# Uncertainty measurement with belief entropy on interference effect in
Quantum-Like Bayesian Networks

Zhiming Huang  Lin Yang  Wen Jiang  School of Electronic and Information,
Northwestern Polytechnical University, Xi’an 710072, China  China Equipment
System Engineering Company, Beijing, 100039

###### Abstract

Social dilemmas have been regarded as the essence of evolution game theory, in
which the prisoner’s dilemma game is the most famous metaphor for the problem
of cooperation. Recent findings revealed people’s behavior violated the Sure
Thing Principle in such games. Classic probability methodologies have
difficulty explaining the underlying mechanisms of people’s behavior. In this
paper, a novel quantum-like Bayesian Network was proposed to accommodate the
paradoxical phenomenon. The special network can take interference into
consideration, which is likely to be an efficient way to describe the
underlying mechanism. With the assistance of belief entropy, named as Deng
entropy, the paper proposes Belief Distance to render the model practical.
Tested with empirical data, the proposed model is proved to be predictable and
effective.

###### keywords:

Bayesian Networks; Quantum Probability;Decision Making; Social dilemmas;
belief entropy; Sure Thing Principle

††journal: APPLIED MATHEMATICS AND COMPUTATION

##  1 Introduction

Prisoner’s dilemma game is a famous metaphor for the problem of cooperation,
which is a critical issue in evolutionary game theory [47, 8]. If two players
all defect, the payoff will be lower than if they all cooperates, as shown in
Table 1. The paradoxical findings are shown in Table 2, where the unknown part
is not equal to the last column. The violation of $The\ Sure\ thing\
Principle$ [57] shows humans break the law of classic probability when making
decision under risk [12]. Many analytical mythologies have been made to the
explanation of this phenomenon but the underlying mechanisms are still
enigmatic. Nevertheless, the quantum theory seems to be a practical method to
uncover the mystery lying behind this incredible phenomenon [32, 4].

The quantum theory has been applied in many filed including information theory
[38], decision making system [50, 49], social and information networks [46,
51]. Busemeyer $et\ al.$ [5, 33] proposed a Quantum Dynamical model based on a
quantum version of a classical dynamical Markov model, which takes the process
of making decisions into account of time evolution. The quantum-like approach
developed by Khrennikov [21] is based on contextual probabilities which can be
applied to many domains like cognitive science economics, game theory, etc
[20, 23, 22]. Masanari $et\ al.$ [2, 1] proposed a quantum-like model to
simulate the brain function.Li $et\ al.$ [26] proposed a quantum strategies
into evolutionary games.

Though there are many models based on quantum probability theory, few of them
are predictable. Inspired by the work [30, 13, 35], we propose a novel
Bayesian Networks model based on quantum probability. This paper does not
consider the noise effect in the quantum information systems [36, 37]. In this
model, the violation of rational decision making in many experiments like
Prisoner’s dilemma game and the Two Stage Gambling game is characterized as
interference effect between competing states. This paper regards man’s mental
beliefs as wave functions. Before the final decision is made, all potential
decisions coexists in man’s mind. Such uncertainty is like superposition state
of wave functions [29]. The interference effect is actually influenced by the
partiality of the man towards to the decisions. Once the interference effect
is determined, the man’s behavior can be predicted and described by quantum
probability theory. This paper proposes Belief Distance to measure the
uncertainty with the assistance of belief entropy, named as Deng entropy.
Uncertainty processing in decision making was firstly developed by Mich le and
Jean Yves [7] and the uncertainty can be measured based on distance [9, 28].
The knowledge to the uncertainty in decision making can help psychologists
predict the behavior of humans with few fit errors. With the ability to
compute the uncertainty of decision, the proposed model is predictable and
simple for calculating.

Table 1: Payoffs table A/B | 0 | 1
---|---|---
0 | 4/4 | 2/5
1 | 5/2 | 3/3

##  2 Organization of this paper

This paper is organized in the following manner. In section 3, basic
mathematical preliminaries will be introduced. In this section, a kind of
belief entropy, called Deng entropy, will be introduced, which plays an
important role in the model. After that the Bayesian model based on quantum
probability will be presented in section 4. Numerical examples will be
illustrated in section to show how this model works. In the end, the proposed
model will be compared with two models proposed in other literature to show
its effectiveness.

Table 2: Experiment results of Prisoner’s dilemma game from literature[30]

Literature | Known to Defect | Known to Collaborate | Unknown | Classical Probability
---|---|---|---|---
Shafir and Tversky, 1992 | 0.9700 | 0.8400 | 0.6300 | 0.9050
Li and Taplin,2002 | 0.8200 | 0.7700 | 0.7200 | 0.7950
Busemeyer et al., 2006a | 0.9100 | 0.8400 | 0.6600 | 0.8750
Hristova and Grinberg, 2008 | 0.9700 | 0.9300 | 0.8800 | 0.9500
Average | 0.8700 | 0.7400 | 0.6400 | 0.8050

  * 1

The second column (Known to Defect) means the probability of the second player
choose to betray when he/she knows the first player has chosen to betray. The
third column(Known to Collaborate) means the probability of the second player
choose to betray when he/she knows the first player has chosen to cooperate.
The fourth column (Unknown) means the probability of the second player choose
to betray without any information about the first player’s action. The final
column(Classical probability) means the probability calculated by the classic
probability theory.

##  3 Preliminaries

###  3.1 Belief Entropy

Many contributions [44, 42, 43] have been made to interpret quantum
probability into Dempster-Shafer probability, in which basic belief assignment
is used to describe the probability of an event[58, 39]. A new belief entropy,
named as Deng entropy [11] is a measure of uncertainty of basic belief
assignment [10, 19]. Basic belief assignment(BBA) is widely used in the field
of information fusion [14, 15, 45] which has been applied in many fields like
Failure Mode and Effect Analysis [18, 16], Fault Diagnose [52, 17] and so on.

###### Definition 3.1

Let $\theta=\left\\{{{H_{1}},{H_{2}},...,{H_{N}}}\right\\}$ be a finite
nonempty set of N elements which is mutually exclusive and exhaustive. Denote
$P(\theta)$ as the power set composed of $2^{N}$ elements of $\theta$. The
basic belief assignments(BBAs) function is defined as a mapping of the power
set $P(\theta)$ to the value between $0$ and $1$. $m:P(\theta)\to[0,1]$, which
satisfies the following conditions:

$\displaystyle m(\emptyset)=0$ (1) $\displaystyle\sum\nolimits_{A\subseteq
P(\theta)}{m(A)=1}$

where the mass m(A) represents the support degree of evidence to event $A$.

Shannon entropy, also named as information entropy, is the expected value of
the information contained in each message which can be modeled by any flow of
information.

###### Definition 3.2

The Shannon entropy is defined as follows:

$H=-\sum\nolimits_{i}{{P_{i}}}{\log_{b}}{P_{i}}$ (2)

where $P_{i}$ satisfies $\sum\nolimits_{i}{P_{i}}=1$, $b$ is base of
logarithm. When $b=2$, the unit of Shannon entropy is bit.

The Belief entropy, named as Deng entropy, is introduced here to measure the
uncertainty degree of BBAs, which is defined by:

###### Definition 3.3

${E_{d}}=-\sum\nolimits_{i}{m(A)}\log\frac{{m(A)}}{{{2^{\left|A\right|}}-1}}$
(3)

Where $m$ is the BBAs function, and $A$ is the element of $P(\theta)$,
$\left|A\right|$ is the cardinality of $A$. When $\left|A\right|$ is equal to
$1$, the belief entropy will degenerate into Shannon entropy. The term
$2^{\left|A\right|}-1$ represents the potential states in A.

Example 1 Assume there is a BBAs function m(a)=1. The Shannon entropy and Deng
entropy are computed as follows:

$H=-1\times log_{2}1=0$

$E_{d}=-1\times log_{2}\frac{1}{2^{1}-1}=1$

This example shows if $\left|A\right|$ is equal to $1$, the belief entropy is
similar with the classic Shannon entropy.

Example 2 Given a set $\Theta=\\{a,b,c\\}$ with $m(\\{a\\})=\frac{1}{2}$ and
$m(\\{b,c\\})=\frac{1}{2}$. The Deng entropy will be:

$E_{d}=-\frac{1}{2}\times log_{2}\frac{\frac{1}{2}}{2^{2}-1}$

The above examples show how Deng entropy works and overcomes the insufficiency
of Shannon entropy when measuring the uncertainty in problems like Example 2.

###  3.2 The Classic Bayesian Network and the Quantum-like Bayesian Model

####  3.2.1 Classic Bayesian Network

Figure 1: An example of Bayesian Network

A classic Bayesian Network is a kind of probabilistic directed acyclic
graphical model, which has been successfully applied in the field of decision
making [31]. In this model, a set of random variables and their conditional
dependencies are represented via a directed acyclic graph. Each node that
represents a variable is associated with a conditional probability table, as
shown in Fig.1.

###### Definition 3.4

The full joint distribution of a Bayesian Network is defined by:

$\Pr({X_{1}},{X_{2}}...,{X_{n}})=\prod\limits_{i=1}^{n}{\Pr({X_{i}}|Parents({X_{i}}))}$
(4)

where $X$ is the list of variables, $Parents(X_{i})$ means nodes pointing to
$X_{i}$. The model can answer any query with response of $yes$ or $no$ by
using conditional probability formula and summing over all nuisance variables.
For some query $X$, the inference is given by Eq.(5)

$\begin{split}\Pr(X|e)=\alpha[\sum\limits_{y\in Y}{\Pr(X,e,y)}]\\\
where\quad\alpha=\frac{1}{{\sum\nolimits_{x\in
X}{{{\Pr}_{c}}(X=x,e)}}}\end{split}$ (5)

where $e$ is the list of observed variables (nodes) and $y$ is the remaining
unobserved variables(nodes) in the network, the $\alpha$ is the normalization
factor for the distribution $\Pr(X|e)$ [34].

