# Holographic dark energy in modified Barrow cosmology

Ahmad Sheykhi<EMAIL_ADDRESS>Department of Physics, College of
Sciences, Shiraz University, Shiraz 71454, Iran  Biruni Observatory, College
of Sciences, Shiraz University, Shiraz 71454, Iran  Maral Sahebi Hamedan
Department of Physics, College of Sciences, Shiraz University, Shiraz 71454,
Iran  Biruni Observatory, College of Sciences, Shiraz University, Shiraz
71454, Iran

###### Abstract

Thermodynamics-Gravity conjecture implies that there is a deep connection
between the gravitational field equations and the first law of thermodynamics.
Therefore, any modification to the entropy expression, directly modifies the
field equations. By considering the modified Barrow entropy associated with
the apparent horizon, the Friedmann equations get modified as well. In this
paper, we reconsider the holographic dark energy (HDE) model when the entropy
is in the form of Barrow entropy. This modification to the entropy not only
changes the energy density of HDE, but also modifies the Friedmann equations.
Therefore, one should take into account the modified HDE in the context of
modified Friedmann equations. We study the Hubble horizon and the future event
horizon as IR cutoffs, and investigate the cosmological consequences of this
model. We also extend our study to the case where dark matter (DM) and dark
energy (DE) interact with each other. We observe that Barrow exponent $\delta$
significantly affects the cosmological behavior of HDE, and in particular the
equation of state (EoS) parameter can cross the phantom line $(w_{de}<-1)$.
Also adding $\delta$ remarkably affects the deceleration parameter, and shifts
the time of universe phase transition.

##  I Introduction

A huge number of cosmological observations support that our Universe is
currently experiences a phase of accelerated expansion Riess ; perl ; abbot .
Understanding the cause of acceleration of the cosmic expansion has been one
of the primarily unsolved problems in modern cosmology. The component which is
responsible for this acceleration is usually dubbed DE which disclosing its
nature and origin has been on the hottest topics of research in the past two
decades (see dark1  for a comprehensive review). The simplest possible
candidate for DE is the cosmological constant, denoted as $\Lambda$ in
Einstein’s field equations of general relativity. The resulting model is
termed as $\Lambda$CDM, that is the most acceptable model for describing both
accelerated expansion as well as solving the DM puzzle. Nevertheless, recent
observations indicate a tension with the $\Lambda$CDM model and reveal that
$\Lambda$CDM isn’t the best fit to some data set luso . Many alternative
theories have been proposed to modify the matter/energy sector of the
Einstein’s field equations. Among these proposals there is a supposition which
plays an important role in finding the nature of DE based on the holographic
principle susskind1 ; susskind2 ; bousso  and is known as the holographic DE
(HDE) hsu . This model is based on the fact that the entropy associated with
the boundary is proportional to the area which was first pointed out by
Bekenstein and Hawking for the black holes beken1 . According to the
Bekenstein-Hawking area law, the entropy of a black hole is given by

$S=\cfrac{A}{4L_{p}^{2}},$ (1)

where $A$ is the horizon area and $L_{p}^{2}$ denotes the Planck area. HDE
models have received a lot of attentions in the literatures Cohen ; Li ;
huang1 ; Wang1 ; Pavon ; Wang2 ; Setare1 ; Setare2 ; Jamil ; Shey1 ; Shey2 ;
Gong ; Shey3 ; Shey4 ; Shey5 ; Wang3 ; Swang ; kim .

Inspired by the Covid-$19$ virus structure, J. D. Barrow argued that quantum-
gravitational effects may deform the geometry of the black hole horizon
leading to an intricate, fractal features Barrow . He discussed that the area
law of the black hole entropy get modified and is given by

$\displaystyle S=\left(\frac{A}{A_{0}}\right)^{1+\delta/2},$ (2)

where $A$ is the black hole horizon area and $A_{0}$ is the Planck area. The
exponent $\delta$ ranges as $0\leq\delta\leq 1$ and represents the amount of
the quantum-gravitational deformation effects. The area law is reproduced in
case of $\delta=0$, and $A_{0}\rightarrow 4L_{p}^{2}=4G$. On the other hand,
the most intricate and fractal structure of the horizon is obtained by
$\delta=1$. In the cosmological setup, the effects of Barrow entropy on the
cosmic evolution have been investigated from different viewpoints. For
example, modification of the area law leads to a new HDE model based on Barrow
entropy Emm1 ; Ana . On the other hand, it was recently proven that Barrow
entropy as well as any other known entropy (Tsalis, Renyi, Kaniadakis, etc) is
just sub-case of generalized entropy expression introduced in Odin1 ; Odin2 .
Other studies on the cosmological consequences of the Barrow entropy can be
carried out in Emm3 ; Abr1 ; Mam ; Abr2 ; Bar2 ; Sri ; Das ; Sha ; Pra ; Odin3
; Odin4 .

Nowadays, it is a general belief that there is a deep correspondence between
the gravitational field equations and the laws of thermodynamics (see e.g. Jac
; Pad ; Pad2 ; Pad3 ; Fro ; Wang0 ; Ver ; Cai5 ; CaiLM ; sheyECFE ; SheyLog ;
SheyPL ; Odin5 ; Abr  and references therein). In the background of cosmology,
“gravity-thermodynamics” conjecture, translates to the correspondence between
Friedmann equations describing the evolution of the universe and the first law
of thermodynamics on the apparent horizon. It has been confirmed that one can
always rewrite the Friedmann equations in any gravity theory in the form of
the first law of themodynamics on the apparent horizon and vice versa CaiKim ;
Cai2 ; Sheyk1 ; Sheyk2 .

In the present work, we are going to investigate HDE when the entropy
associated with the apparent horizon is in the form of Barrow enetropy given
in (2). According to the thermodynamics-gravity conjecture, any modification
to the entropy expression leads to the modified Friedmann equations. The
modified Friedmann equations through Barrow entropy was introduced in Emm2 ;
SheBFE1 . A cosmological scenario based on Barrow entropy was introduce in
SheBFE2 , where it was argued that the exponent $\delta$ cannot reproduce any
term which may play the role of DE and one still needs to take into account DE
component in the modified Friedmann equations to reproduce the accelerated
universe. On the other hand, the HDE density, which is based on holographic
principle, get modified due to the modification of entropy. Although, the
authors of Emm1  modify the energy density of HDE, they still perform their
calculations in the setup of standard Friedmann equations. This is indeed
inconsistent with the thermodynamics-gravity conjecture which states that any
modification to the entropy should modify the field equations of gravity. Our
study differs from Emm1 ; Ana , in that we modify both energy density of HDE
as well as the Friedmann equations describing the evolution of the universe.
Throughout this paper we set $\kappa_{B}=1=c=\hbar$, for simplicity.

