# Time-dependent stochastic basis adaptation for uncertainty quantification

Ramakrishna Tipireddy<EMAIL_ADDRESS>Panos Stinis  Pacific
Northwest National Laboratory, Richland, WA  Alexandre M. Tartakovsky  Pacific
Northwest National Laboratory, Richland, WA; Department of Civil and
Environmental Engineering, University of Illinois Urbana-Champaign, Urbana, IL

###### Abstract

We extend stochastic basis adaptation and spatial domain decomposition methods
to solve time varying stochastic partial differential equations (SPDEs) with a
large number of input random parameters. Stochastic basis adaptation allows
the determination of a low dimensional stochastic basis representation of a
quantity of interest (QoI). Extending basis adaptation to time-dependent
problems is challenging because small errors introduced in the previous time
steps of the low dimensional approximate solution accumulate over time and
cause divergence from the true solution. To address this issue we have
introduced an approach where the basis adaptation varies at every time step so
that the low dimensional basis is adapted to the QoI at that time step. We
have coupled the time-dependent basis adaptation with domain decomposition to
further increase the accuracy in the representation of the QoI. To illustrate
the construction, we present numerical results for one-dimensional time
varying linear and nonlinear diffusion equations with random space-dependent
diffusion coefficients. Stochastic dimension reduction techniques proposed in
the literature have mainly focused on quantifying the uncertainty in time
independent and scalar QoI. To the best of our knowledge, this is the first
time-dependent dimension reduction approach for time-dependent stochastic
PDEs.

###### keywords:

Stochastic basis adaptation, time-dependent problems, domain decomposition

##  1 Introduction

Uncertainty quantification for stochastic partial differential equations SPDEs
with a large number of input random parameters is computationally expensive.
Various stochastic dimension reduction methods e.g. basis adaptation [1, 2,
3], active subspace [4] and sliced inverse regression [5] have been developed
to represent a scalar QoI in a low dimensional stochastic basis. In our prior
work [6, 7] we have combined stochastic basis adaptation and spatial domain
decomposition [8, 9] to represent and compute the solution of a stochastic
steady state PDE using a low dimensional basis. The low dimensional stochastic
basis in each subdomain was obtained by Hilbert KL expansion [10]. Most
physical systems are partially observed and evolve in time, and can be modeled
by nonlinear time-dependent SPDEs. However, the existing stochastic dimension-
reduction methods are limited to steady-state problems. Extending the
dimension reduction methods to time-dependent problems is challenging because
small errors in the low dimensional approximate solution introduced in the
previous time steps accumulate over time and cause divergence from the true
solution. To address this challenge, we extend the basis adaptation and domain
decomposition methods [2, 1, 6, 7] to the case of time-dependent SPDEs. In the
proposed approach, the basis adaptation varies at every time step so that the
low dimensional basis is adapted to the solution at that time step.

As a first application to time-dependent problems, we apply it to a one-
dimensional linear and a one-dimensional nonlinear (Richards) stochastic
diffusion equations. Unlike the two- and three-dimensional computational
domains where the interface between two subdomains is a line and a surface
respectively, algorithms for domain decomposition in one-dimensional domain
can be greatly simplified because the interface between two subdomains is a
point. Hence, the unknown quantities at each interface will be two scalar
quantities namely the solution and the flux. These scalar quantities can be
obtained by solving a set of simple linear algebraic equations in the case of
linear diffusion equation because of the superposition principle. However, for
the nonlinear Richards equation we use an iterative approach to obtain these
quantities. The proposed time-dependent basis adaptation can be extended to
two and three-dimensional problems following the iterative domain
decomposition algorithms such as Neumann-Neumann algorithm shown in [7].

The paper is organized as follows. In Section 2, we introduce the general type
of SPDE that we are interested in, along with the necessary expansions for the
source of stochasticity and the solution. In section 3, we introduce the time-
dependent basis adaption method. In Section 4 we illustrate it through a
linear time-dependent diffusion equation and a nonlinear time-dependent
diffusion equation whose coefficients are modeled as log-normal random fields.
Section 5 contains numerical results. Section 6 contains conclusions and
discussion of future work.

##  2 Time Dependent Stochastic PDEs

Let $D$ be an open subset of $\mathbb{R}^{n}$ and $\Omega$ a sample space. We
want to find find a random field,
$u(x,\omega):D\times\Omega\rightarrow\mathbb{R}$ such that:

$\mathcal{L}(x,u(t,x,\omega);a(x,\omega))=f(x,\omega)\;\;\rm{in}~{}D\times\Omega,$
(1)

subject to the boundary condition

$\mathcal{B}(x,u;a(x,\omega))=b(x,\omega)\;\;\rm{on}~{}\partial
D\times\Omega,$ (2)

where $\mathcal{L}$ is a differential operator and $\mathcal{B}$ is a boundary
operator. We model the uncertainty in the stochastic PDE by treating the
coefficient, $a(x,\omega)$ in the differential operator as a random field and
propagate the uncertainty to the solution field, $u(x,\omega)$. To solve the
stochastic PDE numerically, we discretize the random fields, $a(x,\omega)$ and
$u(x,\omega),$ both in the spatial and stochastic domains. We model
$a(x,\omega)$ with a lognormal random field, such that
$a(x,\omega)=\exp[g(x,\omega)]$, where $g(x,\omega)$ is a Gaussian random
field whose mean and covariance function are known.

###  2.1 Representation of the random coefficient and the solution

The Gaussian random field $g(x,\omega)$ can be approximated through a
truncated KL expansion

$g(x,\omega)\approx\tilde{g}(x,\xi(\omega))=g_{0}(x)+\sum_{i=1}^{M}\sqrt{\lambda_{i}}g_{i}(x)\xi_{i}(\omega),$
(3)

where $g_{0}(x)$ is the mean of the random field $g(x,\omega)$ and
$(\lambda_{i},g_{i}(x))$ are the eigenvalues and eigenvectors obtained by
solving the integral eigenvalue problem

$\int_{D}C_{g}(x_{1},x_{2})g_{i}(x_{2})dx_{2}=\lambda_{i}g_{i}(x_{1}).$ (4)

The eigenvalues are positive and non increasing, and the eigenfunctions
$g_{i}(x)$ are orthonormal,

$\int_{D}g_{i}(x)g_{j}(x)dx=\delta_{ij},$ (5)

where, $\delta_{ij}$ is the Kronecker delta. The random variables $\xi_{i}$ in
Eq. 3 are uncorrelated with zero mean. We model the random variables with
uniform distribution. We further assume that they are independent.

We approximate $a(x,\omega)$ and the solution field, $u(t,x,\omega)$ with
truncated PC expansions with multi-variate orthogonal polynomials

$a(x,\omega)\approx\tilde{a}(x,\boldsymbol{\xi}(\omega))=a_{0}(x)+\sum_{i=1}^{N_{\xi}}a_{i}(x)\psi_{i}(\boldsymbol{\xi}),$
(6)

and

$u(t,x,\omega)\approx\tilde{u}(t,x,\boldsymbol{\xi}(\omega))=u_{0}(t,x)+\sum_{i=1}^{N_{\xi}}u_{i}(t,x)\psi_{i}(\boldsymbol{\xi}),$
(7)

where, $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{M})^{T}$, $u_{0}(x)$ is the mean
of the solution filed, $u_{i}(x)$ are polynomial chaos coefficients and
$\\{\psi_{i}(\boldsymbol{\xi})\\}$ are multi-variate Hermite polynomials.
These polynomials are orthonormal with respect to the inner product defined by
the expectation in the stochastic space,

$\langle\psi_{i}(\boldsymbol{\xi}),\psi_{j}(\boldsymbol{\xi})\rangle\equiv\int_{\Omega}\psi_{i}(\boldsymbol{\xi}(\omega))\psi_{j}(\boldsymbol{\xi}(\omega))dP(\omega)=\delta_{ij}.$
(8)

##  3 Time-dependent basis adaptation

In this section we describe the proposed approach that combines spatial domain
decomposition and time-dependent stochastic basis adaptation.

###  3.1 Basis adaptation at each time step

We will expand on our prior work on uncertainty quantification [2, 1, 6, 7],
which combines stochastic basis adaptation and spatial domain decomposition to
the solution of time-dependent problems. For time-independent problems, the
reduced stochastic basis is constructed only once and used to compute a low-
dimensional representation of the stochastic solution. In the case of time-
dependent problems, the stochastic basis constructed at one time instant is
not enough for the accurate low-dimensional representation of the stochastic
solution at a different time instant. If one insists on using the same basis
for all times, then the accumulation of errors leads to divergence from the
true solution. Hence, a new stochastic basis adapted to the solution at each
time instant is required.