Example: Fig.1 shows an example of Bayesian Network. Assume there are two
servers $S1$ and $S2$ transmitting data packets to $User$. Apparently, the
parent nodes of $User$ are $S1$ and $S2$ and the parent node of $S2$ is $S1$.
Each node has a conditional probability table which represents if a packet is
transmitted successfully. If there is a query, for example, what is the
probability when $user$ successively receives one data packet. The inference
is computed by Eq.(5) as follows:

$\displaystyle\Pr(One\ Packet)=\alpha\\{Pr(S2=T|S1=F)*Pr(S1=F)$ (6)
$\displaystyle+Pr(S2=F|S1=T)*Pr(S1=T)\\}=\alpha(0.3*0.1+0.3*0.9)=0.3\alpha$
$\displaystyle\Pr(two\ or\ zero\ Packets)=\alpha\\{Pr(S2=T|S1=T)*Pr(S1=T)$
$\displaystyle+Pr(S2=F|S1=F)*Pr(S1=F)\\}=\alpha(0.7*0.9+0.7*0.1)=0.7\alpha$
$\displaystyle\alpha=\frac{1}{\Pr(One\ Packet)+\Pr(two\ or\ zero\ Packets)}=1$

The above example shows the basic idea of Bayesian network and procedure of
deriving inferences according to some queries.

####  3.2.2 Quantum-like Bayesian Model

Bayesian networks can split complex problem into small modules that can be
combined to perform inferences [24, 3]. The quantum-like Bayesian Model [30]
replaces the real probability numbers in the classic probability Bayesian
Network model with quantum probability amplitudes [41, 25].

The corresponding part of quantum-like Bayesian Network model to the
application of Born’s rule to Eq.(4) is:

$\Pr({X_{1}},...,{X_{n}})=|\prod\limits_{i=1}^{n}{\psi({X_{i}}|Parents({X_{i}}))}{|^{2}}$
(7)

The quantum application of Born’s rule to the classic marginal probability
distribution Eq.(5) is defined by the equation below:

$\displaystyle\Pr(X|e)=\partial{\rm{|}}\sum\limits_{Y}{\prod\limits_{x}^{N}{\psi({X_{x}}|Parents({X_{x}}),e,y)}}{{\rm{|}}^{2}}$
(8) $Where\quad\partial=\frac{1}{{\sum\nolimits_{x\in
X}{{{\Pr}_{c}}(X=x,e)}}}=1$ (9)

A quantum marginalization formula with interference effects [29] emerges when
the Eq.(8) expands, as shown in below,

$\displaystyle\Pr(X|e)=\partial\sum\limits_{i=1}^{|Y|}{|\prod\limits_{x}^{N}{\psi({X_{x}}|Parents({X_{x}}),e,y=i)}{|^{2}}+2\cdot
Interference}$ (10) $\displaystyle
Interference=\sum\limits_{i=1}^{|Y|-1}{\sum\limits_{j=i+1}^{|Y|}{|\prod\limits_{x}^{N}{\psi({X_{x}}|Parents({X_{x}}),e,y=i)}|\cdot}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad{{|\prod\limits_{x}^{N}{\psi({X_{x}}|Parents({X_{x}}),e,y=j)}|}\cdot\cos({\theta_{i}}-{\theta_{j}})}$

Example: Fig.2 shows an instance of Quantum-like Bayesian Network.

This network can only answer queries with $yes$ or $no$ answer, which are
regarded as base vectors $|0>$ and $|1>$. Fig.3 shows any actions the node
will take can be seen as wave functions characterized by base vectors $|0>$
and $|1>$, as defined by:

$\displaystyle|T>\ =\ cos\theta_{T}|1>\ +\ sin\theta_{T}|0>\ =\
e^{j\theta_{T}}$ (11) $\displaystyle|F>\ =\ cos\theta_{F}|1>\ +\
sin\theta_{F}|0>\ =\ e^{j\theta_{F}}$

Thus, the decision vector for node $A$ is defined by:

$|{\phi_{A}}>\ =\ {\psi_{A=T}}{\rm{|}}{{\rm{T}}_{A}}>{\rm{\ +\
}}{\psi_{A=F}}|{F_{A}}>\ =\ \psi_{A=T}\cdot e^{j\theta_{T_{A}}}\ +\
\psi_{A=F}\cdot e^{j\theta_{F_{A}}}\\\ $ (12)

where the action states $|{F_{A}}>$ and $|{T_{A}}>$ means the actions the node
can take. The index $A$ in $|{F_{A}}>$ and $|{T_{A}}>$ represents this
decision is made by node $A$.

Figure 2: An example of a Quantum-like Bayesian Network

In the same way, Decision vector for node $B$ is

$|{\phi_{B}}>\ =\ {\psi_{B=T}}{\rm{|}}{{\rm{T}}_{B}}>{\rm{\ +\
}}{\psi_{B=F}}|{F_{B}}>\ =\ \psi_{B=T}\cdot e^{j\theta_{T_{B}}}\ +\
\psi_{B=F}\cdot e^{j\theta_{F_{B}}}\\\ $ (13)

For a query ”what is the probability for B to adopt action T ?”, the inference
is computed by Eq.(8):

$\displaystyle\Pr(T|A)$ $\displaystyle\ =\ \partial|\psi_{B=T}\cdot
e^{j\theta_{T_{B}}}\cdot\psi_{A=T}\cdot e^{j\theta_{T_{A}}}\ +\
\psi_{B=T}\cdot e^{j\theta_{T_{B}}}\cdot\psi_{A=F}\cdot
e^{j\theta_{F_{A}}}|^{2}$ (14) $\displaystyle\ =\
\partial|\psi_{B=T}\cdot\psi_{A=T}\cdot e^{j\theta_{1}}\ +\
\psi_{B=T}\cdot\psi_{A=F}\cdot e^{j\theta_{2}}|^{2}$ $\displaystyle\ =\
\partial|\psi_{B=T}\cdot\psi_{A=T}\cdot e^{j\theta_{1}}\ +\
\psi_{B=T}\cdot\psi_{A=F}\cdot e^{j\theta_{2}}|$ $\displaystyle\ \ \ \ \
\cdot|\psi_{B=T}\cdot\psi_{A=T}\cdot e^{j\theta_{1}}\ +\
\psi_{B=T}\cdot\psi_{A=F}\cdot e^{j\theta_{2}}|^{*}$ $\displaystyle\ =\
\partial|\psi_{B=T}\cdot\psi_{A=T}|^{2}\ +\ |\psi_{B=T}\cdot\psi_{A=F}|^{2}\ $
$\displaystyle\ \ \ \ +\ \psi_{B=T}\cdot\psi_{A=T}\cdot
e^{j\theta_{1}}\cdot\psi_{B=T}\cdot\psi_{A=F}\cdot e^{-j\theta_{2}}$
$\displaystyle\ \ \ \ +\ \psi_{B=T}\cdot\psi_{A=F}\cdot
e^{j\theta_{2}}\cdot\psi_{B=T}\cdot\psi_{A=T}\cdot e^{-j\theta_{1}}$
$\displaystyle\ =\ \partial|\psi_{B=T}\cdot\psi_{A=T}|^{2}\ +\
|\psi_{B=T}\cdot\psi_{A=F}|^{2}\ +$ $\displaystyle\ \ \ \
2\cdot|\psi_{B=T}\cdot\psi_{A=T}\cdot\psi_{B=T}\cdot\psi_{A=F}|\cdot
cos(\theta_{1}-\theta_{2})$ $\displaystyle\Pr(F|A)$ $\displaystyle\ =\
\partial|\psi_{B=F}\cdot\psi_{A=T}|^{2}\ +\ |\psi_{B=F}\cdot\psi_{A=F}|^{2}\
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (15) $\displaystyle\ \ \ \
2\cdot|\psi_{B=F}\cdot\psi_{A=T}\cdot\psi_{B=F}\cdot\psi_{A=F}|\cdot
cos(\theta_{3}-\theta_{4})$

where $\theta_{1}=\theta_{T_{B}}+\theta_{T_{A}}$
,$\theta_{2}=\theta_{T_{B}}+\theta_{F_{A}}$,
$\theta_{3}=\theta_{F_{B}}+\theta_{T_{A}}$,
$\theta_{4}=\theta_{F_{B}}+\theta_{F_{A}}$.

Therefore, the answer to query is $\Pr(T|A)$ once the $\partial$ is determined
by Eq.(9). This example illustrates the definition of Quantum-like Bayesian
Network and the detail derivation of Eq.(10).

##  4 The proposed model

Unlike the method in the literature [30], this paper proposes a new way to
calculate the interference value in the quantum-like Bayesian Network model.
The biggest difference is this paper replaces the term
$\cos(\theta_{i}-\theta_{j})$ in Eq.(10) with the uncertainty degree value
$E_{d}$ calculated by Deng entropy. Deng entropy, also named as Belief
entropy, is a powerful tool to measure the belief degree. The term
$\cos(\theta_{i}-\theta_{j})$ in Eq.(10) is a degree of belief uncertainty in
the quantum interference term. When prisoner has no information about the
other prisoner, his/her decision is influenced by his/her belief about the
rival’s decision. That’s why the classic probability framework can not
describe the game properly because to some extant the prisoner is not totally
”ignorant” about the other prisoner but has his/her own belief about the other
part. This uncertain belief causes the interference term in the Bayesian
Network model, which seems to be variable because the human’s mind is
changeable and is hard to measure. However, many experiments in literatures
have revealed that the human’s belief was inclined to certain degree, which
means the interference term has a tendency value. Once the degree of belief
uncertainty could be measured, the model can be established to describe the
behavior that violates the Sure Thing Principle. Here the Deng entropy is
introduced to measure the belief degree and the results turns out to be fit
for the model to describe the game.