This paper is outlined as follows. In section II, we study the HDE in Barrow
cosmology when the IR cutoff is the Hubble radius. In section III, we consider
the future event horizon as IR cutoff, and explore the cosmological
consequences of the HDE through modified cosmology based on Barrow entropy.
The last section is devoted to closing remarks.

##  II HDE with Hubble horizon as IR cutoff

We consider a homogeneous and isotropic flat universe which describes by the
line element

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j},$ (3)

where $a(t)$ is the scale factor. Applying Barrow entropy (2) to the
holographic framework, one can obtain the HDE density in the form Emm1

$\rho_{de}=CL^{\delta-2},$ (4)

where $L$ is the holographic horizon length and $C$ is a parameter with
dimension $[L]^{-2-\delta}$. For latter convenience, we choose
$C=3c^{2}M^{2}_{\rm eff}$, where $c^{2}$ is the holographic model parameter of
order one Li  and $M^{2}_{\rm eff}$ is the effective Planck mass which we
shall introduce latter. Note that in the case where $\delta=0$, we have
$M^{2}_{\rm eff}\rightarrow M^{2}_{p}$, and the standard HDE density,
$\rho_{de}=3c^{2}M^{2}/L^{2}$ is restored.

The modified Friedmann equation based on Barrow entropy, in a flat universe,
is given by SheBFE1 ; SheBFE2

$H^{2-\delta}=\frac{8\pi G_{\rm eff}}{3}(\rho_{m}+\rho_{de}),$ (5)

where $H={\dot{a}}/{a}$ is the Hubble parameter, $\rho_{m}$ and $\rho_{de}$
are the energy density of pressureless matter and DE, respectively. Here
$G_{\rm eff}$ stands for the effective Newtonian gravitational constant
SheBFE1

$G_{\rm
eff}\equiv\frac{A_{0}}{4}\left(\frac{2-\delta}{2+\delta}\right)\left(\frac{A_{0}}{4\pi}\right)^{\delta/2}.$
(6)

If we define $M^{2}_{\rm eff}=(8\pi G_{\rm eff})^{-1}$, then the Friedmann
equation (5) can be written

$H^{2-\delta}=\frac{1}{3M^{2}_{\rm eff}}(\rho_{m}+\rho_{de}),$ (7)

Taking the Hubble radius as IR cutoff, $L=H^{-1}$, the HDE density can be
written

$\rho_{de}=3c^{2}M^{2}_{\rm eff}H^{2-\delta},$ (8)

In the remaining part of this section, we shall consider the case without
interaction between dark sectors and the interacting case, separately.

###  II.1 Non-interacting case

When two dark components of the universe evolve separately, they satisfy two
independent energy conservation equations,

$\displaystyle\dot{\rho}_{de}+3H(1+w_{de})\rho_{de}=0,$ (9)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (10)

where $w_{de}=p_{de}/\rho_{de}$ stands for the EoS parameter of HDE. It is
also convenient to introduce the dimensionless density parameters as

$\displaystyle\Omega_{de}=\cfrac{\rho_{de}}{\rho_{c}}=\cfrac{\rho_{de}}{3M^{2}_{\rm
eff}H^{2-\delta}},$ (11)
$\displaystyle\Omega_{m}=\cfrac{\rho_{m}}{\rho_{c}}=\cfrac{\rho_{m}}{3M^{2}_{\rm
eff}H^{2-\delta}},$ (12)

and thus the Friedmann Eq. (7) can be written

$\displaystyle\Omega_{m}+\Omega_{de}=1,$ (13)

where $\rho_{c}=3M^{2}_{\rm eff}H^{2-\delta}$ is the effective critical energy
density. Substituting Eq.(8) in Eq.(11) one finds $\Omega_{de}=c^{2}$. Taking
the time derivative of the Friedmann equation (7) and using the continuity
Eqs. (9) and (10), we arrive at

$\frac{\dot{H}}{H^{2}}=-\frac{3}{2-\delta}(1+\Omega_{de}w_{de}).$ (14)

If we take the derivative of Eq. (8) respect to cosmic time, and using Eqs.
(9) and (14), we find $w_{de}=0$. This result is similar to the HDE with
Hubble horizon as IR cutoff in standard cosmology, and cannot lead to an
accelerated universe Li ; Pavon .

###  II.2 Interacting case

When the two dark sectors of the universe interact with each other, they do
not satisfy the independent conservation equations, instead they satisfy the
semi energy conservation equations as

$\displaystyle\dot{\rho}_{de}+3H(1+w_{de})\rho_{de}=-Q,$ (15)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (16)

where $Q=3b^{2}H\rho_{de}$ represents the interaction term. Taking the time
derivative of Eq. (8) and using Eqs. (14) and (15), we find

$w_{de}=-\frac{b^{2}}{1-c^{2}},$ (17)

which is independent of the exponent $\delta$, and is similar to the case of
standard cosmology Pavon ; Shey3 . In order to reach $w_{de}<0$ we should have
$c^{2}<1$. Besides, for $b^{2}=1-c^{2}$, this model mimics the cosmological
constant. Moreover, for $b^{2}>1-c^{2}$, this model can cross the phantom
line. The EoS parameter of total energy and matter is defined by

$w_{\rm tot}=\Omega_{de}w_{de}=-\frac{b^{2}c^{2}}{1-c^{2}}.$ (18)

It was argued that in modified Barrow cosmology, the condition for the
accelerated expansion reads SheBFE1

$w_{\rm tot}<-\frac{(1+\delta)}{3}.$ (19)

Combining condition (19) with (18) implies $3b^{2}c^{2}>(1+\delta)(1-c^{2})$
for an accelerated universe. The deceleration parameter can be easily derived

$\displaystyle q$ $\displaystyle=$
$\displaystyle-1-\frac{\dot{H}}{H^{2}}=-1+\frac{3}{2-\delta}[1+\Omega_{de}w_{de}]$
(20)
$\displaystyle=-1+\frac{3}{2-\delta}\left(1-\frac{b^{2}c^{2}}{1-c^{2}}\right).$

which is a constant but depends on $\delta$. In an accelerating universe $q<0$
which translates to $3b^{2}c^{2}>(1+\delta)(1-c^{2})$, consistent with
previous condition (19).