We assume that the stochasticity of the solution of the stochastic PDE at
every time instant can be accounted for through a, usually expensive, sparse
grid collocation method. For each time instant, we can also use a low level
sparse grid to obtain only the linear terms in a PC expansion of the solution
in the entire spatial domain. The linear PC expansion can be used to obtain
the solution in each subdomain. Then, as is done in stochastic basis
adaptation, we can use a Hilbert KL expansion in order to obtain a stochastic
basis for each subdomain.

###  3.2 Time-dependent basis adaptation in each subdomain

We begin with the low level sparse grid solution, which allows us to compute
the linear terms in the PC expansion of the solution in the entire spatial
domain $D$ for the time duration of interest $T$ by solving (17), i.e.,

$u_{g}(t,x,\boldsymbol{\xi}(\omega))=u_{0}(t,x)+\sum_{i=1}^{d}u_{i}(t,x)\xi_{i},$
(9)

where, $u_{i}(t,x)$ are the time-dependent PC coefficients and
$\xi,\;i=1,\ldots,d$ are $d$ (sparse grid) collocation points. The
computational cost of obtaining $u_{g}(t,x)$ is small compared to computing
the full solution $u(t,x).$

We decompose the spatial domain $D$ into $K$ non-overlapping subdomains,
$D^{(i)},i=1,\ldots,K$ such that,

$D=\cup_{i=1}^{K}D^{(i)},\quad D^{(i)}\cap D^{(j)}=\emptyset$ (10)

Let the linear part of the solution in subdomain $D_{s}$ be
$u^{s}_{g}(t,x,\boldsymbol{\xi}(\omega)).$ We have

$u^{s}_{g}(t,x,\boldsymbol{\xi}(\omega))=u_{g}(t,x,\boldsymbol{\xi}(\omega))\mathbb{I}_{D_{s}}(x),$
(11)

where $\mathbb{I}_{D_{s}}(x)$ is the indicator function so that for any set
$D_{s},$ $\mathbb{I}_{D_{s}}=1$ if $x\in D_{s},$ and $\mathbb{I}_{D_{s}}=0$ if
$x\notin D_{s}$.

We use the Hilbert space KL expansion [11, 12, 13, 14, 15] of $u(t,x,\omega),$
to find the reduced basis $\boldsymbol{\eta}^{s}_{t}$ in each subdomain
$D_{s}.$ To use the KL expansion, the solution should satisfy certain
regularity and smoothness conditions, such as the solution being a subset of
$L_{2}(\omega)$ [10].

To find the time-dependent low dimensional basis in $D_{s},$ we construct the
covariance function of $u^{s}_{g}(t,x,\boldsymbol{\xi})$ in each subdomain
$D_{s}$ as follows:

$C^{s,t}_{u_{g}}(x_{1},x_{2})=\sum_{i=1}^{d}u_{i}(t,x_{1})u_{i}(t,x_{2}),\quad
x_{1},x_{2}\in{D_{s}},t\in[0,T].$ (12)

The Hilbert space KL expansion of $u^{s}_{g}(t,x,\boldsymbol{\xi})$ [10]) in
subdomain $D_{s}$ can be written as

$u^{s}_{g}(t,x,\boldsymbol{\xi}(\omega))=u^{s}_{0}(t,x)+\sum_{i=1}^{d}\sqrt{\mu^{s}_{i}(t)}\phi^{s}_{i}(t,x)\eta^{s}_{i}(t,\omega),\quad
x\in{D_{s}},t\in[0,T],$ (13)

where $(\mu^{s}_{i}(t),\phi^{s}_{i}(t,x))$ are time-dependent eigenvalue and
eigenfunction in the Hilbert KL expansion obtained by solving the eigenvalue
problem at a time step $t$

$\int_{D_{s}}C^{s,t}_{u_{g}}(x_{1},x_{2})\phi^{s}_{i}(t,x_{1})dx_{1}=\mu^{s}_{i}(t)\phi^{s}_{i}(t,x_{2}),\quad
i=1,2,\ldots,d.$ (14)

The random variable $\eta^{s}_{i}$ can be written as

$\displaystyle\eta^{s}_{i}(t)$
$\displaystyle=\frac{1}{\sqrt{\mu^{s}_{i}(t)}}\int_{D_{s}}\left(u^{s}_{g}(t,x,\boldsymbol{\xi})-u^{s}_{0}(t,x)\right)\phi^{s}_{i}(t,x)dx,$
$\displaystyle=\frac{1}{\sqrt{\mu^{s}_{i}(t)}}\int_{D_{s}}\left(u_{0}(t,x)+\sum_{j=1}^{d}u_{j}(t,x)\xi_{j}-u^{s}_{0}(t,x)\right)\phi^{s}_{i}(t,x)dx.\quad
i=1,2,\ldots,d.$ (15)

By construction, $u_{0}(t,x)=u^{s}_{0}(t,x)$ for $x\in D_{s},$ and we find

$\displaystyle\eta^{s}_{i}(t)$
$\displaystyle=\frac{1}{\sqrt{\mu^{s}_{i}(t)}}\int_{D_{s}}\left(\sum_{j=1}^{d}u_{j}(t,x)\xi_{j}\right)\phi^{s}_{i}(t,x)dx,\quad
x\in D_{s},t\in[0,T],i=1,2,\ldots,d$
$\displaystyle=\sum_{j=1}^{d}\left(\frac{1}{\sqrt{\mu^{s}_{i}(t)}}\int_{D_{s}}u_{j}(t,x)\phi^{s}_{i}(t,x)dx\right)\xi_{j},\quad
x\in D_{s},t\in[0,T],i=1,2,\ldots,d$
$\displaystyle=\sum_{j=1}^{d}a^{s}_{ij}(t)\xi_{j},$ (16)

where
$a^{s}_{ij(t)}=\frac{1}{\sqrt{\mu^{s}_{i}(t)}}\int_{D_{s}}u_{j}(t,x)\phi^{s}_{i}(t,x)dx,i,j=1,\ldots,d.$
The quantities $a^{s}_{ij}(t)$ provide a time-dependent linear map between the
random variables $\eta^{s}_{i}(t)$ and the original random variables
$\xi_{j},$ for $j=1,\ldots,d.$ In each subdomain, and at each time instant
$t$, we can use the reduced dimensional basis $\\{\eta^{s}_{i}\\}$ to solve
the stochastic PDE in question using sparse grid collocation which is a non-
intrusive method.

##  4 Domain decomposition and basis adaptation for one-dimensional diffusion
equations

In this section, we present two test problems namely, a linear stochastic
diffusion equation and nonlinear stochastic diffusion (Richards) equation for
demonstrating the proposed time dependent basis adaptation and domain
decomposition methods.

###  4.1 Linear diffusion equation

Here, we use a linear 1D time-dependent stochastic diffusion equation to
illustrate the construction. We consider

$\displaystyle\frac{\partial u(t,x,\omega)}{\partial
t}-\frac{\partial}{\partial x}\left[a(x,\omega)\frac{\partial}{\partial
x}u(t,x,\omega)\right]=f,\quad\quad 0<x<L$ (17)

with boundary conditions

$\displaystyle u(t,0)=u_{0}\quad\text{and}\quad u(t,L)=u_{L},$ (18)

and initial condition $u(0,x)=u_{0}(x).$ Due to the linearity of the problem
we will use the superposition principle to compute the solution after domain
decomposition.

For each of the non-overlapping subdomains, $D^{(i)},i=1,\ldots,K$ we solve

$\displaystyle\frac{\partial u^{(i)}(t,x,\omega)}{\partial
t}-\frac{\partial}{\partial
x}\left[a(x,\boldsymbol{\xi})\frac{\partial}{\partial
x}u^{(i)}(t,x,\omega)\right]=f,\quad\quad x\in D_{s}$ (19)

with interface and boundary conditions

$\displaystyle\mathcal{C}^{(ij)}(x,u^{(i)},u^{(j)};a(x,\omega))=\beta(x,\omega)\;\;\rm{on}~{}\partial
D^{(i)}\cap\partial D^{(j)}\times\Omega,$ (20) $\displaystyle
u^{(i)}(t,0)=u_{0}\quad\text{and}\quad u^{(i)}(t,L)=u_{L},\rm{on}~{}\partial
D^{(i)}\cap\partial D\times\Omega.$ (21)

Since this is a one-dimensional linear equation, the domain decomposition
method can be simplified because the solution at the interface is a scalar
quantity. The solution in each subdomain can be obtained using $0$ and $1$
boundary conditions at the interfaces to obtain the solutions. Then, we can
use the superposition principle and apply interface conditions to get the
unknown solution at the interfaces.