###  4.1 Acquisition of Belief Degree

This paper presents such a concept that the existing of interference term is
because the prisoner’s belief to the other prisoner. According to the classic
probability theory, as analysed in the above section, the probability of a
prisoner to defect the other under unknown condition should be equal to
$\frac{1}{2}(Pr(P2=Defect|P1=Cooperate)+Pr(P2=Defect|P1=Defect))$. But the
experiment results in literature denied this, which means the interference
term truly affects. That’s because actually the prisoner is not totally
”ignorant” about the other, for he/she will predict the other prisoner’s
decision from his/her own perspective and then make self’s decision. For every
individual, every one has his/her own characteristics. When predicting other’s
decision from self’s perspective, the result seems to be diverse. But it is
known that there are something that is common for everyone called human nature
which results in most people that they tend to have a same predication
tendency.

###### Definition 4.1

Belief Degree is defined by:

$\displaystyle D_{b}=cos(\theta_{i}-\theta_{j})$ (16)

where $\theta_{i}$ and $\theta_{j}$ are angles in interference term in
Eq.(10). Belief degree represents people’s predication toward their opponents
and their belief tendency to certain actions in prisoners’ dilemma game.

This predication tendency or Belief Degree determines the value of
interference term. According to the previous experiments shown in Table 2, the
value of interference term is inclined to a certain value, which means there
indeed exists predication tendency or Belief Degree. Hence the Belief Degree
can be determined as shown in following.

The quantum marginalization formula comprises two parts, the classic
probability term and interference term, as Eq.(10) shows. It is the
interference term that equips the model with ability to accommodate the
violation of Sure Thing Principle. In this section, Deng entropy will be
introduced to calculate the belief uncertainty to obtain the interference
value. Notice that the Eq.(11) has two basis states, as shown in Fig.3.

Figure 3: vector representation

There always be two vectors representation of Eq.(10), for the variable $X$
has two alternative value, $T$ and $F$.

$\left[{\begin{array}[]{*{20}{c}}{{\alpha_{T}}}\\\
{{\beta_{T}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{{P_{N}}=T}}\cdot{\psi_{{P_{Parents}}=T}}}\\\
{{\psi_{{P_{N}}=T}}\cdot{\psi_{{P_{Parents}}=F}}}\end{array}}\right]{\rm{}}\left[{\begin{array}[]{*{20}{c}}{{\alpha_{F}}}\\\
{{\beta_{F}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{{P_{N}}=F}}\cdot{\psi_{{P_{Parents}}=T}}}\\\
{{\psi_{{P_{N}}=F}}\cdot{\psi_{{P_{Parents}}=F}}}\end{array}}\right]$ (17)

Before applying Deng entropy to obtain the uncertain term
$cos(\theta_{i}-\theta_{j})$ in the interference term, we should process the
data in Eq.(17) firstly. The vector representation of Eq.(17) is shown in
Fig.4. As can be inferred from Eq.(14) and Eq.(15), two $\theta$ in the Fig.4
have the same value. Though we have known the value of two pairs of $\alpha$
and $\beta$, the value of $\theta$ can hardly be determined through existing
methods. One possible solution is just to regard $cos(\theta_{i}-\theta_{j})$
as an uncertain variable, which can be replaced by belief degree $D_{b}$.
Hence once the belief degree is determined, the interference term is settled.
The belief degree can be determined through belief entropy, which can
calculate the uncertainty from Belief Distance.

Figure 4: vector representation of $\alpha$ and $\beta$

###### Definition 4.2

The Belief Distance is defined by:

$\displaystyle
B_{d_{X}}=|{\alpha_{X}}{\rm{+}}\frac{\alpha_{X}-\beta_{X}}{|\alpha_{X}+\beta_{X}-1|}|$
(18) $\displaystyle where\quad|{\alpha_{X}}-0.5|<|{\beta_{X}}-0.5|$

If $|{\alpha_{X}}-0.5|\geq|{\beta_{X}}-0.5|$, the position of $\alpha_{X}$ and
$\beta_{X}$ should be switched.

The Belief Distance measuring the deviation from $0.5$. If no information is
provided, the value of $\alpha$ and $\beta$ would be $0.5$ because node A has
two actions with each amplitude $\sqrt{5}$ and so does node B.
$\frac{|\alpha_{X}-\beta_{X}|}{|\alpha_{X}+\beta_{X}-1|}$ is actually a
derivation of
$\frac{|\alpha_{X}-0.5|-|\beta_{X}-0.5|}{|\alpha_{X}-0.5|+|\beta_{X}-0.5|}$ .

###### Lemma 4.1

With the relative deviation information provided, Belief degree can be
computed by Eq.(18) and Eq.(3):

$D_{b}=-{E_{d}}=\sum\nolimits_{x}{{{\rm{B}}_{{d_{x}}}}}\log\frac{{{{\rm{B}}_{{d_{x}}}}}}{{{2^{\left|{{A_{i}}}\right|}}-1}}$
(19)

${{\left|{{A_{i}}}\right|}}$ means the number of unobserved variables.

In quantum mechanics, the $cos(\theta_{1}-\theta_{2})$ is given by the inner
product between two wave functions[6], which describes the subtraction of
phases of the two wave function. Because it is difficult to compute
$cos(\theta_{i}-\theta_{j})$ from geometric perspective, this paper just
regards it as a variable which can be computed through belief entropy.

##  5 Numerical example

In this section, the proposed method will be applied in the Bayesian Network
model to analyze the average results presented in Table 2. The process could
be summarized as below.

Step 1: Create the model for the problem: If nothing is told, the first
participant in the Prisoner’s dilemma game will choose $Defect$ or $Cooperate$
with probability of $0.5$. The reason we assume the probability equals to
$0.5$ is that the first participant in the model do not have $parents$ and
nothing is told to him/her. However, in the real situation, the participants
will wonder the other participant’s action and make decisions based on the
judgement. Therefore the assumption that the probability $0.5$ is uncertain.
In the Eq.(19), ${{\left|{{A_{i}}}\right|}}$ means the number of variables
whose decision are not sure. Under this situation, the first participant’s
decision assumed by us is not exactly certain, so the term
${{\left|{{A_{i}}}\right|}}$ will equal to $1$. With the data from Table 2, we
can establish a model as shown in Fig.5;

Figure 5: Bayesian Network model for the Prisoners’ dilemma game with the
average results from Table 2

Step 2: Compute the Belief distance: According to Fig.5, the Eq.(17) can be
paraphrased as below:

$\begin{array}[]{l}\left[{\begin{array}[]{*{20}{c}}{{\alpha_{T}}}\\\
{{\beta_{T}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{{P_{N}}=T}}\cdot{\psi_{{P_{Parents}}=T}}}\\\
{{\psi_{{P_{N}}=T}}\cdot{\psi_{{P_{Parents}}=F}}}\end{array}}\right]{\rm{=}}\left[{\begin{array}[]{*{20}{c}}{\sqrt{0.5}\cdot\sqrt{0.26}}\\\
{\sqrt{0.5}\cdot\sqrt{0.13}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{0.3606}\\\
{0.2550}\end{array}}\right]\\\
\left[{\begin{array}[]{*{20}{c}}{{\alpha_{F}}}\\\
{{\beta_{F}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{{P_{N}}=F}}\cdot{\psi_{{P_{Parents}}=T}}}\\\
{{\psi_{{P_{N}}=F}}\cdot{\psi_{{P_{Parents}}=F}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{\sqrt{0.5}\cdot\sqrt{0.74}}\\\
{\sqrt{0.5}\cdot\sqrt{0.87}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{0.6083}\\\
{0.6595}\end{array}}\right]\end{array}$ (20)

In this way, one can calculate the belief distance with Eq.(18). Here we take
the calculation process of ${\alpha_{F}}$ and ${\beta_{F}}$ for example:
notice that $|{\beta_{T}}-0.5|>|{\alpha_{T}}-0.5|$. The Belief Distance for
${\alpha_{F}}$ and ${\beta_{F}}$ is:

${B_{{d_{T}}}}=|0.6083{\rm{+}}\frac{0.6083-0.6595}{|0.6083+0.6595-1|}|=0.41711$
(21)

And the Belief distance for ${\alpha_{T}}$ and ${\beta_{T}}$ can be computed
in the same way:

${B_{{d_{F}}}}=|0.3606{\rm{+}}\frac{0.3606-0.2550}{|0.3606+0.2550-1|}|=0.63531$
(22)

Step 3: Calculate the belief degree using Deng entropy: In the Step 2 we
obtain the Belief Distance $B_{{d_{T}}}$ and $B_{{d_{F}}}$. The Belief
Distance represents the inner connection between two actions decided by the
participants, as a reflection of a prisoner’s belief to the other. The Deng
entropy is an efficient tool to reveal this connection. In Step 1, we have
analyzed that the term $|A_{i}|$ in the Eq.(19) equals to 1. Hence the results
of Eq.(19) is:

$D_{b}=-{E_{d}}=0.41711\cdot\log\frac{{0.41711}}{{{2^{1}}-1}}+{\rm{0}}{\rm{.63531}}\cdot\log\frac{{{\rm{0}}{\rm{.63531}}}}{{{2^{1}}-1}}=-0.9420$
(23)

The $D_{b}$ will replace the term $cos(\theta_{i}-\theta_{j})$ in the Eq.(10)
to perform the probabilistic interference.

$\displaystyle\Pr(P2=Defect)=\partial[|{\psi_{P2=D|P1=D}}{|^{2}}+|{\psi_{P2=D|P1=C}}{|^{2}}$
(24)
$\displaystyle+2\cdot|{\psi_{P2=D|P1=D}}|\cdot|{\psi_{P2=D|P1=C}}|\cdot\cos(\theta_{1}-\theta_{2})]$
$\displaystyle=\partial[0.5\times 0.87+0.5\times 0.74+2\cdot\sqrt{0.5\times
0.87}\cdot\sqrt{0.5\times 0.74}\cdot-0.9420]$

$\Pr(P2=Cooperate)$ can be obtained i the same way:

$\displaystyle\Pr(P2=Defect)=\partial 0.04917$ (25)
$\displaystyle\Pr(P2=Cooperate)=\partial 0.02182$

And the final result is:

$\displaystyle\Pr(P2=Defect)=\frac{\partial 0.04917}{\partial 0.04917+\partial
0.02182}=0.6926$ (26) $\displaystyle\Pr(P2=Defect)=\frac{\partial
0.02182}{\partial 0.04917+\partial 0.02182}=0.3074$

Compare the result with probability in Table 2, the model this paper proposes
produces a result with fit error percentage of ${8.2{\%}}$.