##  III HDE with future event horizon as IR cutoff

In this section we consider the future event horizon as IR cutoff Li

${L=R_{h}=a\int_{t}^{\infty}\cfrac{dt}{a}=a\int_{t}^{\infty}\cfrac{da}{Ha^{2}}},$
(21)

and therefore the HDE density parameter becomes

$\Omega_{de}=c^{2}\left(HR_{h}\right)^{\delta-2},\Rightarrow
HR_{h}=\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{{1}/{(\delta-2)}}.$ (22)

Taking the time derivative of Eq. (21) we find

$\dot{R}_{h}=HR_{h}-1,$ (23)

Also, taking the time derivative of energy density $\rho_{de}=3c^{2}M^{2}_{\rm
eff}R_{h}^{\delta-2}$, yields

$\dot{\rho}_{de}=3(\delta-2)c^{2}M^{2}_{\rm
eff}\dot{R}_{h}{R}_{h}^{\delta-3}=\frac{\delta-2}{R_{h}}\dot{R}_{h}\rho_{de}.$
(24)

Again, we shall consider the non-interacting and interacting cases,
separately.

###  III.1 Non-interacting case

Combining Eqs. (9), (23) and (24), one can obtain

$w_{de}=-\cfrac{1+\delta}{3}+\cfrac{\delta-2}{3}\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2-\delta}},$
(25)

for the EoS parameter of HDE. Note that for $\delta=0$, one restores the EoS
parameter of HDE in standard cosmology Wang2 . We can also obtain the
evolution of density parameter of HDE. Differentiating $\Omega_{de}$ respect
to time and using relations $\dot{\Omega}_{de}=H{\Omega^{\prime}_{de}}$ and
(14), we arrive at

$\Omega^{\prime}_{de}=\Omega_{de}\left\\{(\delta-2)\left[1-\left(\frac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2-\delta}}\right]+3(1+\Omega_{de}w_{de})\right\\},$
(26)

where the prime denotes derivative with respect to $x=\ln{a}$. Substituting
$w_{de}$ from Eq. (25) into the above relation, yields

${\Omega^{\prime}_{de}}=\Omega_{de}(1-\Omega_{de})\left\\{(1+\delta)+(2-\delta)\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2\delta}}\right\\}.$
(27)

This equation shows the evolution of the density parameter of HDE in the
context of modified Barrow cosmology. We plot evolution of $\Omega_{de}$ in
Barrow cosmology in Fig. 1. From this figure we see that at each $z$, the
value of $\Omega_{de}$ decreases with increasing $\delta$. Besides, the
difference between $\Omega_{de}$ in standard cosmology and modified Barrow
cosmology increases at higher redshifts.

Figure 1: Evolution of $\Omega_{de}$ as a function of redshift $z$ for non-
interacting HDE. Here we have set $c^{2}=1$ and $\Omega_{de,0}=0.72$. Figure
2: Evolution of $w_{de}$ as a function of redshift $z$ for non-interacting
HDE. Here we set $c^{2}=1$, $M^{2}_{\rm eff}=1$.

The evolution of EoS parameter $w_{de}$ for HDE versus $z$ is plotted in Fig.
2 for different values of $\delta$. We see that at the present time, the
values of $w_{de}$ is independent of $\delta$, while its behavior differs for
early times. For $z=0$ we have $w_{de}\approx-0.9$, and increasing $\delta$
will decrease $w_{de}$ in all times. For example, for $\delta=0.3$ at $z=0$ we
have $w_{de}=-0.91$ which is closer than the standard model to observational
data. Therefore, in modified Barrow cosmology, $w_{de}$ of HDE at the present
time, is compatible with observations.

Next, we examine the deceleration parameter, $q=-\ddot{a}/(aH^{2})$. Using
Eqs. (14) and (25), one finds

$\displaystyle q$ $\displaystyle=$
$\displaystyle-1-\frac{\dot{H}}{H^{2}}=-1+\cfrac{3}{2-\delta}\left(1+\Omega_{de}w_{de}\right)$
(28) $\displaystyle=$
$\displaystyle\cfrac{1+\delta}{2-\delta}(1-\Omega_{de})-\Omega_{de}\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2-\delta}}.$

The evolutionary behavior of the deceleration parameter is plotted in Fig. 3.
From this figure we see that with increasing the Barrow exponent $\delta$, the
transition from deceleration to acceleration takes place in lower redshift.

Figure 3: The behaviour of the deceleration parameter $q$ as a function of
redshift $z$ for non-interacting HDE in Barrow cosmology, where we set
$c^{2}=1$.

###  III.2 Interacting case

Next, we investigate the case for FRW universe filled with HDE and DM
exchanging energy. We consider the future event horizon as IR cutoff. In
interacting case the conservation equations are in the form of Eqs.(15) and
(16). Following the method of the previous section, we can find the EoS
parameter as

$w_{de}=-b^{2}-\cfrac{\delta+1}{3}+\cfrac{\delta-2}{3}\left(\cfrac{c^{2}}{\Omega_{de}}\right)^{\frac{1}{\delta-2}}.$
(29)

In this case the evolutionary equation for $\Omega_{de}$ becomes

$\displaystyle{\Omega^{\prime}_{de}}$ $\displaystyle=$
$\displaystyle\Omega_{de}(1-\Omega_{de})\left\\{(1+\delta)+(2-\delta)\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2\delta}}\right\\}$
(30) $\displaystyle-3b^{2}\Omega_{de}^{2},$

while the deceleration parameter reads

$\displaystyle
q=-1-\frac{\dot{H}}{H^{2}}=-1+\cfrac{3}{2-\delta}\left(1+\Omega_{de}w_{de}\right)$
(31) $\displaystyle=$
$\displaystyle\cfrac{1+\delta}{2-\delta}(1-\Omega_{de})-\Omega_{de}\left(\cfrac{\Omega_{de}}{c^{2}}\right)^{\frac{1}{2-\delta}}-3\frac{b^{2}\Omega_{de}}{2-\delta}.$

We plot the evolution of $\Omega_{de}$ for different values of $\delta$ in
Fig. 4.