###  4.2 Nonlinear diffusion equation

Here, we consider the nonlinear diffusion (Richards) equation with the van
Genuchten model [16]:

$\displaystyle\frac{\partial\psi(t,x,\omega)}{\partial
t}-\frac{\partial}{\partial x}\left[K(x,\omega)\left(\frac{\partial}{\partial
x}\psi(t,x,\omega)+1\right)\right]=0,\quad\quad 0<x<L$ (22)

with boundary conditions

$\displaystyle\psi(t,0)=\psi_{0}\quad\text{and}\quad\psi(t,L)=\psi_{L},$ (23)

and initial condition $\psi(0,x)$, where $\psi$ is the pressure head. The
water retention $S_{e}$ and hydraulic conductivity $K$ are related to the
pressure head $\psi$ [7] through

$S_{e}=\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}}=\left[\frac{1}{1+(\alpha_{vg}|\psi|)^{n}}\right]^{m},$
(24)

and

$K=K_{s}\sqrt{S_{e}}[1-(1-S_{e}^{1/m})^{m}]^{2},$ (25)

where $m=1-1/n;$ $\theta_{r}$ and $\theta_{s}$ are the residual and saturated
water contents respectively; $\alpha_{vg}$ and $n$ are the van Genuchten model
parameters. In this equation, we model the uncertain hydraulic conductivity as
a log-normal random field $K_{s}(x,\boldsymbol{\xi}).$ We note that since the
Richards equation is nonlinear, obtaining the value at the interface between
subdomains cannot be achieved through the superposition principle. Instead, we
employ an iterative algorithm where we start with an initial guess for the
interface solution and update it at each iteration in order to satisfy the
governing equation.

##  5 Numerical results

In this section, we present numerical solutions of a linear stochastic
diffusion equation and a nonlinear stochastic diffusion equation with the
domain decomposition and time-dependent basis adaptation method and provide
comparison with the solutions obtained with the time-independent basis
adaptation method. For the nonlinear diffusion equation, we also present
solutions obtained without domain decomposition in order to assess the
importance of domain decomposition.

###  5.1 Linear diffusion equation

Consider (17) on the spatial domain $x\in(0,1)$ with the boundary conditions

$\displaystyle u(t,0)=2.0\quad\text{and}\quad u(t,1)=1.0,$ (26)

and initial conditions

$\displaystyle u(0,x)=\begin{cases}2.0,&\text{if }0.375\leq x\leq 0.625\\\
1.0,&\text{otherwise}.\end{cases}$ (27)

Figure 1: Spatial domain decomposed into four subdomains

We model the random coefficient $a(x,\omega)=\exp[g(x,\omega)]$ using a log-
normal distribution with mean $a_{0}=0.5$ and standard deviation
$\sigma_{a}=0.25.$ The correlation length of $a(x,\omega)$ is set to $0.2$.
For the reference solution, we approximate the random coefficient
$a(x,\omega)$ using KL expansion with $d=10$ random variables, i.e.
$\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{10}).$ To obtain the reference
solution, we use the sparse-grid collocation method with sparse-grid level
$5.$ The total number of deterministic simulations required to compute the
reference solution is $8761.$ For the time-dependent basis adaptation, we
decompose the spatial subdomain into four subdomains (see Fig. 1) and in each
subdomain we approximate the solution using a stochastic basis of dimension
$r=3.$ To obtain the time-dependent basis, we need to solve the time dependent
SPDE at a few collocation points using a low sparse-grid level in full
dimension. Here, we use the sparse-grid level 2 and $r=10$, which contain $21$
collocation points. We use the solution at these collocation points to compute
the linear transformation matrix $A$ needed to obtain the reduced basis
$\boldsymbol{\eta}=A\boldsymbol{\xi}.$

Fig. 2 shows the probability density function (pdf) of the solution at $x=0.7$
and time $t=0.2,0.8$ and $1.6.$ We can see that for early times, the estimated
pdf of the solution using time-dependent basis adaptation and domain
decomposition (dashed blue line) matches well with the pdf of the reference
solution (solid black line) whereas the pdf of the solution obtained with
fixed basis adaptation and domain decomposition deviates (dashed red line).
For longer times, the pdf estimates of the time-dependent and fixed basis
adaptation converge. This is to be expected since the pdf of the diffusion
equation converges to a steady state. In particular, while the choice of the
basis adaptation is important for short times, eventually the contracting
nature of the linear diffusion evolution operator makes both time-dependent
and fixed basis adaptations to perform similarly. However, we note that the
relatively inexpensive determination of the time-dependent basis adaptation
allows us to follow the evolution of the pdf accurately for both short and
long times. This can be seen clearly in Fig.2 where the accuracy of the time-
dependent basis adaptation remains practically the same (note the change in
scale in the vertical axis as time progresses).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Linear diffusion equation. Probability density function of $u$ at
$x=0.7$ and a). $t=0.2$, b). $t=0.3$, c). $t=0.4$, d). $t=0.8$, e). $t=1.2$
and f). $t=1.6.$ The reference solution (solid black) is computed with
dimension $d=10$ and the reduced dimensional solutions (dashed red and blue)
are computed with $r=3$.

Figs. 3(a) and 3(b) show the color map of the mean and standard deviation of
the reference solution as a function of $x$ (horizontal axis) and $t$
(vertical axis). Figs. 3(c) and 3(d) show the absolute point error in the mean
and standard deviation of the solution using fixed basis adaptation. Figs.
3(e) and 3(f) show the absolute point error of the mean and standard deviation
of the solution using time-dependent basis adaptation compared to the
reference solution.

While both fixed and time-dependent basis adaptation methods result in similar
relative error for the mean, time-dependent basis adaptation results in
significantly lower relative error for the standard deviation. This is
important because a smaller error in the standard deviation estimate improves
the confidence in the prediction of the mean.

(a) Mean

(b) Standard deviation

(c) Abs. point. error in mean

(d) Abs. point error in standard deviation

(e) Abs. point error in mean

(f) Abs. point error in standard deviation

Figure 3: a) mean of reference solution ($d=10$) and b) standard deviation of
reference solution ($d=10$) c) error in mean of low dimensional solution
($r=3$) with fixed basis adaptation and d) error in standard deviation of low
dimensional solution ($r=3$) with fixed basis adaptation e) error in mean of
low dimensional solution ($r=3$) with time-dependent basis adaptation and f)
error in standard deviation of low dimensional solution ($r=3$) with time-
dependent basis adaptation as a function of $x$ (horizontal axis) and $t$
(vertical axis).

###  5.2 Nonlinear diffusion equation

In Eqs (22)-(25), we set $L=10$, $\psi_{0}=0$ and $\psi_{L}=-0.35$ and
$\psi(0,x)=\frac{\psi_{L}-\psi_{0}}{x_{L}-x_{0}}x$. We model the coefficient
$K_{s}(x,\boldsymbol{\xi})$ as a log-normal random field,
$K_{s}(x,\boldsymbol{\xi})=\exp[g_{s}(x,\boldsymbol{\xi})]$, where $g_{s}$ is
a Gaussian random field with the mean $\mu_{k}=5.0$, standard deviation
$\sigma_{k}=0.5$, and the exponential correlation function with the
correlation length $L/4$. The corresponding coefficient of variation is
$\sigma_{k}/\mu_{k}=0.1$ indicating a weak heterogeneity. In this section, we
demonstrate that even for such a small coefficient of variation, a significant
error in the standard basis adaptation method (with constant basis)
accumulates over time, while a solution obtained using the time-dependent
basis matches well with the reference solution for the considered time
interval.

Figs. 4(a-f) show the pdf of $u$ at $x=6.15$ and $t=0.005,10,20,30,40$ and
$50$ respectively. The reference pdf (solid black line) is obtained by solving
the PDE (22) with a $10$-dimensional random vector $\boldsymbol{\xi}$ using
sparse grid level $5.$ The total number of simulations needed for this
solution is $8761.$ In the basis adoption method with time-independent basis
(the PDF is represented by the dashed red lines), $r=5$ dimensions are used
and the basis adaptation is performed only once at $t=0.005.$ The same number
of dimensions ($r=5$) are used in the basis adaptation method with the time-
dependent basis (PDFs are denoted by dashed blue lines). The new basis are
computed at each time step prior to solving the PDE. Fig. 4 demonstrates that
the pdf predicted with time-dependent basis is significantly more accurate
than the PDF obtained with fixed basis.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Nonlinear diffusion equation: Probability density function of $u$ at
$x=6.15$ and a). $t=0.005$, b). $t=10$, c). $t=20$, d). $t=30$, e). $t=40$ and
f). $t=50.$ The reference solution (solid black) is computed with dimension
$d=10$ and the reduced dimensional solutions (dashed red and blue) are
computed with $r=5$.

Figs. 5(a) and 5(b) show the the mean and standard deviation of the reference
solution as a function of $x$ (horizontal axis) and $t$ (vertical axis). Figs.
5(c) and 5(d) present the absolute point error in the mean and standard
deviation of the solution using fixed basis adaptation. Figs. 5(e) and 5(f)
display the absolute point error in the mean and standard deviation of the
solution of the Richards using time-dependent basis adaptation compared to the
reference solution. While the point errors of the mean are similar in the the
time-independent and time-dependent basis adaptation methods, the point errors
in the standard deviation are approximately 50% smaller with the time-
dependent basis.