Fig.6 shows the comparison of results from literature and prediction of model,
from which we can see that the model prediction is coincident to the
probability observed with little fit errors.

Figure 6: Comparison of results from the literature and results predicted by
the model

##  6 Conclusion

Quantum Bayesian Network inherits inference ability from classic Bayesian
network and has the ability to explain the violation of $Sure\ Thing\
Principle$. The model proposed by this paper successfully described the
paradoxical phenomenon in Prisoners’ dilemma game. Unlike other existing
methods, the proposed model regards the violation as an effect of interference
and utilizes the concept of ”Belief Degree” to make prediction though belief
entropy. the model is compared with two other Quantum models. The first model
(model 1) is the Quantum Prospect Decision Theory(QPDT) model developed by
Yukalov and Sornette[56, 55, 53, 54]. In (QPDT) model, a static heuristic is
used to predict the results. The second model is the Quantum-Like Bayesian
Network proposed by Moreira[30], in which a dynamic heuristic is used to
predict the results.

Table 3: Comparison between proposed model and other two models in literature

Literature |  | Pr(Defect)
---
Observed
| Pr(Defect)
---
Computed(Model 1)
| Fit errors
---
model 1
| Pr(Defect)
---
Computed(model 2)
| Fit errors
---
model 2
| Pr(Defect)
---
Computed(Proposed model)
| Fie errors
---
Proposed model
Li and Taplin,2002 1 | 0.8667 | 0.6334 | 0.2692 | 0.8113 | 0.0639 | 0.8623 | 0.0051
Li and Taplin,2002 2 | 0.7000 | 0.5333 | 0.2381 | 0.7006 | 0.0009 | 0.6691 | 0.0441
Li and Taplin,2002 3 | 0.7667 | 0.5500 | 0.2826 | 0.7159 | 0.0663 | 0.7005 | 0.0863
Busemeyer et al.,2006a | 0.6600 | 0.6250 | 0.0531 | 0.7995 | 0.2113 | 0.6069 | 0.0805
Hristova and Grinberg, 2008 | 0.8800 | 0.7000 | 0.2045 | 0.8968 | 0.0191 | 0.9045 | 0.0279
Average fit error | - | - | 0.2095 | - | 0.0723 | - | 0.04878

  * Li and Taplin,2002 1

we use 3 experiments from literature[27] and number them with 1,2 and 3 after
the authors’ name.

From Table 3 we can clearly notice that the average fit errors of proposed
model is smaller than other two models, which shows the new method for Quantum
Bayesian Network proposed by this paper is effective and reliable.

Figure 7: Visualization of comparison results

Fig.7 visualizes the results from Table 2, from which we can see that the
result predicted by the proposed method is occupying the least area of the
bar.

The dilemma situation considered in this paper is prisoner’s dilemma game with
two strategies [40]. The dilemma strength [48] of the game discussed in this
paper is $Dg^{\prime}=Dr^{\prime}=1$. There are also cases with dilemma
strength different from $Dg^{\prime}=Dr^{\prime}=1$, depending on the payoff
table to the players. Admittedly, the method proposed in this paper is
designed to accommodate the paradoxical findings in dilemma strength
$Dg^{\prime}=Dr^{\prime}=1$. Nevertheless, the method can well predict
behaviours the player will take, as shown in Table 3. Comparing with other
similar methods, the proposed Bayesian Network works with the least fit
errors. In the prisoner’s dilemma game, the prisoner will predict the other’s
action if he/she knows little about the rival. Therefore the probability of
the prisoner to choose $Defect$ under unknown case will be smaller than the
value computed from the classic way. The $Sure\ Thing\ Principle$ is violated
because the belief in the prisoner’s mind affects. On the other hand, the
belief degree is not totally irregular. Lots of evidence have examined the
value is closed within a small range. The advantages of Quantum-like Bayesian
Network is it regards two strategies in people’s mind as two wave functions,
which will produce the interference effect. Hence, this paper proposes Belief
Degree to represent the interference effect and utilizes Belief Distance to
calculate the deviation from totally uncertainty. The belief entropy will
produce a corresponding Belief Degree according to Belief Distance. We analyze
the Prisoners’ dilemma game with the model that applied our method and the
prediction results are close to the observed probability with little fit
error. In the end, we compare the model with two models which use a parameter
called heuristic to predict the probability. The comparison results shows the
effectiveness and reliability of our method.

##  7 Acknowledgments

The work is partially supported by National Natural Science Foundation of
China (Grant No. 61671384), Natural Science Basic Research Plan in Shaanxi
Province of China (Program No. 2016JM6018), Aviation Science Foundation
(Program No. 20165553036).

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${\dagger}$${\dagger}$affiliationtext: Research Fellow of the Japan Society
for the Promotion of Science (DC1)

galaxies: evolution – galaxies: star formation – galaxies: starburst –
infrared: galaxies – radio continuum: galaxies

# Estimation of the Star Formation Rate of Galaxies with Radio Continuum
Obtained with Murchison Widefield Array

Tsutomu T. Takeuchi11affiliation: Division of Particle and Astrophysical
Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464–8602, Japan
22affiliation: The Research Center for Statistical Machine Learning, the
Institute of Statistical Mathematics, 10–3 Midori-cho, Tachikawa, Tokyo
190–8562, Japan Shuntaro A. Yoshida11affiliationmark:  Luca
Cortese33affiliation: International Centre for Radio Astronomy Research
(ICRAR), University of Western Australia, Crawley, Perth, Western Australia
6009, Australia 44affiliation: ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D), Australia
O. Ivy Wong55affiliation: CSIRO Space & Astronomy, PO Box 1130, Bentley,
Western Australia 6102, Australia 33affiliationmark:  Barbara
Catinella33affiliationmark:  44affiliationmark:  and Suchetha
Cooray11affiliationmark:  †<EMAIL_ADDRESS>
###### Abstract

We investigate the correlation between the integrated low-frequency and
infrared (IR) emissions of star-forming galaxies extracted from the Herschel
Reference Survey. By taking advantage of the GaLactic Extragalactic All-sky
MWA (GLEAM) survey operated by the Murchison Widefield Array (MWA) we examine
how this correlation varies at a function of frequency across the 20 GLEAM
narrow bands at $72\mbox{--}231\;[\mbox{MHz}]$. These examinations are
important for ensuring the reliability of the radio luminosity as a SFR
indicator. In this study, we focus on 18 star-forming galaxies whose radio
emission is detected by the GLEAM survey. These galaxies show that a single
power-law is sufficient to characterise the far-infrared-to-radio correlation
across the GLEAM frequency bands and up to $1.5\;[\mbox{GHz}]$. Thus, the
radio continuum in this wavelength range can serve as a good dust extinction-
free SFR estimator. This is particularly important for future investigation of
the cosmic SFR independently from other estimators, since the radio continuum
can be detected from $z=0$ to high redshifts ($z\sim 5\mbox{--}10$) in a
coherent manner.

##  1 Introduction

Star formation is one of the most fundamental aspects of the evolution of
galaxies. Observationally, only viable way to estimate the star formation rate
(SFR) in a galaxy is to detect its electromagnetic flux (or flux density),
calculate the luminosity from the flux, and convert it to the number
luminosity of ionizing photon. We can then estimate the number of OB stars
from the number of ionizing photons through stellar spectral energy
distribution models (e.g., Kennicutt, 1983). This method is based on the fact
that the lifetime of OB stars ($\sim 10^{6}$ yr) is much shorter than typical
age of galaxies ($\sim 10^{10}$ yr), namely, we measure the “instantaneous”
number of newly formed stars from OB stars with an assumed initial mass
function (e.g., Kroupa, 2002).

Thus, the crucial point is to measure an observable that can be directly
connected to OB stars. Because of their high luminosity and temperature,
ionizing ultraviolet (UV) radiation is the most straightforward measure of the
number of OB stars. Specifically, the shortest timescale indicator SFR is the
number of ionizing UV photons emitted from the most massive O stars. This can
be determined indirectly via hydrogen recombination lines from Hii regions,
such as H$\alpha$, making them the most commonly used as a primary SFR
estimator (e.g., Kennicutt, 1998). Non-ionizing UV radiation is the next
available observable that can be used to estimate the number of OB stars
(e.g., Takeuchi et al., 2005a; Buat et al., 2007; Takeuchi et al., 2010).
However, since UV photons are strongly scattered and finally absorbed by dust
grains, UV alone does not provide a complete census of SFR. This is why the
mid- and far-infrared (MIR and FIR, respectively) radiation also serve as an
efficient estimator of the SFR (e.g., Kennicutt, 1998; Takeuchi et al., 2005a,
2010, 2013), and shows how a SFR indicator using both UV and FIR may provide a
more robust SFR estimate. Recently such indicators are proposed and become
popular (e.g., Kennicutt et al., 2009; Takeuchi et al., 2010; Murphy et al.,
2011). Many indirect SFR indicators are also proposed: X-ray luminosity
originated from X-ray binaries, forbidden lines from Hii regions (e.g.,
[Oii]), polycyclic aromatic hydro-carbon (PAH) band emission, etc. (e.g.,
Kennicutt, 1998).

