are
iteratively optimized by the EM algorithm. In the E-step, we update the
evidence weights $\mathbf{z}_{i}$ using the ground-truth explanations. In the
M-step, we use the back-propagation algorithm to optimize the dual encoder-
decoder model’s parameters $\mathbf{\Theta}$ given the expected values of
$\mathbf{z}_{i}$. During inference, we only iteratively update the evidence
weights $\mathbf{z}_{i}$ based on the trained explanation generation model to
gain insight into the contribution of each evidence paragraph in generating
the explanation. The weights of the evidence paragraphs $\mathbf{z}_{i}$ of an
input example are uniformly initialized as $\frac{1}{k}$ unless mentioned
otherwise.

####  2.3.1 Expectation-Maximization during Training

E-step: In the E-step at each time step $t$, given input features
$\mathbf{X}_{i}$, the parameters of the dual encoder-decoder model at time
step $t-1$, and evidence representation
$\mathbf{{E}}_{i}=[\mathbf{{E}}_{i,1},\mathbf{{E}}_{i,2},\dots,\mathbf{{E}}_{i,k}]$,
we generate explanation $\overline{\mathbf{Exp}}_{i,j}^{t-1}$. We first
compute the inverse cross-entropy between the generated and ground truth
explanations:

$\displaystyle
c_{i,j}^{t}=\frac{1}{-\sum_{m=1}^{L_{d}}P(Exp_{i,m}^{t-1})log(P(\overline{Exp}_{i,j,m}^{t-1}))},$
(4)

where $P(\overline{Exp}_{i,j,m})$ is the generated token probability at
position $m$ of sample $i$ when only evidence $j$ is used, and $P(Exp_{i,m})$
denotes the ground truth token at the same position of sample $i$, which is
just a one-hot encoding in our case.

The expected values of $\mathbf{z}_{i}^{t}$ are then computed as:

$\displaystyle z_{i,j}^{t}$
$\displaystyle=\frac{\exp^{\frac{c_{i,j}^{t}}{\lambda}}}{\sum_{j=1}^{k}\exp^{\frac{c_{i,j}^{t}}{\lambda}}},$
(5)

where $\lambda$ is a temperature parameter that controls the softness of the
evidence weights distribution. We gradually reduce $\lambda$ using
$\lambda=e^{-0.01\times\text{iteration}}$, so that weights will increasingly
concentrate on the most important pieces of evidence.

Intuitively, a piece of evidence should have a high weight if using it
produces an generated explanation that is close to the ground truth one. We
leverage the inverse cross entropy to measure the similarity. The lower the
value of this metric is, the closer the distance between the generated
explanation and the ground-truth is, meaning that the model should pay more
attention to this evidence paragraph by assigning it a larger weight.

M-step: In the M-step, at iteration step $t$ given the input features
$\mathbf{X}_{i}$, and the output ground truth explanation $Exp_{i}^{t}$, we
optimize the dual encoder-decoder model’s parameters $\mathbf{\Theta}^{t}$
with backpropagation using the cross entropy (CE) loss commonly used in text
generation:

$\displaystyle\mathcal{L}$
$\displaystyle=-\sum_{i=1}^{N}\sum_{m=1}^{L_{d}}P(Exp_{i,m}^{t})logP(\overline{Exp}_{i,m}^{t}),$
(6)

where $L_{d}$ refers to the length of the ground truth sequence. $N$ is the
number of training instances. Note that during explanation generation, the
decoder integrates the evidence weights computed in the E-step in the way
described in section 2.2.2.

The EM iteration stops when the KL divergence between the evidence weights of
iteration $t$ and those of iteration $t-1$ is smaller than a defined threshold
$\epsilon$.

####  2.3.2 Inference

When generating an explanation at test time, the model parameters
$\mathbf{\Theta}^{t}$ are fixed, but we still need to estimate the weights of
the evidence paragraphs $\bf{z}_{i}$ for the test question. Here, we use the
generated explanations to infer the weights of the evidence paragraphs. In
each iteration $t$, we calculate the expected weight of the evidence in a
manner similar to the E-step used during training. However, we modify this
step by evaluating the cross-entropy of the generated explanation and using
the predicted answer from a trained question-answering system, namely
BioLinkBERT [18]. The global explanation generated at $t-1$:

$\displaystyle c_{i,j}^{t}$
$\displaystyle=\frac{1}{-\sum_{m=1}^{L_{d}}P(\overline{Exp}_{i,m}^{t-1})logP(\overline{Exp}_{i,j,m}^{t-1})},$
(7)

$\displaystyle z_{i,j}^{t}$
$\displaystyle=\frac{\exp^{\frac{c_{i,j}^{t}}{\lambda}}}{\sum_{j=1}^{k}\exp^{\frac{c_{i,j}^{t}}{\lambda}}}.$
(8)

The stopping criterion is the same as in the training phase. After
$\bf{z}_{i}$ has converged, we generate the final explanation.