In the above examples, the solutions are obtained with the domain
decomposition method by dividing the domain into $4$ non-overlapping
subdomains and in each subdomain the solution is computed using $781$
simulations corresponding to the reduced dimension $r=5$ and sparse-grid level
$5.$ The total number of simulations is $781\times 4=3124$, which is
significantly smaller than $8761$ simulations in the reference solution. The
iterative method for computing values at the interface between the subdomains
does not add significantly to the computational cost.

(a) Mean

(b) Standard deviation

(c) Abs. point error in mean

(d) Abs. point error in standard deviation

(e) Abs. point error in mean

(f) Abs. point error in standard deviation

Figure 5: Richards equation: a) mean of reference solution ($d=10$) and b)
standard deviation of reference solution ($d=10$) c) error in mean of low
dimensional solution ($r=5$) with fixed basis adaptation and d) error in
standard deviation of low dimensional solution ($r=5$) with fixed basis
adaptation e) error in mean of low dimensional solution ($r=5$) with time-
dependent basis adaptation and f) error in standard deviation of low
dimensional solution ($r=5$) with time-dependent basis adaptation as a
function of $x$ (horizontal axis) and $t$ (vertical axis).

Finally, in Fig. 6 we present errors in the solution obtained with the time-
dependent basis adaptation without domain decomposition. These errors are
smaller than in the time-independent basis adaptation method but larger than
in the time-dependent basis adaptation method with the domain decomposition.

(a) Abs. point error in mean

(b) Abs. point error in standard deviation

(c) Abs. point error in mean

(d) Abs. point error in standard deviation

Figure 6: Richards equation: Basis adaptation without domain decomposition a)
error in mean of low dimensional solution ($r=5$) with fixed basis adaptation,
b) error in standard deviation of low dimensional solution ($r=5$) with fixed
basis adaptation c) error in mean of low dimensional solution ($r=5$) with
time-dependent basis adaptation and d) error in standard deviation of low
dimensional solution ($r=5$) with time-dependent basis adaptation as a
function of $x$ (horizontal axis) and $t$ (vertical axis).

##  6 Conclusions

We proposed a dimension reduction method with time-dependent basis adaption
and the spatial domain decomposition for stochastic PDEs. The time-dependent
basis adaptation is computed using an inexpensive calculation of the full
dimensional stochastic solution in the whole spatial domain using lower
sparse-grid level (2 in this work).

We have used the proposed approach for solving the time-dependent one-
dimensional linear and nonlinear diffusion equations with random diffusion
coefficient. Our results show that the proposed approach significantly
outperforms the time-independent basis adaptation method. Although the time-
dependent basis adaptation method alone is quite effective, combining with the
domain decomposition method further improved the results. To the best of our
knowledge, this is the first time-dependent dimension reduction approach for
time-dependent stochastic PDEs. In our future work, we will extend our
approach to two and three dimensional stochastic PDEs.

##  7 Acknowledgments

This research was supported by the U.S. Department of Energy, Office of
Science, Office of Advanced Scientific Computing Research as part of the
“Uncertainty Quantification For Complex Systems Described by Stochastic
Partial Differential Equations” project. Pacific Northwest National Laboratory
is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.

## References

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# Can multiple-choice questions really be useful
in detecting the abilities of LLMs?

###### Abstract

Multiple-choice questions (MCQs) are widely used in the evaluation of large
language models (LLMs) due to their simplicity and efficiency. However, there
are concerns about whether MCQs can truly measure LLM’s capabilities,
particularly in knowledge-intensive scenarios where long-form generation (LFG)
answers are required. The misalignment between the task and the evaluation
method demands a thoughtful analysis of MCQ’s efficacy, which we undertake in
this paper by evaluating nine LLMs on four question-answering (QA) datasets in
two languages: Chinese and English. We identify a significant issue: LLMs
exhibit an order sensitivity in bilingual MCQs, favoring answers located at
specific positions, i.e., the first position. We further quantify the gap
between MCQs and long-form generation questions (LFGQs) by comparing their
direct outputs, token logits, and embeddings. Our results reveal a relatively
low correlation between answers from MCQs and LFGQs for identical questions.
Additionally, we propose two methods to quantify the consistency and
confidence of LLMs’ output, which can be generalized to other QA evaluation
benchmarks. Notably, our analysis challenges the idea that the higher the
consistency, the greater the accuracy. We also find MCQs to be less reliable
than LFGQs in terms of expected calibration error. Finally, the misalignment
between MCQs and LFGQs is not only reflected in the evaluation performance but
also in the embedding space. Our code and models can be accessed at
https://github.com/Meetyou-AI-Lab/Can-MC-Evaluate-LLMs.

Keywords: Natural Language Processing, Large Language Model, Question
Answering, Text Generation, Evaluation Methods

Can multiple-choice questions really be useful
in detecting the abilities of LLMs?

Wangyue Li†♢, Liangzhi Li†◆††thanks:  Corresponding author. , Tong Xiang†◆,
Xiao Liu†, Wei Deng♢♡, Noa Garcia◆
---
†Meetyou AI Lab, ♢Southwestern University of Finance and Economics, ◆Osaka
University
♡Chongqing University of Posts and Telecommunications
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Over the past few years, large language models (LLMs) have exhibited
remarkable performance on a wide range of question-answering (QA) tasks (Brown
et al., 2020; Kadavath et al., 2022; Robinson and Wingate, 2023). The
evaluation of LLMs’ strengths and limitations often relies on diverse
benchmarks presented in different formats (Singhal et al., 2023; Liu et al.,
2023b), domains (Jin et al., 2019; Zhong et al., 2020), and languages (Petroni
et al., 2019; Bang et al., 2023). As previous research has shown (Liang et
al., 2022; Chang et al., 2023; Li et al., 2023a; Chia et al., 2023),
evaluation using benchmarks is essential for the detection and mitigation of
various issues such as misinformation (Zheng et al., 2021; Gao et al., 2022),
hate speech (ElSherief et al., 2021; Lu et al., 2023), and malicious uses (Xu
et al., 2021; Ganguli et al., 2022; Shaikh et al., 2023; Zou et al., 2023).
Such mechanisms are critical for safeguarding against harmful content and
promoting responsible usage of LLMs in various contexts.

QA benchmarks come in a variety of formats, including True/False questions
(TFQs) in which models predict whether a statement in the question is correct
or not, multiple-choice questions (MCQs), in which multiple candidate answers
accompany the input question, and long-form generation questions (LFGQs), in
which a generated answer could span multiple sentences. Among these, multiple-
choice is the most popular format as it allows a simple and quick assessment
of model performance (Bhakthavatsalam et al., 2021; Ramamurthy and Aakur,
2022; Liu et al., 2023a; Huang et al., 2023). However, MCQs also present
several limitations, such as potential misalignment with real-world use cases
where LLMs are often required to answer questions in long-from generation
format (Nuance, 2023; Bommasani et al., 2021). In addition, LLMs have been
shown to be affected by changes in the position of the candidate answers
(Zheng et al., 2023; Wang et al., 2023) and their contents (Pezeshkpour and
Hruschka, 2023) when answering MCQs. The aforementioned problems highlight the
limitations of MCQs benchmarks in evaluating LLMs, which could potentially
lead to overestimation of LLMs capabilities.

With the above issues in mind, our motivation is to explore the limitations
and characteristics of both MCQs and LFGQs as main evaluation formats in QA
tasks. We aim to answer the following research questions:

  1. 1.

How does the arrangement of options in MCQs influence LLMs’ selection of
responses?

  2. 2.

What methodologies can be employed to conduct comprehensive comparative
experiments between MCQs and LFGQs? Additionally, what specific aspects should
be considered when conducting comparative tests?

The answers to these questions contribute to the understanding and comparison
between MCQs and LFGQs as evaluation formats in QA tasks. Given the prevalence
of MCQs as the dominant evaluation format, our aim is to thoroughly examine
their efficacy. This begins with a detailed exploration of MCQs’ capabilities
and subsequently extends to a comparative analysis with LFGQs, providing a
comprehensive assessment of both formats.

We address the first question by conducting a series of experiments (§3) to
reveal the sensitivity of LLMs to answering MCQs by applying slight
perturbations to the positional order of the options. We find significant
differences between the answers in multiple LLMs (§3.1). We also identify
specific patterns of the selected answer according to its position that varies
among different LLMs (§3.2). For the second question, we conduct comparative
experiments (§4) to quantify the misalignment between MCQs and LFGQs on three
different spaces: direct output space (§4.1), token logits space (§4.2), and
hidden embedding space (§4.3). By doing this, we aim to gain a deeper
understanding of the unique characteristics between the two types of
questions.

Our key findings reveal that:

  * •

LLMs exhibit order sensitivity in bilingual MCQs, favoring answers at the
first position.