Among them, long-wavelength radio continuum is expected to be a promising
pathway to explore high-$z$ galaxy evolution (Haarsma et al., 2000; Murphy et
al., 2011; Takeuchi et al., 2016). In general, radio frequency emission is
produced by the free-free and synchrotron radiation processes (Condon, 1992).
The free-free emission is caused by electrons in Hii regions around OB star
clusters. Synchrotron emission is produced by energetic electrons spiraling
around the interstellar magnetic field. The energetic electrons are
accelerated during supernovae explosions, the final stage of massive stars.
Thus, both radiation processes should be related, even if indirectly to the
number of OB stars. The advantage of using radio continuum over UV-optical
indicators is that it is free from dust extinction. Therefore, radio continuum
is suitable to trace the star formation activities including the dust-
enshrouded ones. Especially, the advent of the Square Kilometre Array (SKA)
will bring us an unprecedented amount of information on the SF history of the
Universe through the radio window (e.g., Murphy et al., 2015; Takeuchi et al.,
2016).

However, it is important to remember that the radio calibration of the SFR is
entirely empirical, contrary to the H$\alpha$ or UV ones, though the physical
process behind it is simple (Murphy et al., 2011). Then, the calibration of
the radio continuum by other wavelength is necessary. This is mainly done
through the IR–radio correlation ubiquitously seen in galaxies (e.g., Condon,
1992). Many observational and statistical works on the IR–radio correlation
have been done. From 80s to early 2000s, space IR facilities brought a large
sample to explore the IR–radio correlation, by IRAS, ISO, Spitzer, AKARI
(e.g., Hummel, 1986; Devereux & Eales, 1989; Bicay & Helou, 1990; Pierini et
al., 2003; Appleton et al., 2004; Murphy et al., 2006; Pepiak et al., 2014;
Solarz et al., 2019). More recently, Herschel data contributed significantly
to the extensive studies on the IR–radio correlation (e.g., Jarvis et al.,
2010; Smith et al., 2014). Though there have been many attempts to explore its
evolution, it remains still controversial (e.g., Bourne et al., 2011; Mao et
al., 2011; Schleicher & Beck, 2013; Schober et al., 2016; Delvecchio et al.,
2021; Molnár et al., 2021). In spite of such uncertainties, the potential use
of the radio continuum as an SFR estimator has attracted attention (e,g.,
Bell, 2003; Murphy et al., 2011; Matthews et al., 2021). Particularly, the
recent development of long-wavelength facilities is a driving force to push
forward such studies, gearing at the SKA (e.g., Shao et al., 2018; Read et
al., 2018; Tisanić et al., 2022).

Thus, a careful investigation of the basic properties of the radio SFR
estimator in relation to the IR–radio correlation remains an important task.
So far, many important works have been done as listed above. Particularly, a
breakthrough has been brought by the Low Frequency Array (LOFAR: e.g., Gürkan
et al., 2018; Wang et al., 2019; Smith et al., 2021; Bonato et al., 2021). In
this work, we show a first exercise from the long-wavelength radio continuum
from galaxies observed by the Murchison Widefield Array (MWA: Tingay et al.,
2013) in Western Australia. The MWA provided us with radio continuum data
densely sampled in frequency (20 bands). We use the sample obtained by the
GLEAM survey (Wayth et al., 2015; Hurley-Walker et al., 2017) to calibrate the
SFR with radio continuum flux density together with the FIR indicator of the
SFR. We also explore the effect of the spectral energy distribution (SED) at
radio wavelengths and the power-law index of the SED to estimate the SFR.

This paper is organized as follows. We introduce the properties of sample
galaxies in Section 2. In Section 3, we discuss the method of the SFR
calibration with the FIR indicator and the FIR–radio correlation. Then we
discuss the result further in Section 5. Section 6 is devoted to summary.
Images and SEDs are presented in Appendix.

##  2 The Sample

Table 1:  This table shows 18 HRS galaxies selected in Section 2. Column (1) NGC (2) R.A. (3) Dec. (5) D25 (6) Dist. are from Cortese et al. (2012), column (4) Type is from Ciesla et al. (2014) (Cortese et al. 2012 for only HRS 163) and column (7) GLEAM ID refers to GLEAM catalog. “AGN” is identified from the BPT diagram (e.g. Baldwin et al., 1981; Kewley et al., 2001; Kauffmann et al., 2003; Schawinski et al., 2007) with emission lines (Boselli et al., 2015).  HRS ID | NGC | R.A. | Dec | Type | D25 | Dist. | GLEAM ID | AGN
---|---|---|---|---|---|---|---|---
|  |  |  |  | [arcmin] | [Mpc] |  |
HRS 25 | 3437 | 10:52:35.75 |  $+$22:56:02.9 | Sc | 2.51 | 18.24 | J105236$+$225606 | –
HRS 36 | 3504 | 11:03:11.21 |  $+$27:58:21.0 | Sab | 2.69 | 21.94 | J110311$+$275812 | –
HRS 50 | 3655 | 11:22:54.62 |  $+$16:35:24.5 | Sc | 1.55 | 21.43 | J112254$+$163522 | –
HRS 77 | 4030 | 12:00:23.64 |  $-$01:06:00.0 | Sbc | 4.17 | 20.83 | J120023$-$010607 | –
HRS 102 | 4254 | 12:18:49.63 |  $+$14:24:59.4 | Sc | 6.15 | 17 | J121850$+$142515 | –
HRS 114 | 4303 | 12:21:54.90 |  $+$04:28:25.1 | Sbc | 6.59 | 17 | J122154$+$042827 | –
HRS 122 | 4321 | 12:22:54.90 |  $+$15:49:20.6 | Sbc | 9.12 | 17 | J122255$+$154939 | –
HRS 144 | 4388 | 12:25:46.82 |  $+$12:39:43.5 | Sb | 5.1 | 17 | J122548$+$123917 | Seyfert
HRS 163 | 4438 | 12:27:45.59 |  $+$13:00:31.8 | Sb | 8.12 | 17 | J122744$+$130020 | Seyfert
HRS 190 | 4501 | 12:31:59.22 |  $+$14:25:13.5 | Sb | 7.23 | 17 | J123159$+$142503 | –
HRS 201 | 4527 | 12:34:08.50 |  $+$02:39:13.7 | Sbc | 5.86 | 17 | J123408$+$023909 | –
HRS 203 | 4532 | 12:34:19.33 |  $+$06:28:03.7 | Im (Im/S) | 2.6 | 17 | J123420$+$062758 | –
HRS 204 | 4535 | 12:34:20.31 |  $+$08:11:51.9 | Sc | 8.33 | 17 | J123418$+$081157 | –
HRS 205 | 4536 | 12:34:27.13 |  $+$02:11:16.4 | Sbc | 7.23 | 17 | J123427$+$021114 | –
HRS 220 | 4579 | 12:37:43.52 |  $+$11:49:05.5 | Sb | 6.29 | 17 | J123743$+$114909 | LINER
HRS 247 | 4654 | 12:43:56.58 |  $+$13:07:36.0 | Scd | 4.99 | 17 | J124355$+$130801 | –
HRS 251 | 4666 | 12:45:08.59 |  $-$00:27:42.8 | Sc | 4.57 | 21.61 | J124508$-$002747 | –
HRS 306 | 5363 | 13:56:07.21 |  $+$05:15:17.2 | pec | 4.07 | 16.23 | J135607$+$051516 | –

The Herschel Reference Survey (HRS) catalog (Boselli et al., 2010) contains
322 nearby galaxies observed in a wide frequency range from FUV to the radio
at $1500\;[\mbox{MHz}]$ (e.g., Cortese et al., 2012; Ciesla et al., 2014;
Boselli et al., 2015). The sample is volume-limited, i.e., it contains sources
with distances between 15 and 25 Mpc. All late-type spirals and irregulars
(Sa–Sd–Im–BCD) with a 2MASS $K$ band total magnitude $K_{\rm Stot}\leq 12$ and
all ellipticals and lenticulars (E, S0, S0a) with $K_{\rm Stot}\leq 8.7$ were
selected. The $K$-band selection intended to minimize the selection effects
associated with dust and with young high-mass stars and to introduce a
selection in stellar mass. These galaxies cover a wide range of morphology
types (early-type, late-type and peculiar), and a galaxy environment from the
center of the Virgo cluster to the isolated field. These galaxies are suitable
for finding the trend of general property in galaxies, since HRS galaxies
represent a general sample of nearby galaxies.

The GaLactic Extragalactic All-sky MWA (GLEAM) survey (Wayth et al., 2015;
Hurley-Walker et al., 2017) observed a whole southern and a northern sky up to
$+30^{\circ}$ ($\sim 25,000\,\mathrm{\deg}^{2}$) using Murchison Widefield
Array (MWA: Tingay et al., 2013) in Western Australia. The GLEAM survey
covered a low-frequency range of $72\mbox{--}231\;[\mbox{MHz}]$ with 20 narrow
bands (each band has $7.6\;[\mbox{MHz}]$ bandwidth). The sensitivity and
angular resolution at $200\;[\mbox{MHz}]$ are $\sim 7\,\mathrm{mJy}$ and $\sim
2\,\mathrm{arcmin}$, respectively.

After the cross-matching with these survey catalogs within a radius of
$120^{\prime\prime}$, we find 39 HRS galaxies have a potential radio
counterpart in the GLEAM catalog. We selected the subsample out of these 39
galaxies, according to the following procedures.

  1. 1.

Remove blended samples (sample size: $39\rightarrow 33$),

  2. 2.

Remove elliptical galaxies ($33\rightarrow 29$),

  3. 3.

Select objects with at least two high quality MWA flux densities
($\mbox{SNR}>5$) ($29\rightarrow 18$).

Step (i) is imposed because it is very difficult to separate radio sources for
the blended objects. We removed elliptical galaxies from the sample [Step
(ii)], because their radio emission is most likely not tracing star formation
activity, but due to a non star-forming origin (likely AGNs), which is not
suitable for the current analysis. Step (iii) allows us to fit the SED model
properly within the MWA bands. This procedure yielded a sample of 18 star-
forming galaxy samples. A summary of the properties of the 18 objects in our
final sample is presented in Table 1. Their radio contours superposed on the
SDSS $i$-band image are shown in Figure 5 in Appendix A.