Figure 4: Evolution of $\Omega_{de}$ as a function of redshift $z$ for interacting HDE in Barrow cosmology. Here we take $c^{2}=1$, $M^{2}_{\rm eff}=1$ and $b^{2}=0.1$. Figure 5: The evolution of EoS parameter $w_{de}$ as a function of redshift $z$ for interacting HDE. Here we take $c^{2}=1$, $M^{2}_{\rm eff}=1$, $b^{2}=0.1$. $w_{de}$ | $b^{2}=0.03$ | $b^{2}=0.06$ | $b^{2}=0.1$
---|---|---|---
$\delta=0$ | $-0.92909$ | $-0.95909$ | $-0.99909$
$\delta=0.1$ | $-0.92944$ | $-0.95944$ | $-0.99944$
$\delta=0.2$ | $-0.92991$ | $-0.95991$ | $-0.99991$
$\delta=0.3$ | $-0.93049$ | $-0.96049$ | $-1.00049$
Table 1: Numerical results for EoS parameter of interacting HDE at the present
time, $z=0$.

We can also plot the EoS and the deceleration parameter for the interacting
HDE in modified Barrow cosmology. From Fig. 5 we see that with increasing
$\delta$, the EoS parameter decreases as well. We also observe from Fig. 6
that with increasing the Barrow exponent $\delta$ the transition from
deceleration to acceleration occurs at lower redshift.

Figure 6: Evolution of the deceleration parameter $q$ as a function of
redshift $z$ for interacting HDE. We have taken $c^{2}=1$, $M^{2}_{\rm
eff}=1$, $b^{2}=0.1$.

Evolution of the total EoS parameter for different values of $\delta$ in Fig.
7 indicates that there is no significant difference between $w_{tot}$ of HDE
in standard cosmology ($\delta=0$) and in modified Barrow cosmology
($\delta\neq 0$).

Figure 7: Evolution of $w_{tot}$ as a function of redshift $z$ for interacting
HDE, where $c^{2}=1$, $M^{2}_{\rm eff}=1$, $b^{2}=0.1$.

By fixing the Barrow component $\delta$ and changing the values of $b^{2}$ we
can reveal the effects of interaction term on the cosmological behavior of our
model. In Fig. 8, we plot the evolution of $\Omega_{de}$ for interacting HDE
in modified Barrow cosmology. It is seen that at each time, the value of
$\Omega_{de}$ is higher compared to standard cosmology.

In Fig. 9 we can see that $w_{de}$ of the interacting HDE decreases with
increasing coupling constant $b^{2}$. For $\delta=0.2$ and $b^{2}=0.1$ we see
that $w_{de}$ crosses the phantom regime.

Figure 8: Evolution of $\Omega_{de}$ versus $z$ for modified Barrow HDE, where
$c^{2}=1$, $M^{2}_{\rm eff}=1$, $\delta=0.2$. Figure 9: Evolution of EoS
parameter $w_{de}$ as a function of redshift $z$ for interacting HDE. Here we
take $c^{2}=1$, $M^{2}_{\rm eff}=1$, and $\delta=0.2$.

As one can see in Fig. 10, changing $b^{2}$ has affected on the phase
transition of interacting HDE in Barrow cosmology. We observe that unlike the
effects of adding the values of $\delta$ on deceleration parameter, increasing
$b^{2}$ will shift the transition to higher redshifts and it means that the
expansion of the universe changed from decelerating phase to an accelerating
phase in much earlier times. By comparing the effects of changing $\delta$ and
$b^{2}$, given by Figs. 6 and 10, we can see that the best values for the
parameters are in the range $0.06<b^{2}<0.1$ and $0.2<\delta<0.3$.

Figure 10: Evolution of the deceleration parameter $q$ as a function of
redshift $z$ for interacting HDE in modified Barrow cosmology. Here we have
taken $c^{2}=1$, $M^{2}_{\rm eff}=1$, and $\delta=0.2$.

Again by using our previous results for $\Omega_{de}$ and $w_{de}$, we can
plot the total EoS parameter, for different values of $b^{2}$ in case of
interacting HDE which is shown in Fig. 11. Adding coupling constant will
decrease $w_{tot}$ significantly during all times and reveals the difference
between interacting and non-interacting HDE in the framework of Barrow
cosmology.

Figure 11: Evolution of $w_{tot}$ as a function of redshift $z$ for
interacting HDE in modified Barrow cosmology. Here we have taken $c^{2}=1$,
$M^{2}_{\rm eff}=1$, and $\delta=0.2$.

##  IV Closing remarks

According to thermodynamics-gravity conjecture, any modification to the
entropy modifies the energy density of HDE as well as the Friedmann equations.
Based on this and using the modified Friedmann equations, we investigated the
HDE when the entropy is in the form of Barrow entropy. We first have taken the
Hubble radius as IR cutoff and showed that the accelerated universe can be
achieved only in the interacting case. This behavior is similar to HDE in
standard cosmology with Hubble cutoff. We then considered the future event
horizon as IR cutoff and investigated both non-interacting and interacting HDE
in a flat universe.

We examined the effects of Barrow exponent $\delta$ on the cosmological
evolution of HDE. We observed that for $\delta=0$ the EoS parameter of non-
interacting HDE lies completely in the quintessence regime, while for the
interacting case it can cross the phantom line. In both cases, with increasing
$\delta$, the EoS parameter $w_{de}$ decreases at any time. Another
interesting result we found in this work is that the presence of $\delta$ can
change the time of phase transition of the universe from deceleration to the
acceleration. Indeed, with increasing $\delta$ the phase transition occurs at
lower redshifts. This behavior is seen for both non-interacting and
interacting cases. On the other hand, for a fixed value of $\delta$, if we
increase the coupling constant of interaction term, the transition occurs in
higher redshift.