  * •

Answers obtained from MCQs and LFGQs for identical questions have a low
correlation.

  * •

Higher consistency does not indicate better model performance.

  * •

The misalignment between MCQs and LFGQs is evident in the evaluation
performance as well as in the embedding space.

Overall, our study aims to provide a better understanding of the difference in
QA formats in LLM evaluation, uncover underlying patterns, and shed light on
the improvement of current methods.

## 2\. Experimental Details

##### Models

We use different models on different experiments, tailoring our choices based
on the specific goals of each experiment, as summarized in Table 1. To check
whether LLMs are sensitive to the order of the candidate answers (§3), we
evaluate three models: ChatGLM-6B (Zeng et al., 2023; Du et al., 2022) and two
models from the GPT family, namely GPT-3.5-turbo (OpenAI, 2023b) and GPT-4
(OpenAI, 2023a). In comparing MCQs and LFGQs (§4), we again use different
models on the three different spaces. For the direct output space (§4.1), we
use GPT-3.5-turbo, GPT-4, and ChatGLM-6B, considering both their diversity and
performances. For the token logits space (§4.2), we only test GPT-3.5-turbo,
as it is the only model that can output token probabilities (Manakul et al.,
2023) within three models. Finally, in the embedding space (§4.3), we conduct
experiments with models from multiple popular LLM families across various
sizes, including StableLM-Tuned-Alpha-3/7B (Stability-AI, 2023), RedPajama-
INCITE-Instruct-3B-v1 (Computer, 2023), Llama-2-7b-chat-hf (Touvron et al.,
2023b), Dolly-v2-2/7/12B (Conover et al., 2023), Vicuna-7b-v1.3 (Chiang et
al., 2023), and Open-llama-3/7B (Touvron et al., 2023a).

| Are LLMs sensitive to order? |  | MCQ vs LFGQ
---|---|---|---
Model | Order Sensitivity | Patterns Decomposition |  | Direct Output | Token Logits | Embeddings
GPT-3.5-turbo (OpenAI, 2023b) | ✓ | ✓ |  | ✓ | ✓ |
GPT-4 (OpenAI, 2023a) | ✓ | ✓ |  | ✓ |  |
ChatGLM-6B (Zeng et al., 2023) | ✓ | ✓ |  | ✓ |  |
Stablelm-tuned-$\alpha$ (Stability-AI, 2023) |  |  |  |  |  | ✓
RedPajama-INCITE-3B-v1 (Computer, 2023) |  |  |  |  |  | ✓
Dolly-v2 (Conover et al., 2023) |  |  |  |  |  | ✓
Vicuna-7b-v1.3 (Chiang et al., 2023) |  |  |  |  |  | ✓
Open-llama (Touvron et al., 2023a) |  |  |  |  |  | ✓
Llama-2-7b-chat-hf (Touvron et al., 2023b) |  |  |  |  |  | ✓

Table 1: Summary of the LLMs used in each of our analyses.

##### Datasets

We conduct experiments on four evaluation benchmarks:

  1. 1.

CARE-MI (Xiang et al., 2023): A Chinese benchmark for evaluating LLM
misinformation in the maternity and infant care domain. It includes $1,612$
LFGQs. The questions can also be obtained as MCQs and TFQs from the original
MLEC-QA(Li et al., 2021) and MEDQA (Jin et al., 2020) datasets, according to
the question generation process of CARE-MI, resulting in each question being
formulated in the three formats: MCQ, LFGQ, and TFQ.

  2. 2.

M3KE (Liu et al., 2023a): A dataset with $20,477$ standard Chinese questions
for $71$ tasks, encompassing all major levels of Chinese education system,
including humanities, history, politics, law, education, psychology, science,
technology, art and religion in MCQ format. Each question presents four
candidate answers.

  3. 3.

ARC (Clark et al., 2018): A dataset with natural, grade-school science
questions (authored for human tests) in English. It is the largest public-
domain set of this kind with $7,787$ questions. Each question contains four
candidate answers.

  4. 4.

MATH: A synthetic dataset randomly generated by a script with simple
mathematical questions in English. Each question has four candidate answers.

Dataset | Size | Lang. | Format
---|---|---|---
CARE-MI | $344$ | ZH | MCQ/LFGQ/TFQ
M3KE | $299$ | ZH | MCQ
ARC | $291$ | EN | MCQ
MATH | $300$ | EN | MCQ
Total | $1,234$ | - | -
Table 2: Summary of the evaluation datasets. Lang. stands for the language of
the datasets.

As shown in Table 2, we ensure data balance by using a similar number of
samples from each dataset.

For the first research question (§3), we use all the datasets: CARE-MI, M3KE,
ARC, and MATH. In the second research question (§4), we use the CARE-MI
dataset for the direct output (§4.1) and token logits analysis (§4.2) as it is
the only dataset offering the three different QA formats. The ARC dataset is
used on the embedding space analysis (§4.3), wherein we extend its MCQs to
LFGQs by not presenting the candidate answers to the LLMs.

The selection of benchmarks is guided by three specific criteria: (1) Source
diversity: we aim to conduct our analyses across different domains. (2)
Language: We presume language is a potential factor influencing LLMs
evaluation performance, so we conduct experiments on two high-resource
languages, i.e., Chinese and English. (3) Performance-level: By incorporating
benchmarks with varying levels of LLMs demonstrated performance, we aim to
better understand how model proficiency influences results in the different QA
formats. Additionally, in the token logits space (§4.2), we investigate
changing the number of candidate answers to explore their impact on the
expected calibration error.

##### Prompt design

To encourage the generation of concise responses, we provide LLMs with prompts
both prior to (pre-prompt) and following (post-prompt) each question in any
dataset format. Additionally, for MCQs, each question is accompanied by four
candidate options, with only one being correct. The pre-prompt for MCQs is
“Please select a correct option", while the post-prompt “Only one option can
be selected. No explanation is allowed". For LFGQs, the post-prompt is “Just
answer in one sentence", and for TFQs, it is “Just answer ‘yes’ or ‘no’". We
find that this prompt design facilitates the generation of brief content,
aiding subsequent accuracy evaluation (§4.1) and automatic confidence
calculation (§4.2).

## 3\. Are LLMs sensitive to the order of candidate answers?

We first investigate how the arrangement of candidate answers in MCQs datasets
affects the evaluation of LLMs. We find that LLMs consistently exhibit a
strong preference for specific positions when presented with options in
different orders, as illustrated in Figure 1.

Figure 1: Example of order sensitivity experiments, in which the correct
answer is D. In the ARC dataset, when predicting a wrong option (A, B, or C),
the LLM prefers the option located in the first position.

### 3.1. Order Sensitivity

To check whether there are significant differences in LLMs’ answers when the
candidate options are arranged in a different order, we employ the chi-squared
test (McHugh, 2013). To isolate the influence of the correct answer, we
designate option D as the only correct option for all the questions. Then, we
establish two scenarios: in CASE1, the option order is ’ABCD’, and in CASE2,
it is ’BACD’. Importantly, when arranging the option order, we also rearrange
the contents and positions of each candidate option accordingly, rather than
simply altering the numbering, as shown in Figure 1.

In the chi-squared test, we set the null hypothesis, $H_{0}$, stating that the
responses in CASE1 and CASE2 originate from the same distribution. The chi-
squared statistic is calculated as

$X^{2}=\sum\limits_{i=0}^{N}\frac{(O_{i}-R_{i})^{2}}{R_{i}},$ (1)

where $\sum\limits_{i=0}^{N}$ is the sum of $N$ candidate options , $O_{i}$
the frequency of each option in CASE1, and $R_{i}$ the frequency of each
option in CASE2. With the significance test, we can get the p-value $P$ from
the chi-squared probabilities based on the chi-squared statistic and the
degrees of freedom, $N-1=3$. Additionally, we can calculate the accuracy gap,
which represents the difference between the original accuracy and the accuracy
after reordering.

|  | GPT3.5 | GPT4 | ChatGLM
---|---|---|---|---
CARE-MI
| $X^{2}$ | $144.192$ | $15.660$ | $27.605$
| $P$ | **$0.000$ | *$0.001$ | **$0.000$
| Acc | $0.203$ | $0.637$ | $0.378$
| Gap | $-0.043$ | $-0.029$ | $-0.116$
M3KE
| $X^{2}$ | $90.308$ | $20.829$ | $12.377$
| $P$ | **$0.000$ | **$0.000$ | **$0.006$
| Acc | $0.381$ | $0.632$ | $0.411$
| Gap | $-0.030$ | $-0.017$ | $+0.014$
ARC
| $X^{2}$ | $36.515$ | $2.681$ | $10.511$
| $P$ | **$0.000$ | $0.443$ | *$0.015$
| Acc | $0.512$ | $0.935$ | $0.553$
| Gap | +$0.148$ | -$0.031$ | -$0.116$
MATH
| $X^{2}$ | $25.129$ | $4.513$ | $90.566$
| $P$ | **$0.000$ | $0.211$ | **$0.000$
| Acc | $0.597$ | $0.780$ | $0.480$
| Gap | +$0.000$ | -$0.023$ | -$0.043$
Table 3: LLMs’ order sensitivity results. The rearrangement of options makes
LLMs output different answers.* indicates $P<0.05$, ** indicates $P<0.001$,
and bold indicates larger than significance level $\alpha$.