##  3 Method: IR–Radio Correlation

In order to calibrate the radio continuum as a SFR indicator and examine its
performance, we made use of the IR emission as the primary SFR indicator. A
tight correlation between the $60\,\mu{\rm m}$ and $1.4\;[\mbox{GHz}]$ flux
densities were found by Helou et al. (1985) and Condon et al. (1991). They
defined $q$ as the logarithmic IR-to-radio flux density ratio. In this work,
we use $q_{\nu}$, a generalized version of $q$ to apply to various frequency
datasets, and explore the reliability of radio continuum as a SFR indicator
with high-quality MWA data.

###  3.1 IR-to-radio flux density ratio $q_{\nu}$

Helou et al. (1985) and Yun et al. (2001) have found that $q_{1400\,{\rm
MHz}}=2.14\pm 0.14$ and $2.34\pm 0.26$ for local star-forming galaxies,
respectively. Bell (2003) have shown that $q_{1400\,{\rm MHz}}=2.64\pm 0.26$
if we use the total IR emission $\left(8-1000\,\mu{\rm m}\right)$ instead of
FIR emission. A recent study shows that this relation also holds at
$151\,[\mbox{MHz}]$: $q_{151\,{\rm MHz}}=1.72\pm 0.529$ (Calistro Rivera et
al., 2017), $q_{151\,{\rm MHz}}=1.37$ (Wang et al., 2019)), though with a
scatter larger than that of $q_{1400\,{\rm MHz}}$.

In this paper, we adopt the following definition of $q_{\nu}$ (Calistro Rivera
et al., 2017):

$\displaystyle q_{\nu}$ $\displaystyle\equiv$
$\displaystyle\log\left(\frac{L_{8-1000\,\mu{\rm m}}}{3.75\times
10^{12}\;[\mathrm{erg\,s^{-1}\,Hz^{-1}}]}\right)$ (1)
$\displaystyle-\log\left(\frac{L_{\mathrm{radio},\nu}}{[\mathrm{erg\,s^{-1}\,Hz^{-1}}]}\right)$

where $L_{8-1000\,\mu{\rm m}}$ is the total rest-frame IR luminosity
integrated over $8\mbox{--}1000\,\mu{\rm m}$, which reflects the total dust
emission (e.g., Takeuchi et al., 2005b) and $3.75\times 10^{12}$ is equivalent
to the frequency of $80\,\mu{\rm m}$ introduced to change units.
$L_{\mathrm{radio},\nu}$ is the radio luminosity at the frequency of $\nu$.

Here, we derive $L_{8-1000\,\mu{\rm m}}$ using the calibration equation in
Galametz et al. (2013) with IR fluxes from Spitzer, PACS
($24,\,70,\,100,\,160\,\mu{\rm m}$: Bendo et al. 2012; Cortese et al. 2014)
and Herschel ($250\,\mu{\rm m}$: Ciesla et al. 2012). Although Ciesla et al.
(2014) have already estimated the total IR luminosity for most of HRS galaxies
using SED fitting, we recalculate the $L_{8-1000\,\mu{\rm m}}$ with discrete
broadband IR flux densities (Takeuchi et al., 2005b; Galametz et al., 2013)
because one of the flux densities is missing for HRS163 in the sample. Thus,
for consistency, we adopt $L_{8-1000\,\mu{\rm m}}$ calculated here for all the
sample galaxies. The difference of the determination of the total IR
luminosity can be evaluated from Fig. 2 of Ciesla et al. (2014). Our IR
luminosity calibration and that of Ciesla et al. (2014) do not differ
systematically within the quoted error of Ciesla et al. (2014), either.

###  3.2 SFR from the radio emission

We investigate the frequency dependence of $q_{\nu}$ across the frequency
range covered by GLEAM. In this work, we assume a simple power-law and fit the
following equation to the data at MWA frequencies and $1500\;[\mbox{MHz}]$
(Boselli et al., 2015).

$\displaystyle q_{\nu}=-\gamma\log{\nu}+\beta$ (2)

where $\gamma$ is the power-law index, and $\beta$ is the section at the
ordinate. In Section 4.1, we show the distribution of the estimated $\gamma$
of the current sample.

In order to estimate SFR from the low-frequency radio emission, we start with
the following equations:

$\displaystyle\mathrm{SFR}_{\mathrm{IR}}$ $\displaystyle=$ $\displaystyle
3.88\times 10^{-44}L_{8-1000\,\mu{\rm m}}\;[\mathrm{erg}\,{\rm s}^{-1}]\;,$
(3) $\displaystyle q_{\nu}$ $\displaystyle=$ $\displaystyle q_{1500\,[{\rm
MHz}]}+\log{\left(\frac{\nu}{1500\,[\mbox{MHz}]}\right)}^{\gamma}$ (4)

Equation (3) is the estimate of SFR from the total IR emission (Murphy et al.,
2011), and equation 4 relates between $q_{\nu}$ at a certain frequency $\nu$
and that at $1500\;[\;[\mbox{MHz}]]$.

Substituting equations (1) and (4) into equation (3) yields the following
formula linking the SFR to the radio luminosity at frequency $\nu$:

$\displaystyle\mathrm{SFR}_{\mathrm{radio},\,\nu}$
$\displaystyle\quad=\left(1.46\times 10^{-31}\right)10^{q_{1500\,[{\rm
MHz}]}}{\left(\frac{\nu}{1500\,[\mbox{MHz}]}\right)}^{-\gamma}$
$\displaystyle\qquad\times L_{\mathrm{radio},\,\nu}$ (5)

In Figures 3 and 4 we compare ${\rm SFR}_{\rm radio}$ with other SFR
indicators presented in Boselli et al. (2015).

##  4 Results

Table 2:  The estimated properties of the 18 HRS galaxies selected in the
sample. Columns are (1) $\gamma_{\mathrm{MWA}}$, (2)
$\gamma_{\mathrm{MWA+1500\,MHz}}$, (3)
$\mathrm{SFR_{\mathrm{radio,\,151\;[\mbox{MHz}]}}}$, (4)
$\mathrm{SFR}_{\mathrm{IR}}$, and (5) ${\rm SFR}_{{\rm FUV+24}\mu{\rm m}}$.
HRS ID | $\gamma_{\mathrm{MWA}}$ | $\gamma_{\mathrm{MWA+1500MHz}}$ | ${\rm SFR}_{\mathrm{radio}}$ | ${\rm SFR}_{\mathrm{IR}}$ | ${\rm SFR}_{{\rm FUV+24}\mu{\rm m}}$
---|---|---|---|---|---
|  |  | $\left[M_{\odot}\,\mathrm{yr}^{-1}\right]$ | $\left[M_{\odot}\,\mathrm{yr}^{-1}\right]$ | $\left[M_{\odot}\,\mathrm{yr}^{-1}\right]$
HRS 25 | $0.33\pm 0.47$ | $-0.66\pm 0.07$ | $0.96\pm 0.28$ | $1.41\pm 0.08$ | $1.50$
HRS 36 | $-0.42\pm 0.10$ | $-0.53\pm 0.05$ | $5.15\pm 1.35$ | $4.09\pm 0.16$ | $6.25$
HRS 50 | $-0.38\pm 0.20$ | $-0.63\pm 0.04$ | $1.45\pm 0.41$ | $1.91\pm 0.07$ | $1.59$
HRS 77 | $-0.57\pm 0.07$ | $-0.63\pm 0.04$ | $3.55\pm 0.94$ | $4.81\pm 0.19$ | $3.96$
HRS 102 | $-0.70\pm 0.05$ | $-0.70\pm 0.03$ | $8.19\pm 2.12$ | $6.47\pm 0.25$ | $6.37$
HRS 114 | $-0.57\pm 0.05$ | $-0.58\pm 0.04$ | $5.49\pm 1.43$ | $6.15\pm 0.23$ | $6.63$
HRS 122 | $-0.77\pm 0.07$ | – | $5.45\pm 0.22$ | $4.28\pm 1.14$ | $5.35$
HRS 144 | $0.22\pm 0.22$ | $-0.73\pm 0.08$ | $3.09\pm 0.85$ | $1.66\pm 0.06$ | $2.95$
HRS 163 | $-0.81\pm 0.35$ | – | $0.69\pm 0.03$ | $2.13\pm 0.66$ | $0.39$
HRS 190 | $-0.67\pm 0.12$ | $-0.72\pm 0.05$ | $4.92\pm 1.31$ | $4.37\pm 0.18$ | $3.16$
HRS 201 | $-0.54\pm 0.06$ | $-0.56\pm 0.04$ | $2.71\pm 0.71$ | $4.55\pm 0.24$ | $3.79$
HRS 203 | $-0.55\pm 0.13$ | $-0.60\pm 0.04$ | $1.52\pm 0.43$ | $0.99\pm 0.04$ | $1.33$
HRS 204 | $-0.71\pm 0.10$ | – | $2.60\pm 0.11$ | $2.69\pm 0.75$ | $2.85$
HRS 205 | $-0.61\pm 0.04$ | $-0.57\pm 0.02$ | $2.63\pm 0.69$ | $3.58\pm 0.14$ | $4.51$
HRS 220 | $-0.42\pm 0.18$ | $-0.77\pm 0.05$ | $2.32\pm 0.65$ | $1.56\pm 0.08$ | $1.11$
HRS 247 | $-0.27\pm 0.29$ | $-0.66\pm 0.06$ | $1.65\pm 0.49$ | $2.59\pm 0.11$ | $2.52$
HRS 251 | $-0.58\pm 0.03$ | $-0.58\pm 0.02$ | $9.11\pm 2.35$ | $9.20\pm 0.33$ | $6.66$
HRS 306 | $-0.61\pm 0.08$ | $-0.51\pm 0.04$ | $1.75\pm 0.46$ | $0.27\pm 0.03$ | $0.18$

Now we present the results of the analysis. All the estimated quantities used
in this work are summarized in Table 2.