To sum up, the incorporation of modified Friedmann equations to HDE improves
the phenomenology comparing to the standard Friedmann equation while keeping
the Barrow exponent $\delta$ to smaller values. This is an advantage of this
scenario, since in a more realistic case we expect the Barrow exponent to be
closer to the standard Bekenstein-Hawking value and have the results
compatible with observations.

###### Acknowledgements.

We thank Mahya Mohammadi for helpful discussion and valuable comments.

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Uncertainty Quantification for nonparametric regression using Empirical Bayesian neural networks

1]Stefan Franssen
2]Botond Szabó
[1]Delft Institute of Applied Mathematics
[2]Department of Decision Sciences, Bocconi University

Bocconi Institute for Data Science and Analytics (BIDSA)

We propose a new, two-step empirical Bayes-type of approach for neural networks. We show in context of the nonparametric regression model that the procedure (up to a logarithmic factor) provides optimal recovery of the underlying functional parameter of interest and provides Bayesian credible sets with frequentist coverage guarantees. The approach requires fitting the neural network only once, hence it is substantially faster than Bootstrapping type approaches. We demonstrate the applicability of our method over synthetic data, observing good estimation properties and reliable uncertainty quantification.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

abstractBig Data are radically changing biomedical research. The unprecedented advances in automated collection of large-scale molecular and clinical data pose major challenges to data analysis and interpretation, calling for the development of new computational approaches. The creation of powerful systems for the effective use of biomedical Big Data in Personalized Medicine (a.k.a. Precision Medicine) will require significant scientific and technical developments, including infrastructure, engineering, project and financial management. We review here how the evolution of data-driven methods offers the possibility to address many of these problems, guiding the formulation of hypotheses on systems functioning and the generation of mechanistic models, and facilitating the design of clinical procedures in Personalized Medicine.
journaltitleSystems Biology • Nanobiotechnology
shortjournalCurrent Opinion in Biotechnology
titleBig Data Analytics for Personalized Medicine
family=S. Ramos,
family=S. Gehrig,
family=P. Pinggera,
family=U. Franke,
family=C. Rother,

booktitle2017 IEEE Intelligent Vehicles Symposium (IV)
eventtitle2017 IEEE Intelligent Vehicles Symposium (IV)
titleDetecting Unexpected Obstacles for Self-Driving Cars: Fusing Deep Learning and Geometric Modeling

New York, NY, USA

Association for Computing Machinery

abstractArtificial Intelligence (AI) is revolutionizing the modern society. In the automotive industry, researchers and developers are actively pushing deep learning based approaches for autonomous driving. However, before a neural network finds its way into series production cars, it has to first undergo strict assessment concerning functional safety. The chances and challenges of incorporating deep learning for self-driving cars are presented in this paper.
booktitleProceedings of the 1st International Workshop on Software Engineering for AI in Autonomous Systems
seriesSEFAIS '18
shorttitleDeep Learning for Self-Driving Cars
titleDeep Learning for Self-Driving Cars: Chances and Challenges

abstractDeep learning has been widely applied in computer vision, natural language processing, and audio-visual recognition. The overwhelming success of deep learning as a data processing technique has sparked the interest of the research community. Given the proliferation of Fintech in recent years, the use of deep learning in finance and banking services has become prevalent. However, a detailed survey of the applications of deep learning in finance and banking is lacking in the existing literature. This study surveys and analyzes the literature on the application of deep learning models in the key finance and banking domains to provide a systematic evaluation of the model preprocessing, input data, and model evaluation. Finally, we discuss three aspects that could affect the outcomes of financial deep learning models. This study provides academics and practitioners with insight and direction on the state-of-the-art of the application of deep learning models in finance and banking.
journaltitleFrontiers of Business Research in China
shortjournalFrontiers of Business Research in China
titleDeep Learning in Finance and Banking: A Literature Review and Classification

abstractThe growing Internet of Things (IoT) provides significant resources to be integrated with critical infrastructures to enable cyber-physical systems. More specifically, the deployment of smart meters for electricity usage monitoring in the smart grid can provide granular and detailed information from which power load forecasting can be carried out. However, the accurate prediction of long-term power usage remains a challenging issue. In light of many recent advances, deep learning has the potential to significantly improve the ability to assess data and make predictions, and is already rapidly changing the world we live in. As such, in this paper, we consider the use of deep learning, via Recursive Neural Network (RNN) and Long Short-Term Memory layers, for the long-term prediction of localized power consumption. In particular, we consider the optimization of both data feature sets and neural network models, developing three model-feature combinations to maximize prediction accuracy and minimize error. Through detailed experimental evaluation, our results demonstrate the ability to achieve highly accurate predictions over periods as large as 21 days through the integration of correlated features.
journaltitleProceedings of the 9th International Conference of Information and Communication Technology [ICICT-2019] Nanning, Guangxi, China January 11-13, 2019
shortjournalProcedia Computer Science
titleDeep Learning-Based Power Usage Forecast Modeling and Evaluation
Cyber-Physical Systems,Deep Learning,Internet of Things,Smart Grid

titleElectric Energy Consumption Prediction by Deep Learning with State Explainable Autoencoder

journaltitleAdvances in neural information processing systems
titleImagenet Classification with Deep Convolutional Neural Networks

journaltitleIEEE Signal processing magazine
titleDeep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups

cambridge university press Cambridge

titleNeural Network Learning: Theoretical Foundations

booktitleProceedings of the AAAI Conference on Artificial Intelligence
titleWhen and Why Are Deep Networks Better than Shallow Ones?

journaltitleInternational Journal of Automation and Computing
titleWhy and When Can Deep-but Not Shallow-Networks Avoid the Curse of Dimensionality: A Review

abstractDownload Citation | Nonparametric regression using deep neural networks with ReLU activation function | Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU... | Find, read and cite all the research you need on ResearchGate
journaltitleThe Annals of Statistics
titleNonparametric Regression Using Deep Neural Networks with ReLU Activation Function

titleAdaptivity of Deep ReLU Network for Learning in Besov and Mixed Smooth Besov Spaces: Optimal Rate and Curse of Dimensionality

Red Hook, NY, USA

Curran Associates Inc.