Results are shown in Table 3, from which we note the following observations:

  1. 1.

There is a considerable disparity in LLMs’ outputs across the two scenarios.
Except for two instances,111GPT4 model on the ARC ($X^{2}=2.681$, $P=0.443$)
and the MATH ($X^{2}=4.513$, $P=0.211$) datasets. all the results have a
p-value $P<0.05$, rejecting the null hypothesis and implying that the
distribution of answers predicted by the model varies significantly when
options A and B are interchanged. This indicates that the order of options
significantly influences LLMs’ predictions in MCQs datasets.

  2. 2.

Among the GPT family, the rearrangement of options has a more pronounced
effect on GPT-3.5-turbo, with bigger accuracy gaps, than on GPT-4.

  3. 3.

Higher accuracy can mitigate significant differences in the order arrangement
to some extent. Results from GPT-4 on the ARC and the MATH datasets indicate
that high accuracies ($\geq 0.780$) can lead to not rejecting the null
hypothesis.

  4. 4.

There is no evident correlation between the accuracy gap and the original
accuracy. A higher accuracy does not necessarily imply a lower gap between the
two scenarios.

### 3.2. Pattern Decomposition

Next, we further explore the pattern decomposition of LLMs to investigate
potential patterns underlying their sensitivity to order. We propose the
following two hypotheses for exploration: (1) LLMs may have different
positional preferences due to their different model bases; (2) LLMs may have
different positional preferences depending on whether they have previously
memorized the contents of the datasets.

|  |  | GPT-3.5-turbo |  | GPT-4 |  | ChatGLM-6B
---|---|---|---|---|---|---|---
Dataset | CASE |  | 1st | 2nd | 3rd |  | 1st | 2nd | 3rd |  | 1st | 2nd | 3rd
CARE-MI | C1 |  | $41$ | $33$ | $26$ |  | $44$ | $37$ | $19$ |  | $37$ | $42$ | $21$
C2 |  | $83$ | $8$ | $9$ |  | $61$ | $26$ | $13$ |  | $61$ | $23$ | $16$
M3KE | C1 |  | $25$ | $35$ | $40$ |  | $49$ | $26$ | $25$ |  | $33$ | $39$ | $28$
C2 |  | $75$ | $1$ | $24$ |  | $49$ | $31$ | $30$ |  | $46$ | $32$ | $22$
ARC | C1 |  | $47$ | $27$ | $26$ |  | $37$ | $31$ | $32$ |  | $33$ | $28$ | $39$
C2 |  | $52$ | $16$ | $32$ |  | $21$ | $36$ | $43$ |  | $33$ | $36$ | $31$
MATH | C1 |  | $22$ | $60$ | $18$ |  | $41$ | $30$ | $29$ |  | $17$ | $69$ | $14$
C2 |  | $61$ | $3$ | $36$ |  | $47$ | $28$ | $25$ |  | $22$ | $57$ | $11$
Table 4: Pattern decomposition results. C1 refers to CASE1, and C2 to CASE2.
The numbers indicate the percentage (%) of incorrect options (A, B, or C) that
each model selects for each position (1st position, 2nd position, and 3rd
position). Deeper background indicates a higher preference for that position.

We use the same LLMs as in §3.1, as they can provide concise answers to the
questions and come from different model bases. Regarding the datasets, CARE-
MI, M3KE, and ARC are derived from website documents, while the MATH dataset,
synthetically generated by us, ensures that the LLMs have not been exposed to
identical questions during training. Results are presented in Table 4, from
which we can extract the following conclusions:

  1. 1.

Within the GPT family, GPT-3.5-turbo and GPT4 exhibit different behavior. When
predicting incorrect options (A, B, or C), GPT4 shows a stronger inclination
towards the option positioned first compared to GPT-3.5-turbo. Specifically,
when option B is presented first, GPT-3.5-turbo tends to lean towards
selecting option B. Furthermore, ChatGLM-6B showcases a certain preference for
the first two options.

  2. 2.

The behavior of the LLMs remains consistent across datasets originating from
different languages and sources. Hence, we can conclude that the dataset’s
language or source, regardless of whether the models were previously exposed
to them or not, is not the underlying cause of the models’ positional
preferences.

### 3.3. Yes, LLMs are sensitive to ordering

Our experiments showed that the order of candidate answers in MCQs
significantly impacts LLMs outputs. GPT-3.5-turbo and GPT4 exhibited different
preferences, while ChatGLM-6B showed a certain preference for the first two
positions. In addition, the positional preferences in each LLM seemed to
remain consistent across datasets originating from different languages and
sources.

These findings are problematic because they reveal potential biases and
inconsistencies in LLM outputs, which can affect the reliability and accuracy
of their responses. Failure to understand and address these preferences may
lead to biased recommendations, inaccurate information retrieval, and flawed
decision-making. It is important to develop methods to mitigate these effects,
as well as to investigate evaluation protocols that are less impacted by
positional preferences. In light of these observations, in the next section,
we compare MCQs and LFGQs evaluation methods.

## 4\. Multiple Choice vs
Long Form Generation

To compare QA evaluation formats and gain a deeper understanding of LLMs
evaluation protocols, we expose several LLMs to the same questions presented
in different formats. Then, we analyze and compare the results in three
spaces: the direct output space (§4.1), the token logits space (§4.2), and the
embedding space (§4.3).

### 4.1. Direct Output

In the direct output space, which refers to the responses generated by the
LLMs, accuracy is one of the most common evaluation metrics used for
benchmarking purposes and performance assessment. The difference in accuracy
between the MCQs and LFGQs formats is the first aspect we consider (§4.1.1).
Additionally, we study the relationship between consistency and accuracy
(§4.1.2) by exploring whether LLMs, if familiar with a particular concept,
tend to generate responses that are similar and encompass consistent factual
information (Manakul et al., 2023).

#### 4.1.1. Accuracy

We randomly select $100$ samples from the CARE-MI dataset and evaluate GPT4,
GPT-3.5-turbo, and ChatGLM-6B on them. For MCQs, accuracy is computed as
usual, i.e. if the predicted answer matches the ground truth, it is considered
correct. For LFGQs, accuracy is determined through human evaluation, with $0$
denoting an incorrect response and $1$ a correct one. Figure 2 (top) compares
the accuracy between MCQs and LFGQs across the three LLMs. Notably, the
accuracies of MCQs are consistently higher than those of LFGQs. This
difference can be attributed to the fact that MCQs offer candidate options,
facilitating the prediction task. To delve deeper into the analysis, in Figure
2 (bottom), we visualize a matrix in which, for the same question, there are
four scenarios: 1) the response is correct in both formats, 2) the response is
incorrect in both formats, 3) the response is correct in MCQs but incorrect in
LFGQs, and 4) the response is correct in LFGQs but incorrect in MCQs. Results
show that there are a relatively large number of questions where the LLMs can
respond correctly in MCQs, but fail in the LFGQs format. Furthermore, we
quantify the differences in accuracy produced by the two formats with Pearson
correlation coefficients. The obtained values are remarkably low: $0.39$ for
GPT4, $0.7$ for GPT-3.5-turbo, and $0.33$ ChatGLM-6B, clearly indicating that
different versions of the same question yield different answers from the LLM.

Figure 2: Comparison between MCQs and LFGQs on the CARE-MI dataset. Top:
accuracy and pearson correlation. Bottom: MCQ vs LFGQs matrix.

#### 4.1.2. Consistency

Next, we explore the relationship between consistency and accuracy.
Consistency stands for the degree to which the LLMs provide the same answer
when asked the same question multiple times. For example, an answer of ‘AAAAB’
is considered more consistent than ‘BCDAB’ when presented with the same
question five times. To conduct this evaluation, we first define the
quantitative measures for consistency and accuracy in a sequence of responses
to a repeated question.