##  3 Experiments

###  3.1 Datasets

In our study, we utilize two medical question-answering explanation datasets,
namely the Medical Diagnosis QAE (MQAE-diag) and the Medical QAE (MQAE)
datasets, in conjunction with a repository of textual medical evidence
documents. The distinction between MQAE-diag and MQAE lies in the nature of
their questions. MQAE-diag is tailored for questions related to patient
diagnosis, whereas MQAE is designed to evaluate fundamental medical knowledge.
222Datasets and code will be made publicly available upon acceptance.

MQAE-diag The MQAE-diag dataset’s samples are derived from the MedMCQA dataset
[13], and two medical examination databases, i.e., the Spanish medical
residency exam (MIR)333https://www.curso-mir.com/index.html and the United
States Medical Licensing Examination (USMLE).444https://www.usmle.org/ We
select samples whose questions contain patients’ descriptions and construct
the MQAE-diag dataset. In this dataset, there are 8038 samples for training,
995 instances for validation and 987 for testing.
MQAE The MQAE dataset is based on the MedMCQA dataset [13]. We extract 15993,
2000, and 1999 clinical instances for training, validation and testing. In
this dataset, the question only contains a single question and does not
include any descriptions of the patient’s case.
Document Collection with Evidence Information.  We leverage the StatPearls
medical book555https://www.ncbi.nlm.nih.gov/books/NBK430685/, which contains
9070 medical documents, as our medical textual knowledge. Evidence retrieval
is discussed in section 2.1.

Models | R1 | R2 | RL | BLEU-4
---|---|---|---|---
T5q+a | 15.48$\pm$1.27 | 4.74$\pm$0.88 | 12.20$\pm$1.02 | 2.18$\pm$0.75
BARTq+a | 18.57$\pm$1.95 | 5.71$\pm$1.16 | 13.02$\pm$1.29 | 0.87$\pm$0.13
SIMIq+a+e | 18.89$\pm$2.35 | 6.17$\pm$2.03 | 13.79$\pm$2.20 | 1.54$\pm$0.64
MHOPq+a+e | 21.35$\pm$2.57 | 6.79$\pm$2.31 | 14.77$\pm$1.98 | 1.83$\pm$0.57
MEANq+a+e | 22.57$\pm$2.50 | 7.13$\pm$1.78 | 16.04$\pm$1.65 | 2.25$\pm$0.93
EMIN |  32.24$\pm$3.21 |  13.23$\pm$2.98 |  23.19$\pm$2.79 |  10.48$\pm$3.22
Table 1: Experimental results in terms of ROUGE (R) and BLEU scores (%) on the
MQAE-diag dataset. “q”, “a”, and “e” represents questions, answers and
evidence, respectively.

###  3.2 Experimental Set-up

Model Settings: The trained QA system is the BioLinkBERT [18]. We employ the
system on the four-options multiple question-answering tasks and it achieves
an accuracy of $60.18\%$ on the MQAE-diag dataset and $55.85\%$ on the MQAE
dataset. The maximum length of questions and evidence spans is $944$ and the
explanations’ maximum length is $128$. The size of the hidden representations
of the model is $768$. The number of selected evidence paragraphs $k$ is set
to 10 (we evaluate the influence of the value of this hyperparameter).

Training and Inference Details: 1) For the training, all baseline models and
our framework are trained with fp16. All of the models utilize the AdamW
optimizer [11] with a learning scheduler initialized at $2e^{-5}$ and linearly
decreased to 0. We apply the early stopping strategy on the baseline models by
monitoring the loss on the validation set while the patience number is set to
$4$. The KL divergence threshold $\epsilon$ for our approach in the training
stage is $0.01$. 2) During the inference stage, we use beam search to generate
text and set the width of the beam to $5$. $\epsilon$ in the inference stage
is also $0.01$.