abstractWe introduce Spike-and-Slab Deep Learning (SS-DL), a fully Bayesian alternative to dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with unknown levels of smoothness. Indeed, we show that the posterior distribution concentrates at the near minimax rate for α-Hölder smooth maps, performing as well as if we knew the smoothness level α ahead of time. Our result sheds light on architecture design for deep neural networks, namely the choice of depth, width and sparsity level. These network attributes typically depend on unknown smoothness in order to be optimal. We obviate this constraint with the fully Bayes construction. As an aside, we show that SS-DL does not overfit in the sense that the posterior concentrates on smaller networks with fewer (up to the optimal number of) nodes and links. Our results provide new theoretical justifications for deep ReLU networks from a Bayesian point of view.
booktitleProceedings of the 32nd International Conference on Neural Information Processing Systems
titlePosterior Concentration for Sparse Deep Learning
family=Michael Kohler,
family=Sophie Langer,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleOn the Rate of Convergence of Fully Connected Deep Neural Network Regression Estimates

Curran Associates, Inc.

booktitleAdvances in Neural Information Processing Systems
titleDeep Exploration via Bootstrapped DQN

Curran Associates, Inc.

booktitleAdvances in Neural Information Processing Systems
titleSimple and Scalable Predictive Uncertainty Estimation Using Deep Ensembles

abstractDeep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of available theoretical results, the literature has been largely silent about uncertainty quantification for deep learning. This paper takes a step forward in this important direction by taking a Bayesian point of view. We study Gaussian approximability of certain aspects of posterior distributions of sparse deep ReLU architectures in non-parametric regression. Building on tools from Bayesian non-parametrics, we provide semi-parametric Bernstein-von Mises theorems for linear and quadratic functionals, which guarantee that implied Bayesian credible regions have valid frequentist coverage. Our results provide new theoretical justifications for (Bayesian) deep learning with ReLU activation functions, highlighting their inferential potential.
booktitleProceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics
seriesProceedings of Machine Learning Research
titleUncertainty Quantification for Sparse Deep Learning

titleEfficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee

abstractOver the last decade, neural networks have reached almost every field of science and became a crucial part of various real world applications. Due to the increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence, i.e. are badly calibrated. To overcome this, many researchers have been working on understanding and quantifying uncertainty in a neural network’s prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. For that, a comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and available implementations. Different examples from the wide spectrum of challenges in the fields of medical image analysis, robotic and earth observation give an idea of the needs and challenges regarding uncertainties in practical applications of neural networks. Additionally, the practical limitations of uncertainty quantification methods in neural networks for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
eprintclasscs, stat
titleA Survey of Uncertainty in Deep Neural Networks
omputer Science - Machine Learning,Statistics - Machine Learning
family=Dennis D. Cox,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleAn Analysis of Bayesian Inference for Nonparametric Regression
family=David Freedman,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleWald Lecture: On the Bernstein-von Mises Theorem with Infinite-Dimensional Parameters

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleFrequentist Coverage of Adaptive Nonparametric Bayesian Credible Sets
coverage,Credible set,uncertainty quantification

abstractWe continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein-von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donsker- and Kolmogorov-Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.
journaltitleThe Annals of Statistics
shortjournalAnn. Statist.
titleOn the Bernstein-von Mises Phenomenon for Nonparametric Bayes Procedures
Mathematics - Statistics Theory

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleSupremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
anisotropic smoothness,mixed partial derivatives,nonparametric multivariate regression,sup-norm posterior contraction,Tensor product B-splines

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleAdaptive Bernstein–von Mises Theorems in Gaussian White Noise
Adaptation,Bayesian inference,conﬁdence set,Credible set,posterior asymptotics

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleOn Coverage and Local Radial Rates of Credible Sets
Credible ball,excessive bias restriction,local radial rate

Institute of Mathematical Statistics

abstractWe investigate the frequentist coverage properties of (certain) Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem (in context of sieve type of priors), we introduce an extra assumption on the functional parameters, the so-called “general polished tail” condition. We then show that under standard assumptions, both the hierarchical and empirical Bayes methods, result in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abstract results to various examples, including the nonparametric regression model, density estimation using exponential families of priors, density estimation using histogram priors and the nonparametric classification model, for which we show that their size is near minimax adaptive with respect to the considered specific pseudometrics.
issn0090-5364, 2168-8966
journaltitleThe Annals of Statistics
titleAsymptotic Frequentist Coverage Properties of Bayesian Credible Sets for Sieve Priors

Rousseau en Szabo - 2019 - Asymptotic frequentist coverage properties of Baye.pdf;C
ierarchical Bayes,Mathematics - Statistics Theory,Nonparametric regression,Posterior contraction rates,sieve prior,uncertainty quantification

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleStatistical Guarantees for Bayesian Uncertainty Quantification in Nonlinear Inverse Problems with Gaussian Process Priors
Bernstein–von Mises theorems,credible sets,Schrödinger equation,X-ray transforms

The Institute of Mathematical Statistics

journaltitleAnnals of Statistics
shortjournalAnn. Statist.
titleConvergence Rates of Posterior Distributions

Institute of Mathematical Statistics and Bernoulli Society

journaltitleElectronic Journal of Statistics
titleOracle Posterior Contraction Rates under Hierarchical Priors

titleTensorFlow: Large-scale Machine Learning on Heterogeneous Systems

Springer Science and Business Media LLC

titleArray Programming with NumPy

titleUncertainty Quantification Using Empirical Bayesian Deep Neural Networks.

titleA Simulation Study of Uncertainty Quantification in Classification Tasks Using Empirical Bayesian Deep Neural Networks

titleWielandt's Theorem

journaltitleMathematische Annalen
shortjournalMathematische Annalen
titleZur Theorie Der Matrices

journaltitleSitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
shortjournalSitzungsber. Königl. Preuss. Akad. Wiss.
titleUeber Matrizen Aus Nicht Negativen Elementen

titleTopics in Random Matrix Theory

Cambridge University Press

abstractThis classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.
seriesCambridge Mathematical Library
titleSpline Functions: Basic Theory