Formally, let $\mathcal{A}=\\{A_{1},A_{2},...,A_{D}\\}$ be a sequence of
answers, where a model is queried $D$ times. Each answer $A_{i}\in\mathcal{A}$
is selected from a set of $N$ unique options
$\mathcal{O}=\\{Opt_{1},Opt_{2},...,Opt_{N}\\}$. From $\mathcal{A}$, we derive
a count sequence
$\mathcal{C}=\\{\textsc{count}(Opt_{1}),\textsc{count}(Opt_{2}),...,\textsc{count}(Opt_{N})\\}$,
where $\textsc{count}(Opt_{i})$ represents the number of occurrences of
$Opt_{i}$ in $\mathcal{A}$, marking $Opt_{\max}$ as the option with the
largest $\textsc{count}(Opt_{i})$. We define sequence consistency $K$ as

$\begin{split}K(\mathcal{A})&=\frac{1}{D}\textsc{count}(Opt_{max})\\\
&+\frac{1}{D}\sum\limits_{Opt_{i}\neq
Opt_{\max}}\max(0,\textsc{count}(Opt_{i})-1).\end{split}$ (2)

As for accuracy, if $A_{ref}\in\mathcal{O}$ is the correct answer, the
accuracy for sequence $\mathcal{A}$ can be defined as

$\text{Acc}(\mathcal{A})=\frac{1}{D}\textsc{count}(A_{ref}).$ (3)

In the experiments, we use the same samples as in §4.1.1 and repeat each
question five times for each LLM. To compute consistency and accuracy on LFQs,
we manually group the long-text generated answers into options so that answers
with similar meanings are grouped together under the same option. We also
explore the impact of different temperatures, which is the parameter that
controls the degree of randomness of the generated text, by using values $0$,
$0.5$, and $1$.

Figure 3 shows GPT-3.5-turbo’s consistency for MCQs and LFGQs across the three
temperature values. Both formats tend to be consistent in their answers. Even
when the temperature is increased, consistency does not decrease notably.
Between the two formats, LFGQs tend to be more consistent than MCQs. We also
calculate the Pearson correlation coefficient between consistency and
accuracy. In the case of MCQs, the Pearson correlation coefficient is $0.32$,
while for LFGQs, the coefficient reaches $0.416$, implying that higher
consistency does not necessarily mean more correct. Our findings suggest that
a higher level of consistency indicates a sharper probability distribution of
specific knowledge, but it does not guarantee the correctness of the
knowledge. Unlike SelfCheckGPT (Manakul et al., 2023), which leverages the
idea that the higher the consistency, the higher the correctness, we do not
find a direct relationship between consistency and accuracy. We believe this
is due to the knowledge required to answer the evaluation dataset. While
SelfCheckGPT is evaluated on information from famous individuals Lebret et al.
(2016), we use specialized professional medical datasets.

Figure 3: GPT-3.5-turbo’s consistency on MCQs and LFGQs with different
temperatures.

### 4.2. Token Logits

To compare MCQs and LFGQs in the token logits space, which is the space of
predicted probabilities, we rely on two techniques: unified confidence
calculation and expected calibration error.

#### 4.2.1. Unified confidence calculation

One of the mainstream approaches used to analyze why LLMs select specific
options when answering MCQs is through token logits (Manakul et al., 2023).
GPT-3.5-turbo, for instance, can generate log probabilities for the most
probable tokens associated with each output
token.222https://platform.openai.com/docs/guides/gpt However, while there are
formulas to calculate confidence for multiple options (Jiang et al., 2021;
Holtzman et al., 2021; Lin et al., 2022a), direct utilization of token
probability calculations for comparing MCQs with LFGQs is not straightforward.

Since our goal is to compare MCQs and LFGQs, we follow Jiang et al. (2021) and
propose a unified confidence calculation applicable to the three QA formats:
MCQs, LFGQs, and TFQs. Let us assume an input question $q$ that makes a LLM
generate the set of answers $\mathcal{A}$.

Each answer $A_{i}\in\mathcal{A}$ contains $|A_{i}|$ tokens. Each token,
denoted as $t_{k}$, with $1\leq k\leq|A_{i}|$, has a corresponding
autoregressive token log probability $P_{log}(t_{k}|q,t_{<k})$. We first
compute the average token log probability of each answer $A_{i}$ as

$P_{avg}(A_{i}|q)=\frac{{\sum_{i=1}^{|A_{i}|}}P_{log}(t_{k}|q,t_{<k})}{\max(1,|A_{i}|)}.$
(4)

From the initial set of answers $\mathcal{A}$, which may contain duplicates,
we consolidate them into $z$ unique answers, denoted as
$\mathcal{A}^{uni}=\\{A_{1}^{uni},A_{2}^{uni},...,A_{z}^{uni}\\}$, where
$z\leq D$. For each unique answer, we select the highest log probability
observed for any instance of that answer in $\mathcal{A}$, denoted as
$P_{log}^{highest}(A_{i}^{uni}|q)$. Subsequently, we rank the first $W$ unique
answers, where $W\leq z$,333$W=4$ in our experiments. from $\mathcal{A}^{uni}$
in descending order by their frequency and the corresponding
$P_{log}^{highest}(A_{i}^{uni}|q)$, to filter out excessively similar
responses and maintain the diversity in the unique answers. We calculate the
standardized confidence for the first $W$ answers as

$C_{N}(A_{w}^{uni})=\frac{e^{p^{higest}_{avg}}(A_{w}^{uni}|q)}{\sum^{W}_{w=1}e^{p^{higest}_{avg}}(A_{w}^{uni}|q)}.$
(5)

Finally, For MCQs, we use regularization matching to combine first $W$ answers
with four candidate options, and get the final label ($0$ or $1$) with the sum
of corresponding standardized confidence. For LFGQs, we directly get the final
label ($0$ or $1$) according to standardized confidence for the first $W$
answers and human labeling for each answer.

#### 4.2.2. Expected Calibration Error

After obtaining the unified confidence, we compute model calibration (Gupta et
al., 2006; Ahmed et al., 2020) to test whether a LLM exhibits good calibration
across different dataset evaluation formats. A well-calibrated model should
provide confidence (i.e., logit) estimates that closely match the actual
probability of the correctness of the answer. Inaccurate predictions should
correspond to low confidence (i.e, logit) values, whereas accurate predictions
should yield high confidence (i.e., logit) values.

In practice, we employ a commonly used metric known as expected calibration
error (ECE) (Niculescu-Mizil and Caruana, 2005) to assess the alignment of
confidence and accuracy. ECE is computed as the weighted average of the
difference between the accuracy and confidence. To measure confidence
quantitatively, we divide the $[0,1]$ interval into multiple bins. Each sample
falls into one of these bins based on the model’s predicted results. The
average model confidence is calculated in each bin, and then compared with the
average accuracy of the sample real label in the bin. The absolute value of
these two differences can measure the model’s confidence. A larger difference
indicates lower model confidence. Formally,

$\text{ECE}=\sum_{b=1}^{\mathcal{B}}\frac{|n_{b}|}{N}|\text{acc}(b)-\text{conf}(b)|.$
(6)

where $b$ represents the $b$-th bin, $\mathcal{B}$ represents the total number
of bins, $n_{b}$ represents the number of samples in the $b$-th bin,
$\text{acc}(b)$ represents the average value of the true label of the sample
in the $b$-th bin, $\text{conf}(b)$ represents the average value of the model
prediction probability in the $b$-th bin. In our experiments, we set
$\mathcal{B}=100$.

#### 4.2.3. Results

Within the CARE-MI dataset, we use the three QA formats, MCQs, LFGQs, and
TFQs, to compute ECE and reliability. We use confidence scores and true labels
to draw reliability diagrams as in (Kängsepp et al., 2022). A reliability
diagram closely aligning with the identity line suggests good model
calibration, while a significant deviation indicates poor calibration. Results
are shown in Figure 4 and in Table 5. LLMs operating on MCQs exhibit the
poorest calibration and highest ECE compared to the other two formats. This
suggests that the LLMs’ predictions in MCQs are not accurately aligned with
the true probability of correct answers, indicating overconfidence in their
responses. Additionally, we observe that the ECE in TFQs ($0.276$), which
contain only two candidate options, is lower than in MCQs ($0.426$), which
have four candidate options. To investigate the impact of the number of
candidate answers, we conduct experiments by varying the number of options in
MCQs across the four datasets. We analyze whether the number of options and
the domain of each dataset affect ECE. Throughout these experiments, we
maintain the correct answer consistently positioned as the last option. As
depicted in Table 5, we do not find a clear correlation between these factors
and ECE, meaning that the number of options and the domain do not seem to
influence LLM’s performance.

Figure 4: Reliability diagrams for MCQs, LFGQs, and TFQs on the CARE-MI dataset. Format | CA | CARE-MI | M3KE | ARC | MATH
---|---|---|---|---|---
MCQ | $4$ | $0.426$ | $0.317$ | $0.492$ | $0.281$
MCQ | $3$ | $0.329$ | $0.364$ | $0.382$ | $0.259$
MCQ | $2$ | $0.414$ | $0.427$ | $0.280$ | $0.257$
LFGQ | - | $0.304$ | - | - | -
TFQ | $2$ | $0.276$ | - | - | -
Table 5: GPT-3.5-turbo’s ECE for different formats and number of candidate
answers. CA stands for the number of candidate answers.