Models | R1 | R2 | RL | BLEU-4
---|---|---|---|---
T5q+a | 14.48$\pm$1.43 | 5.94$\pm$1.01 | 13.33$\pm$1.34 | 0.30$\pm$0.26
BARTq+a | 15.61$\pm$1.44 | 6.49$\pm$1.47 | 13.47$\pm$1.38 | 0.68$\pm$0.24
SIMIq+a+e | 18.18$\pm$2.03 | 7.87$\pm$1.67 | 15.21$\pm$1.94 | 0.82$\pm$0.37
MHOPq+a+e | 20.03$\pm$2.15 | 8.07$\pm$1.98 | 16.78$\pm$1.97 | 1.03$\pm$0.56
MEANq+a+e | 21.17$\pm$2.03 | 9.57$\pm$2.12 | 17.71$\pm$1.84 | 1.35$\pm$0.43
EMIN |  28.03$\pm$2.81 |  12.77$\pm$3.06 |  21.08$\pm$2.34 |  9.17$\pm$2.89
Table 2: Experimental results in terms of ROUGE (R) and BLEU scores (%) on the
MQAE datasets. “q”, “a”, and “e” represents questions, answers and evidence,
respectively.

###  3.3 Baseline Models

Due to the maximum length of the BART encoder and other language model based
encoders, we cannot use the concatenation of the question, answer and and all
selected evidences as the input. Hence the simple baselines below do not use
retrieved evidence paragraphs. Their results are mentioned as a kind of
underbound of the performance. The generated explanation here only relies on
the pretrained language model.

T5 (q+a) [14] passes the concatenation of questions and answers to the T5
language model to generate explanations.

BART (q+a) [10] simply concatenates questions and answers as input, which is
fed into the BART model to generate explanations.

The following models retrieve paragraphs in the same way as done in our model
(see 2.1), and the evidence paragraphs are individually encoded with the BART
encoder (see 2.2.1).

SIMI (q+a+e) removes EMIN’s E-step and computes the evidence weights
$\mathbf{z_{i}^{\prime}}$ with the dot product similarity between the encoder
outputs of question-answer representations and evidence features normalized by
a softmax function. SIMI leverages the EMIN’s decoder to incorporate evidence.

MHOP (q+a+e) Following the literature on Key-Value Memory Networks [12], MHOP
encodes the evidence paragraphs with the BART encoder and attends in multiple
hops to this encoded ”memory”.

MEAN (q+a+e) discards the E-step in EMIN, and uses uniform evidence weights to
fuse the evidence features, so evidence weights are not learned with EM
inference. The dual encoder-decoder is the same as for EMIN.

###  3.4 Results and Discussion

To compare with baseline models, we report the results of models on the ROUGE
(recall oriented metric and coded further as R) and BLEU-4 scores (precision
oriented metric) that compare the ground-truth explanation with the generated
explanation. The numbers in the metric names refer to the size of n-gram of
words considered. ROUGE-L (RL) takes into account the length of the longest
common subsequence of words. Table 1 and Table 2 reports test set performance
for all models on both datasets.

MQAE-diag. Our model outperforms the baseline models by large margins across
all evaluation metrics. Compared to the baseline models that use encoded
evidence paragraphs to generate explanations (i.e., SIMI, MHOP, and MEAN),
EMIN improves the BLEU score by $11.5$, $9.99$, and $9.75$ percentage points,
respectively. MEAN is the best evidence-enhanced model, and EMIN outperforms
it by $9.87$, $6.61$, $7.28$, and $9.73$ percentage points on the R1, R2, RL
and BLEU scores, respectively. EMIN beats the baselines that do not consider
evidence paragraphs by even larger margins.

MQAE. On the MQAE dataset, EMIN improves all evaluation scores, especially the
BLEU score. Our approach outperforms the best no-evidence model, i.e., BART
(q+a), by $14.09$, $7.77$, $10.23$, and $11.35$ percentage points on the R1,
R2, RL and BLEU scores, respectively. In particular, our framework improve the
R1 scores by $9.27$, $8.00$, and $6.45$ percentage points compared with the
evidence-enhanced models (SIMI, MHOP, and MEAN). The results show that EMIN
yields large gains over all competitors and the evidence based approaches are
all much better than the non evidence based ones.