###  4.1 Distributions of $\gamma$

Figure 1:  Distribution of $\gamma$ in eq. (2). Left: the distribution of
$\gamma$ obtained from the fitting only to the MWA data. Right: same as Left
panel but also with a 1500 MHz flux density.  Figure 2:  Comparison between
$\gamma$ obtained from MWA data with and without 1500 MHz flux density. The
errors are from Table 2. Dashed line represents the 1 to 1 diagonal, and
dotted lines are the envelope obtained by shifting the diagonal by a factor
0.5 along the abscissa and ordinate.

We first present the distribution of estimated $\gamma$ in Figure 1. The left
panel shows the $\gamma$ distribution from the fitting obtained solely with
MWA data, and the right panel is with MWA and $1500\;[\mbox{MHz}]$ data
combined. We show the mean and median with the standard and quantile deviation
in both panels. We note that three galaxies in our sample (HRS 122, 163 and
204) are fitted only including MWA flux densities due to the low signal-to-
noise of available $1500\,\;[\mbox{MHz}]$ detections (Boselli et al., 2015).
We clearly see that two galaxies (HRS 25 and 144) are outliers in the left
panel of Fig. 1, while they do not show any deviation from the bulk of the
population when the shorter frequency data are included. We note that the
origins of the radio continuum measurement are the same (Condon et al., 1990)
for all but one galaxy (HRS 306: Condon et al., 2002). Thus, this can hardly
be a source of a systematic error, and indeed the estimated $\gamma$ of HRS
306 is consistent with others within the quoted measurement uncertainty.

In order to examine these outliers, we compare $\gamma$ only from the MWA data
and both from MWA and 1500 MHz data on individual basis in Fig. 2. The two
outliers are prominent in Fig. 2. We also see that the estimated slopes become
$\gamma<-0.5$ for all the samples, even if some of them had a flatter $\gamma$
when estimated only from MWA data. This suggests that there is a critical
frequency where the turnover appears between MWA frequencies and
$1500\;[\mbox{MHz}]$. At low frequencies, the spectral slope is prone to be
flatter due to the free-free absorption (e.g., Calistro Rivera et al., 2017;
Schober et al., 2017; Chyży et al., 2018). In addition, Schober et al. (2017)
show that a Milky Way-like galaxy (similar in SFR) has the critical frequency
one order of magnitude lower than the MWA frequency. However, HRS 25 and 144
have a SFR similar to the Milky Way (Boselli et al., 2015) and a critical
frequency between MWA frequencies and $1500\;[\mbox{MHz}]$. This would be
difficult to explain by the conjecture of Schober et al. (2017). Possible
reasons for this are 1) poor statistics, i.e., fewer data can constrain the
fitted slope only weakly, and 2) physical peculiarity of the galaxy. As for
HRS 25 might really have a flatter slope, but very possibly it is due to the
statistical uncertainty. In contrast, HRS 144 is identified as a Seyfert
galaxy from the BPT diagram (e.g., Baldwin et al., 1981; Kewley et al., 2001;
Kauffmann et al., 2003; Schawinski et al., 2007), and the galactic nuclei
affects the spectral slope. Further, an Hi absorption towards the core of HRC
144 is reported from the VIVA observation (Chung et al., 2009). Hence, it
would be natural that the radio continuum emission is not consistent with that
of a purely star-forming galaxy. In this case, the shallow slope is physical.
To have a firm conclusion for these two galaxies, we would need more data in a
wide frequency range, e.g., from RACS (Hale et al., 2021). We leave this as
our future work.

We adopt the averaged $\gamma$ obtained with the MWA frequencies and
$1500\;[\mbox{MHz}]$ for the estimation of SFR in the following (Section 4.2).
The fitting results are shown in Appendix B.

###  4.2 Star Formation Rate (SFR) from the low-frequency emission

Figure 3:  Comparison between $\mathrm{SFR}_{\mathrm{radio},\,\nu}$ and
$\mathrm{SFR}_{\mathrm{IR}}$ at each MWA frequency. Symbols are the averages
of $\mathrm{SFR}_{\mathrm{radio},\,\nu}$ among the sample galaxies, and the
error bars represent the standard deviation in the sample. Upper panel shows
the SFRs estimated with individually obtained $\gamma$ and $q_{1500\,[{\rm
MHz}]}$ for the calibration of $\mathrm{SFR}_{\mathrm{radio},\,\nu}$. Middle
panel is the same but with averaged $\gamma$ and $q_{1500\,[{\rm MHz}]}$.
Bottom panel compares the SFR ratios from these two approaches.

Here we show the comparison between $\mathrm{SFR}_{\mathrm{radio},\,\nu}$
defined by eq. 3.2 with $\mathrm{SFR}_{\mathrm{IR}}$ (eq. 3), and ${\rm
SFR}_{{\rm FUV}+{\rm 24}\mu{\rm m}}$ (Boselli et al., 2015) described as

$\displaystyle{\rm SFR}_{{\rm FUV}+{\rm 24}\mu{\rm m}}=6.21\times
10^{-44}\left(L_{\rm FUV,\,obs}+3.89L_{24\,\mu{\rm m}}\right)\;,$ (6)

obtained from the calibration of Hao et al. (2011). Here the units of
luminosities are measured by $[\mbox{erg\,s}^{-1}]$.

Figure 3 shows the ratio between $\mathrm{SFR}_{\mathrm{radio},\,\nu}$ and
$\mathrm{SFR}_{\mathrm{IR}}$. We used the $q_{1500\,{\rm MHz}}$ and $\gamma$
of individual samples in eq. 3.2. Symbols are the averages of
$\mathrm{SFR}_{\mathrm{radio},\,\nu}$ among the sample galaxies at each MWA
frequency, and the error bars represent the standard deviation in the sample.
Note that the number of galaxies at each frequency is different because we
restricted to the sample with high $S/N$ for each flux densities ($S/N\geq
5$). The estimated $\mathrm{SFR}_{\mathrm{IR}}$s are consistent within a range
of $\sim 10$ %.

Figure 4: Comparison between ${\rm SFR}_{\rm IR}$, ${\rm SFR}_{{\rm
FUV+}24\mu{\rm m}}$, and ${\rm SFR}_{\rm radio}$. Three panels show the
comparison between ${\rm SFR}_{\rm IR}$ and ${\rm SFR}_{\rm radio}$, ${\rm
SFR}_{{\rm FUV+}24\mu{\rm m}}$ and ${\rm SFR}_{\rm radio}$, and ${\rm
SFR}_{\rm IR}$ and ${\rm SFR}_{{\rm FUV+}24\mu{\rm m}}$, from the top to the
bottom.

We then examine the consistency of the radio SFR with other SFR indicators. We
show the comparison between ${\rm SFR}_{\rm IR}$ and ${\rm SFR}_{\rm radio}$,
${\rm SFR}_{{\rm FUV+}24\mu{\rm m}}$ and ${\rm SFR}_{\rm radio}$, and ${\rm
SFR}_{\rm IR}$ and ${\rm SFR}_{{\rm FUV+}24\mu{\rm m}}$, in Fig. 4. Here ${\rm
SFR}_{\rm radio}$ stands for the radio SFR at 151 MHz,
$\mathrm{SFR}_{\mathrm{radio},\,151\,{\rm MHz}}$. The top panel of Fig. 4
clearly indicates that the SFR estimated from the radio continuum is
consistent and unbiased with respect to that from IR emission, by
construction. We note that the scatter is not negligible (the scatter
$\sigma^{2}=0.089$, and Pearson’s correlation coefficient is $\rho=0.77$).
Note that the correlation analysis and linear regression in the following are
performed in logarithmic scale. The scatter of the correlation is comparable
to the typical measurement error of the radio side, and we may conclude that
it is mainly caused by the observational error.

A similar comparison is presented in the middle panel of Fig. 4, for ${\rm
SFR}_{{\rm FUV+}24\mu{\rm m}}$ and ${\rm SFR}_{\rm radio}$. Again the radio
SFR estimator is consistent with the UV+24$\,\mu$m hybrid estimator of Hao et
al. (2011) used in Boselli et al. (2015). We note that Boselli et al. (2015)
did not provide the error on the UV+24$\,\mu$m estimates. The scatter is
smaller than the top panel ($\sigma^{2}=0.064$ and the correlation coefficient
is $\rho=0.87$). This can be explained as follows. The timescale of the UV
continuum $t_{\rm UV}$ is longer than $t_{\rm OB}$. In contrast, the timescale
of radio continuum $t_{\rm radio}$ is longer in general, $t_{\rm radio}\sim
10^{8}\;[\rm yr]$ (e.g., Condon, 1992), comparable to the radio continuum
$t_{\rm UV}\sim 10^{8}\;[{\rm yr}]\simeq t_{\rm radio}$. Then, as far as the
contribution of the visible star formation is significant (relatively low SFR,
say ${\rm SFR}\lesssim 10\;[M_{\odot}\,\mbox{yr}^{-1}]$), the correlation is
tight. Further, at ${\rm SFR}\simeq 10\;[M_{\odot}\,\mbox{yr}^{-1}]$, it is
well known that the IR dominates the luminosity from star formation activity
(e.g., Takeuchi et al., 2010, 2013). In addition, since the dust opacity
becomes large at high SFR, the apparent dust continuum tends to be emitted at
long wavelengths ($\lambda\gtrsim 60\;[\mu{\rm m}]$). Then, the mid-IR
emission may not be strongly related to the SFR, and the 24-$\mu$m based
hybrid SFR estimator tends to underestimate the SFR (e.g., Takeuchi et al.,
2005b), resulting in the large deviation at high SFR in the middle panel in
Fig. 4.

This effect is also visible in the bottom panel in Fig. 4. The scatter in the
bottom panel is $\sigma^{2}=0.076$ and the correlation coefficient is
$\rho=0.69$, not very tight. This might indicate the limitation of the MIR-
based hybrid estimator. To summarize, the radio continuum SFR estimator works
as an unbiased measure of the SFR, though we should keep in mind the
nonnegligible scatter.

##  5 Discussion

We further compare our results in Section 4.1 with previous works. Extensive
works have been published from LOFAR team.