### 4.3. Embeddings

Up to this point, our analysis reveals that the misalignment between MCQs and
LFGQs answers is evident in both the direct output (§4.1) and the token logits
(§4.2). Next, we investigate whether this difference is also manifested in the
embedding space derived from the hidden states of the models (Burns et al.,
2023). We also explore how the embeddings behave under different question
formats and models. The technique proposed by Li et al. (2023b) enables the
extraction of hidden outputs from the model by collecting the heads of the
attention blocks, and use these heads as index to obtain the hidden outputs of
each layer from the model.444https://github.com/davidbau/baukit We utilize the
hidden outputs of the last token in the input. To thoroughly investigate the
distinctions in the embedding space between MCQs and LFGQs themselves as much
as possible, unlike the prompt design in the previous experiments, we only set
a post-prompt for MCQs, and LFGQs do not contain any prompts. Finally, the
hidden outputs have information on the number of input samples, the number of
hidden layers, the number of attentions, and the dimensions of heads. Refer to
Table 6 in the Appendix (§8) for more details. We randomly select $40$ samples
from the ARC dataset, each with MCQs and LFGQs formats, and plot t-SNE (Van
der Maaten and Hinton, 2008) representations of the hidden embeddings in each
layer. Figure 5 shows the visualizations for Llama-2-7b-chat-hf. Other model
visualizations are provided in the Appendix (§8). The results show that the
embeddings from MCQs and LFGQs display clear separations in some layers of the
hidden states. We observe a consistent trend across the various LLMs: in the
initial layers, embeddings of the two formats show clear separations. However,
as we progress towards the final layers, the embeddings corresponding to MCQs
and LFGQs tend to become closer. Additionally, in certain models, the
embeddings are distinctly separated in specific middle layers. For instance,
in the open-llama-7b model, the embeddings exhibit clear differentiation in
the 14th layer. Finally, the representation of embeddings from the same model
but different sizes can vary, as shown in the embeddings of Dolly-v2-3b and
Dolly-v2-7b in Figures 9 and 10 in the Appendix.

Figure 5: t-SNE visualization for each layer in Llama-2-7b-chat-hf. MCQs in
red, LFGQs in green.

### 4.4. Different QA formats produce different answers

Our experiments showed that different formats of a single question may result
in different performances. Moreover, our results challenged the notion that
greater consistency leads to higher accuracy by closely examining the
relationship between them. When comparing MCQs and LFGQs in expected
calibration error, prompts from MCQs were the most overconfident in their
predictions. Finally, in the embedding space, MCQs and LFGQs representations
were clearly separated in some layers of the hidden states.

## 5\. Related Work

### 5.1. QA Benchmarks

QA is a prevalent evaluation method in natural language processing tasks. With
the surge of LLMs, several QA evaluation benchmarks have emerged to asses
models’ reasoning and fact-retrieval skills (Liang et al., 2022; Chang et al.,
2023; Li et al., 2023a; Chia et al., 2023). These QA benchmarks encompass
divers dataset formats, including multiple-choice questions (MCQs)
(Bhakthavatsalam et al., 2021; Ramamurthy and Aakur, 2022; Liu et al., 2023a;
Huang et al., 2023), long-form generation questions (LFGQs) (Zhang et al.,
2018; Lin et al., 2022b; Xiang et al., 2023) and True/False questions (TFQs)
(Singhal et al., 2023). Many existing QA evaluation benchmarks use relatively
simple MCQs formats, in which models can strongly rely to formulate their
answers. In addition, previous work has primarily focused on evaluations of
MCQs, not considering comparisons between the different formats (Jiang et al.,
2021; Lin et al., 2022a; Robinson and Wingate, 2023). In this paper, we
focused on conducting a comprehensive comparative analysis between MCQs and
LFGQs, thereby enhancing the understanding of the drawbacks and limitations of
the different evaluation methods.

### 5.2. LLMs and Multiple-Choice Questions

Previous work has underscored the sensitivity of LLMs to prompting strategies
(Zhao et al., 2021; Singhal et al., 2023) and positional bias (Wang et al.,
2023), which pose challenges to model assessment. For instance, Zheng et al.
(2023) showed that GPT-4 tends to favor the candidate answer presented in the
first position, leading to unfair evaluation results. Additionally,
Pezeshkpour and Hruschka (2023) observed that GPT-4 and InstructGPTOuyang et
al. (2022) perform differently when answer options are rearranged on various
benchmarks. We expand upon prior work, which focused on a limited number of
models and scenarios, to study and identify general patterns and analyze their
underlying causes across diverse datasets and models.

## 6\. Discussion and Conclusion

This paper focused on testing the effectiveness of MCQs evaluating LLMs.
Motivated by the observation of consistent preference biases across different
datasets with several LLMs, we first conducted a significance test to
determine the position of the candidate answers affect LLMs’ predictions,
resulting in accuracy instability. More specifically, we analyzed how
different LLMs have different positional preference patterns on the same
dataset, while the preference positional patterns of a particular LLM remained
constant across datasets from different sources. In addition, we conducted
comparative experiments between MCQs and LFGs in three different spaces to
ascertain the advantages and disadvantages of each as evaluation benchmarks.

##### Recommendations

Based on our experiments, we offer a few suggestions for utilizing MCQs and
LFGQs formats in LLM evaluation benchmarks:

  1. 1.

The choice of QA format should be aligned with the type of knowledge being
evaluated. Whereas it may be fine to use MCQs for testing general knowledge,
in some professional domains—particularly those carrying legal
responsibilities, such as the medical field, it is advisable to use LFGQs
under human supervision to ensure a more rigorous evaluation.

  2. 2.

When using MCQs for evaluating LLM, adjusting the number of options, whether
decreasing or increasing them, does not necessarily enhance accuracy and
confidence. However, regarding order sensitivity, reordering candidate answers
for each question and repeating questions can enhance the robustness of the
assessment process.

  3. 3.

Our findings do not indicate a strong correlation between consistency and
accuracy in LLMs responses. Therefore, we do not recommend relying on
consistency as a tool to enhance performance in LLMs.

  4. 4.

Given the discrepancy we found between MCQs and LFGQs results, we believe that
LFGQs is the best format for evaluating LLM, as it aligns well with real-world
use cases. We recommend prioritizing LFGQs format and evaluating LLM from
various perspectives, including correctness, completeness, relevance, and
interpretability.

We hope that the results presented in the paper and the investigation about
order sensitivity and comparative analyses between MCQs and LFQs can inspire
future research to improve evaluation benchmarks for LLMs.

## Acknowledgment

This work was partly supported by JSPS KAKENHI No.JP22K12091 and the State Key
Program of National Nature Science Foundation of China No.61936001.

## 7\. Reference

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## 8\. Appendix

In this appendix, we report the visualization of the t-SNE projected
embeddings in the other seven models (Figures 6-13) and the hidden embedding
space details for each LLMs (Table 6).

Model | hidden layers | attention heads | head dim.
---|---|---|---
open-llama-3b | $26$ | $32$ | $100$
open-llama-7b | $32$ | $32$ | $128$
vicuna-7b-v1.3 | $32$ | $32$ | $128$
dolly-v2-3b | $32$ | $32$ | $80$
dolly-v2-7b | $32$ | $32$ | $128$
Llama-2-7b-chat-hf | $32$ | $32$ | $128$
RedPajama-INCITE-Instruct-3B-v1 | $32$ | $32$ | $80$
stablelm-tuned-$\alpha$-3b | $16$ | $32$ | $128$
stablelm-tuned-$\alpha$-7b | $16$ | $32$ | $192$

Table 6: Number of hidden layers, number of attention heads, and the head
dimensionality of the LLMs. Figure 6: The visualization of t-SNE for each
layer in the model Open-llama-3b. The red samples are MCQs, and the samples in
green are LFGQs. Figure 7: The visualization of t-SNE for each layer in the
model Open-llama-7b. The red samples are MCQs, and the samples in green are
LFGQs. Figure 8: The visualization of t-SNE for each layer in the model
Vicuna-7b-v1.3. The red samples are MCQs, and the samples in green are LFGQs.
Figure 9: The visualization of t-SNE for each layer in the model Dolly-v2-3b.
The red samples are MCQs, and the samples in green are LFGQs. Figure 10: The
visualization of t-SNE for each layer in the model Dolly-v2-7b. The red
samples are MCQs, and the samples in green are LFGQs. Figure 11: The
visualization of t-SNE for each layer in the model Stablelm-tuned-alpha-3b.
The red samples are MCQs, and the samples in green are LFGQs. Figure 12: The
visualization of t-SNE for each layer in the model RedPajama-INCITE-
Instruct-3B. The red samples are MCQs, and the samples in green are LFGQs.
Figure 13: The visualization of t-SNE for each layer in the model Stablelm-
tuned-alpha-7b. The red samples are MCQs, and the samples in green are LFGQs.