Cambridge University Press

titleFundamentals of Nonparametric Bayesian Inference

The Institute of Mathematical Statistics and the Bernoulli Society

journaltitleElectronic Journal of Statistics
shortjournalElectron. J. Statist.
titleAdaptive Estimation of Multivariate Functions Using Conditionally Gaussian Tensor-Product Spline Priors

abstractLetybe a random vector in Rn, satisfyingEy⊗y=id.LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by finite sums of the tensorsyi⊗yi. We prove that for some absolute constantCE1M∑i=1Myi⊗yi−id⩽C·lognM·(E‖y‖logM)1/logM,provided that the last expression is smaller than 1. We apply this estimate to improve a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position.
journaltitleJournal of Functional Analysis
shortjournalJournal of Functional Analysis
titleRandom Vectors in the Isotropic Position

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

abstractBig Data are radically changing biomedical research. The unprecedented advances in automated collection of large-scale molecular and clinical data pose major challenges to data analysis and interpretation, calling for the development of new computational approaches. The creation of powerful systems for the effective use of biomedical Big Data in Personalized Medicine (a.k.a. Precision Medicine) will require significant scientific and technical developments, including infrastructure, engineering, project and financial management. We review here how the evolution of data-driven methods offers the possibility to address many of these problems, guiding the formulation of hypotheses on systems functioning and the generation of mechanistic models, and facilitating the design of clinical procedures in Personalized Medicine.
journaltitleSystems Biology • Nanobiotechnology
shortjournalCurrent Opinion in Biotechnology
titleBig Data Analytics for Personalized Medicine
family=S. Ramos,
family=S. Gehrig,
family=P. Pinggera,
family=U. Franke,
family=C. Rother,

booktitle2017 IEEE Intelligent Vehicles Symposium (IV)
eventtitle2017 IEEE Intelligent Vehicles Symposium (IV)
titleDetecting Unexpected Obstacles for Self-Driving Cars: Fusing Deep Learning and Geometric Modeling

New York, NY, USA

Association for Computing Machinery

abstractArtificial Intelligence (AI) is revolutionizing the modern society. In the automotive industry, researchers and developers are actively pushing deep learning based approaches for autonomous driving. However, before a neural network finds its way into series production cars, it has to first undergo strict assessment concerning functional safety. The chances and challenges of incorporating deep learning for self-driving cars are presented in this paper.
booktitleProceedings of the 1st International Workshop on Software Engineering for AI in Autonomous Systems
seriesSEFAIS '18
shorttitleDeep Learning for Self-Driving Cars
titleDeep Learning for Self-Driving Cars: Chances and Challenges

abstractDeep learning has been widely applied in computer vision, natural language processing, and audio-visual recognition. The overwhelming success of deep learning as a data processing technique has sparked the interest of the research community. Given the proliferation of Fintech in recent years, the use of deep learning in finance and banking services has become prevalent. However, a detailed survey of the applications of deep learning in finance and banking is lacking in the existing literature. This study surveys and analyzes the literature on the application of deep learning models in the key finance and banking domains to provide a systematic evaluation of the model preprocessing, input data, and model evaluation. Finally, we discuss three aspects that could affect the outcomes of financial deep learning models. This study provides academics and practitioners with insight and direction on the state-of-the-art of the application of deep learning models in finance and banking.
journaltitleFrontiers of Business Research in China
shortjournalFrontiers of Business Research in China
titleDeep Learning in Finance and Banking: A Literature Review and Classification

abstractThe growing Internet of Things (IoT) provides significant resources to be integrated with critical infrastructures to enable cyber-physical systems. More specifically, the deployment of smart meters for electricity usage monitoring in the smart grid can provide granular and detailed information from which power load forecasting can be carried out. However, the accurate prediction of long-term power usage remains a challenging issue. In light of many recent advances, deep learning has the potential to significantly improve the ability to assess data and make predictions, and is already rapidly changing the world we live in. As such, in this paper, we consider the use of deep learning, via Recursive Neural Network (RNN) and Long Short-Term Memory layers, for the long-term prediction of localized power consumption. In particular, we consider the optimization of both data feature sets and neural network models, developing three model-feature combinations to maximize prediction accuracy and minimize error. Through detailed experimental evaluation, our results demonstrate the ability to achieve highly accurate predictions over periods as large as 21 days through the integration of correlated features.
journaltitleProceedings of the 9th International Conference of Information and Communication Technology [ICICT-2019] Nanning, Guangxi, China January 11-13, 2019
shortjournalProcedia Computer Science
titleDeep Learning-Based Power Usage Forecast Modeling and Evaluation
Cyber-Physical Systems,Deep Learning,Internet of Things,Smart Grid

titleElectric Energy Consumption Prediction by Deep Learning with State Explainable Autoencoder

journaltitleAdvances in neural information processing systems
titleImagenet Classification with Deep Convolutional Neural Networks

journaltitleIEEE Signal processing magazine
titleDeep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups

cambridge university press Cambridge

titleNeural Network Learning: Theoretical Foundations

booktitleProceedings of the AAAI Conference on Artificial Intelligence
titleWhen and Why Are Deep Networks Better than Shallow Ones?

journaltitleInternational Journal of Automation and Computing
titleWhy and When Can Deep-but Not Shallow-Networks Avoid the Curse of Dimensionality: A Review

abstractDownload Citation | Nonparametric regression using deep neural networks with ReLU activation function | Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU... | Find, read and cite all the research you need on ResearchGate
journaltitleThe Annals of Statistics
titleNonparametric Regression Using Deep Neural Networks with ReLU Activation Function

titleAdaptivity of Deep ReLU Network for Learning in Besov and Mixed Smooth Besov Spaces: Optimal Rate and Curse of Dimensionality

Red Hook, NY, USA

Curran Associates Inc.

abstractWe introduce Spike-and-Slab Deep Learning (SS-DL), a fully Bayesian alternative to dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with unknown levels of smoothness. Indeed, we show that the posterior distribution concentrates at the near minimax rate for α-Hölder smooth maps, performing as well as if we knew the smoothness level α ahead of time. Our result sheds light on architecture design for deep neural networks, namely the choice of depth, width and sparsity level. These network attributes typically depend on unknown smoothness in order to be optimal. We obviate this constraint with the fully Bayes construction. As an aside, we show that SS-DL does not overfit in the sense that the posterior concentrates on smaller networks with fewer (up to the optimal number of) nodes and links. Our results provide new theoretical justifications for deep ReLU networks from a Bayesian point of view.
booktitleProceedings of the 32nd International Conference on Neural Information Processing Systems
titlePosterior Concentration for Sparse Deep Learning
family=Michael Kohler,
family=Sophie Langer,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleOn the Rate of Convergence of Fully Connected Deep Neural Network Regression Estimates