Discussion: we compare the results presented in Table 1 and Table 2 to
emphasize the significance of our proposed EM (Expectation-Maximization)
inference method. Unlike the MEAN model, which employs uniform weights
throughout, EMIN iteratively updates evidence weights. Essentially, MEAN can
be seen as the starting point of EMIN, where there is no distinction in
assigning importance to each evidence paragraph when generating explanations.
While both the MHOP and EMIN models iteratively adjust attention weights on
the tokens within individual evidence paragraphs within the dual encoder-
decoder architecture, it is worth noting that MHOP does not capture the
importance of larger evidence paragraphs. Here, EM inference offers an elegant
solution for reasoning at the paragraph level. To demonstrate the efficiency
of EM inference, we conducted a comparison between EM inference and a model
using full attention across concatenated evidence paragraphs. Assume that we
have $m$ paragraphs and each paragraph has $n$ words, the time complexity of
the EM inference and the full attention one are $\mathcal{O}(m^{2}\cdot n)$
and $\mathcal{O}(m^{2}\cdot n^{2})$, respectively. Consequently, EM inference
proves to be more efficient than the full attention model.

###  3.5 Case Study

Table 3 presents a case study on explanation generation in the context of a
trained QA system making a wrong prediction. From the table, we can observe
that the ground-truth explanation and EMIN explanation (based on the ground-
truth answer) are all indicate “metformin increase insulin output”. Even
though the trained QA systems make a incorrect prediction, i.e., “acarbose”,
the EMIN explanation (based on the incorrect answer) still suggests the
“metformin” is the correct drug for the type 2 diabetes patients.

Question: a 52-year-old woman begins pharmacotherapy after being diagnosed
with type 2 diabetes mellitus. four weekslater, her hepatic glucose output is
decreased, and target tissue glucose uptake and utilization are increased.
whichof the following drugs was most likely prescribed for this patient?
---
GT Explanation: metformin is awesome. it decreases hepatic glucose production,
decreases intestinal absorption of glucose, and improves insulin sensitivity
by increasing peripheral glucose uptake and utilization.
EMIN Explanation (GT): metformin (a type 2 diabetes) is a ketoacidotic with a
tendency to produce hepatic dysfunction and increase insulin output. It is a
polypeptide hormone, so it would be unlikely for the patient to be an
insulinoma type 2.As a side effect, it blocks insulin production and decreased
glucose output.
EMIN Explanations (WA): Acarbose is a glycoprotein that binds to glucose
receptors in the liver, causing it to decrease hepatic glucose output and
increase protein synthesis. Inhibitors to glycerol/sulfamethoxazole (a type 2
diabetes antagonist), clofibrate, and mefenamic acid are the most useful drugs
for Type 2 diabetes patients.
Table 3: As an example from the MCQAE dataset The correct answer for this
question is ‘metformin’, but the trained QA system predicts ‘acarbose’ as the
answer. In this context, ‘GT’ refers to the explanation based on the ground-
truth answer, and ‘WA’ represents the explanation based on the wrong answer.
The Italian words in EMIN explanation explanations (WA) indicated that the
“metformin” is the correct drug for the type 2 diabetes patients.

##  4 Related Work

In recent years, significant advancements in medical question-answering
systems have been driven by the introduction of novel models and datasets [1,
16, 13]. However, within the domain of medical inquiry, the role of
explanations has become increasingly critical. Approaches such as segmenting
highlighted textual spans [4] and knowledge-base paths [19] have limitations
when they come to providing explanation which requires substantial mental
effort, particularly because out-of-context spans may be difficult to
interpret. An alternative approach would be to generate a natural language
explanation.

Large Language Models (LLMs) have achieved remarkable performance on medical
question-answering datasets, emphasizing the significance of accurate
predictions. However, user acceptance, particularly among clinicians, is
contingent upon the model’s ability to provide transparent explanations of its
decision-making process. To address this need, Chain-of-thought (CoT)-based
techniques [17, 20] are proposed. In the medical domain, applications deployed
in hospitals must prioritize data privacy, necessitating local server
deployment. However, the sheer size of LLMs precludes their local deployment
in such contexts.

##  5 Conclusions

We propose an approach to collectively reason over all textual evidence while
harnessing the potent attention mechanisms of a transformer architecture. Our
method employs an Expectation-Maximization (EM) algorithm to deduce the
weights of evidence paragraphs as latent variables, enhancing the model
parameters for explanation generation. This gives rise to the EMIN framework,
which notably outperforms existing models, such as those utilizing multi-hop
reasoning, in terms of generating explanations. The explicit evidence weights
we derive unveil which pieces of evidence bolster the generated explanations.
These weights complement the attention weights assigned to tokens within the
evidence paragraphs, all in a scalable manner. This ensures that users gain
comprehensive insights into the significance of the evidence within the
generated explanation.

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