Curran Associates, Inc.

booktitleAdvances in Neural Information Processing Systems
titleDeep Exploration via Bootstrapped DQN

Curran Associates, Inc.

booktitleAdvances in Neural Information Processing Systems
titleSimple and Scalable Predictive Uncertainty Estimation Using Deep Ensembles

abstractDeep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of available theoretical results, the literature has been largely silent about uncertainty quantification for deep learning. This paper takes a step forward in this important direction by taking a Bayesian point of view. We study Gaussian approximability of certain aspects of posterior distributions of sparse deep ReLU architectures in non-parametric regression. Building on tools from Bayesian non-parametrics, we provide semi-parametric Bernstein-von Mises theorems for linear and quadratic functionals, which guarantee that implied Bayesian credible regions have valid frequentist coverage. Our results provide new theoretical justifications for (Bayesian) deep learning with ReLU activation functions, highlighting their inferential potential.
booktitleProceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics
seriesProceedings of Machine Learning Research
titleUncertainty Quantification for Sparse Deep Learning

titleEfficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee

abstractOver the last decade, neural networks have reached almost every field of science and became a crucial part of various real world applications. Due to the increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence, i.e. are badly calibrated. To overcome this, many researchers have been working on understanding and quantifying uncertainty in a neural network’s prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. For that, a comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and available implementations. Different examples from the wide spectrum of challenges in the fields of medical image analysis, robotic and earth observation give an idea of the needs and challenges regarding uncertainties in practical applications of neural networks. Additionally, the practical limitations of uncertainty quantification methods in neural networks for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
eprintclasscs, stat
titleA Survey of Uncertainty in Deep Neural Networks
omputer Science - Machine Learning,Statistics - Machine Learning
family=Dennis D. Cox,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleAn Analysis of Bayesian Inference for Nonparametric Regression
family=David Freedman,

journaltitleThe Annals of Statistics
shortjournalThe Annals of Statistics
titleWald Lecture: On the Bernstein-von Mises Theorem with Infinite-Dimensional Parameters

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleFrequentist Coverage of Adaptive Nonparametric Bayesian Credible Sets
coverage,Credible set,uncertainty quantification

abstractWe continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein-von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donsker- and Kolmogorov-Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.
journaltitleThe Annals of Statistics
shortjournalAnn. Statist.
titleOn the Bernstein-von Mises Phenomenon for Nonparametric Bayes Procedures
Mathematics - Statistics Theory

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleSupremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
anisotropic smoothness,mixed partial derivatives,nonparametric multivariate regression,sup-norm posterior contraction,Tensor product B-splines

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleAdaptive Bernstein–von Mises Theorems in Gaussian White Noise
Adaptation,Bayesian inference,conﬁdence set,Credible set,posterior asymptotics

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleOn Coverage and Local Radial Rates of Credible Sets
Credible ball,excessive bias restriction,local radial rate

Institute of Mathematical Statistics

abstractWe investigate the frequentist coverage properties of (certain) Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem (in context of sieve type of priors), we introduce an extra assumption on the functional parameters, the so-called “general polished tail” condition. We then show that under standard assumptions, both the hierarchical and empirical Bayes methods, result in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abstract results to various examples, including the nonparametric regression model, density estimation using exponential families of priors, density estimation using histogram priors and the nonparametric classification model, for which we show that their size is near minimax adaptive with respect to the considered specific pseudometrics.
issn0090-5364, 2168-8966
journaltitleThe Annals of Statistics
titleAsymptotic Frequentist Coverage Properties of Bayesian Credible Sets for Sieve Priors

Rousseau en Szabo - 2019 - Asymptotic frequentist coverage properties of Baye.pdf;C
ierarchical Bayes,Mathematics - Statistics Theory,Nonparametric regression,Posterior contraction rates,sieve prior,uncertainty quantification

Institute of Mathematical Statistics

journaltitleThe Annals of Statistics
titleStatistical Guarantees for Bayesian Uncertainty Quantification in Nonlinear Inverse Problems with Gaussian Process Priors
Bernstein–von Mises theorems,credible sets,Schrödinger equation,X-ray transforms

The Institute of Mathematical Statistics

journaltitleAnnals of Statistics
shortjournalAnn. Statist.
titleConvergence Rates of Posterior Distributions

Institute of Mathematical Statistics and Bernoulli Society

journaltitleElectronic Journal of Statistics
titleOracle Posterior Contraction Rates under Hierarchical Priors

titleTensorFlow: Large-scale Machine Learning on Heterogeneous Systems

Springer Science and Business Media LLC

titleArray Programming with NumPy

titleUncertainty Quantification Using Empirical Bayesian Deep Neural Networks.

titleA Simulation Study of Uncertainty Quantification in Classification Tasks Using Empirical Bayesian Deep Neural Networks

titleWielandt's Theorem

journaltitleMathematische Annalen
shortjournalMathematische Annalen
titleZur Theorie Der Matrices

journaltitleSitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
shortjournalSitzungsber. Königl. Preuss. Akad. Wiss.
titleUeber Matrizen Aus Nicht Negativen Elementen

titleTopics in Random Matrix Theory

Cambridge University Press

abstractThis classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.
seriesCambridge Mathematical Library
titleSpline Functions: Basic Theory

Cambridge University Press

titleFundamentals of Nonparametric Bayesian Inference

The Institute of Mathematical Statistics and the Bernoulli Society

journaltitleElectronic Journal of Statistics
shortjournalElectron. J. Statist.
titleAdaptive Estimation of Multivariate Functions Using Conditionally Gaussian Tensor-Product Spline Priors

abstractLetybe a random vector in Rn, satisfyingEy⊗y=id.LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by finite sums of the tensorsyi⊗yi. We prove that for some absolute constantCE1M∑i=1Myi⊗yi−id⩽C·lognM·(E‖y‖logM)1/logM,provided that the last expression is smaller than 1. We apply this estimate to improve a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position.
journaltitleJournal of Functional Analysis
shortjournalJournal of Functional Analysis
titleRandom Vectors in the Isotropic Position