###  5.1 Spectral index $\gamma$

First we examine the value of the spectral index $\gamma$. We obtained the
average $\gamma=-0.63\pm 0.07$ with MWA and $1500\,\;[\mbox{MHz}]$ data, while
Calistro Rivera et al. (2017) and Chyży et al. (2018) obtained $-0.78\pm 0.24$
and $-0.56\pm 0.11$, respectively. All are consistent with each other within
the range of quoted errors. The slight difference is due to the different
sample selection: while we used nearby galaxies within $25\;\mbox{Mpc}$,
Calistro Rivera et al. (2017) used 758 galaxies up to $z\sim 2$, and Chyży et
al. (2018) used 118 galaxies up to $z=0.04$. In turn, it would be interesting
to examine the evolution of $\gamma$ with redshifts (e.g., Buat et al., 2011;
Mao et al., 2011; Schleicher & Beck, 2013; Solarz et al., 2016; Delvecchio et
al., 2021; Matthews et al., 2021). Chyży et al. (2018) claimed that the slope
is steeper at higher frequencies ($1.3\mbox{--}5\;[\mbox{GHz}]$). Due to the
redshift of SEDs, the slope of galaxy samples would appear to be steeper at
higher-$z$ in the sample of Calistro Rivera et al. (2017), taken from $z\sim
2$.

###  5.2 Calibration of the radio SFR

We then compare our global calibration of the SFR with other works. In Section
4.2, we discussed the consistency of our SFR calibrations using the low-
frequency emission. To calculate the radio SFR more accurately, we need the
SED of each galaxy (Section 4.2). Since star-forming galaxies have a variety
of $q_{\nu}$ at low frequencies ($\sim 0.53\;\mathrm{dex}$ in Figs. 14 and 15
of Calistro Rivera et al. 2017) and the physical details are still not fully
understood, the radio SFR calibration would have some uncertainty.
Substituting median values $\gamma=-0.63$, $q_{1500\,[{\rm MHz}]}=2.48$ and
$\nu=150\,\;[{\rm MHz}]$ into eq. (3.2) yields the following equation:

$\displaystyle{\rm SFR}_{\rm radio,\,151\,{\rm MHz}}=$
$\displaystyle\quad\left(1.03\pm 0.27\right)\times
10^{-29}L_{\mathrm{radio},\,151\,{\rm
MHz}}\,[\mathrm{erg}\,\mathrm{s}^{-1}]\;.$ (7)

For example, Calistro Rivera et al. (2017) have obtained the coefficient of
$0.76\pm 0.08$ at $z=0$ [their eq. (11)] from their calibration. Thus, we
should keep in mind that the SFR calibration has this amount of uncertainty,
possibly caused by the galaxy selection and the variation of $q_{\nu}$ at low
frequencies.

Gürkan et al. (2018) investigated the relation between the low frequency radio
luminosity obtained by LOFAR at 150 MHz and the SFR for galaxies at $0\lesssim
z\lesssim 0.3$. They used the data in the Herschel Astrophysical Terahertz
Large Area Survey (H-ATLAS: Eales et al. 2010) North Galactic Pole (NGP) field
($\sim 142\mbox{deg}^{2}$. The SFR was estimated from the SED of their sample
by the SED fitting using MAGPHYS (da Cunha et al., 2008). They have made a
classification of sample galaxies by the optical emission line diagnostics by
the SDSS spectra. They adopted the equation

$\displaystyle L_{\rm radio,150\,MHz}=L_{1}{\rm SFR}^{\beta}\;,$ (8)

but used a logarithmic scale for the parameter estimation. They obtained
$\beta=1.07\pm 0.01$, $L_{1}=10^{22.06\pm 0.01}\;[\mbox{W\, Hz}^{-1}]$, and
$\sigma=1.45\pm 0.04$ for their star forming galaxies.

Wang et al. (2019) used the 150-MHz selected LOFAR value-added source
catalogue in the Hobby–Eberly Telescope Dark Energy Experiment (HETDEX) Spring
Field and cross-matched with the 60-$\mu$m selected Revised IRAS Faint Source
Survey Redshift (RIFSC$z$) catalogue at $z\lesssim 0.05$. They have estimated
$L_{\rm radio,150\,MHz}$ for the cross-matched sources and compared it with
${\rm H}\alpha$, $60\;\mu$m, and total IR-based SFR estimators. They obtained

$\displaystyle\log L_{\rm radio,150\,MHz}$ $\displaystyle\quad=\left(1.35\pm
0.06\right)\log{\rm SFR}_{{\rm H}\alpha}+\left(3.20\pm 0.06\right)\;,$ (9)
$\displaystyle\log L_{\rm radio,150\,MHz}$ $\displaystyle\quad=\left(1.31\pm
0.05\right)\log{\rm SFR}_{60\mu{\rm m}}+\left(3.14\pm 0.06\right)\;,$ (10)
$\displaystyle\log L_{\rm radio,150\,MHz}$ $\displaystyle\quad=\left(1.37\pm
0.05\right)\log{\rm SFR}_{{\rm IR}}+\left(3.09\pm 0.05\right)\;.$ (11)

Note that these estimates are for the subsample above their 90-% completeness
limit, roughly corresponding to $\log{\rm SFR}\gtrsim 0.5$. If they use the
whole sample, the slope is significantly flatter and close to unity.

More recently, Smith et al. (2021) presented the result of the LOFAR Surveys
Key Science Project, with the 150 MHz data over the European Large Area
Infrared Space Observatory Survey-North 1 field ($z\lesssim 1$). They obtained

$\displaystyle\log L_{\rm radio,150\,MHz}$ $\displaystyle\quad=(22.221\pm
0.008)+(1.058\pm 0.007)\log{\rm SFR}\;.$ (12)

They pointed out an upward deviation at lower-SFR regime toward higher
redshifts. Bonato et al. (2021) performed LOFAR deep observations of the
Lockman Hole field at 150MHz to examine the relation. They summarized the
related results including above-mentioned works and found that most of them
agree with each other within the range of sample scatter (see their Fig. 4).
They also point out that putting the selection by statistical significance of
the source would bias the relation.

All these results are expressed in the form of SFR–radio luminosity relation,
rather than the reverse, which we believe is practical. Then, it is difficult
to compare these relations with eq. (7). However, since most of them show a
slope close to unity, it would still make sense to compare the relations by
flipping SFR and $L_{\rm radio}$. By this indirect comparison, we safely
conclude that our radio-luminosity SFR estimator is consistent with most of
the preceding LOFAR-based works at $z\simeq 0$.

##  6 Summary

In this study, we examine the correlation between integrated low-frequency
radio and infrared luminosities for a sample of 18 galaxies extracted from
Herschel Reference Survey (HRS) catalog (Boselli et al., 2015). We use flux
densities obtained by the Murchison Widefield Array (MWA) as a part of the
GLEAM survey.

We focused on the radio-to-infrared (IR) correlation coefficient, $q_{\nu}$,
and applied a model as

$\displaystyle q_{\nu}=-\gamma\log{\nu}+\beta$

[eq. (2)]. We found that $q_{\nu}$ from the MWA bands to
$1500\,\;[\mbox{MHz}]$ can be well approximated this single power law model,
and obtained the slope $\gamma$ consistent with previous studies (median
$\gamma=-0.63$, $q_{1500\,{\rm MHz}}=2.48$).

We also investigated the consistency of the radio SFR estimator ${\rm
SFR}_{\rm radio}$ expected to be an extinction-free indicator. The radio SFR
estimator performs stably within the frequency range of MWA. It is consistent
with the total IR estimator by construction, with a scatter due to the details
of the star formation history. The advantage of ${\rm SFR}_{\rm radio}$ is
that it does not require multifrequency observations for normal star-forming
galaxies, unlike the total IR estimator.

For further understanding of the star formation activity as seen from low-
frequency, we need a larger sample with multiband radio observations. Among
such future projects, the updated GLEAM survey will be one of the most
promising one to reveal physical details, with an order-of-magnitude better
angular resolution and sensitivity than the latest survey. Data from RACS
(Hale et al., 2021) will also help further investigation, but we leave this as
our future work.

This work is based on the HRS data together with GLEAM survey maps, both of
which are available from their data archive in public.

This work has been supported by the Japan Society for the Promotion of Science
(JSPS) Grants-in-Aid for Scientific Research (19H05076 and 21H01128). This
work has also been supported in part by the Sumitomo Foundation Fiscal 2018
Grant for Basic Science Research Projects (180923), and the Collaboration
Funding of the Institute of Statistical Mathematics “New Development of the
Studies on Galaxy Evolution with a Method of Data Science”. SC has been
supported by the JSPS (21J23611). Parts of this research were conducted by the
Australian Research Council Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), through project number CE170100013.

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##  Appendix A Galaxy images

Here we show the radio contours superposed on the SDSS $i$-band images of all
the sample galaxies in Figure 5.

Figure 5:  These are the SDSS (Abolfathi et al., 2018) i-band images with
radio contours from the GLEAM survey at $170-231\;[\mbox{MHz}]$ (solid green
lines). These galaxies are selected in Section 2. We draw the green contours
started from the 3$\sigma$ local noise and increase by a factor of the square
root of $2^{n}$ (“n” is an integer). The green star marker shows the location
of a radio source referred to as the GLEAM catalog. The size of this marker
does not mean any feature of observations. The blue circle shows the isophotal
optical size at $25\,\mathrm{mag}\,\mathrm{arcsec}^{-2}$ from Boselli et al.
(2010). On the bottom left in each plot, we show the beam size of the GLEAM
survey.  Figure 5:  — Continued.

##  Appendix B Fitting results

The fitting results of the whole sample is presented in Figure 6.

Figure 6:  These figures show the fitting result. Red points show fluxes at
each MWA frequency, and a yellow point shows the flux at $1500\;[\mbox{MHz}]$
(Boselli et al., 2015). Blue solid line shows the best fitting line, and the
shaded region represents the 95% confidence interval for the fitting to MWA
frequencies. Yellow line and shaded area show the fitting result with
$1500\;[\mbox{MHz}]$ besides MWA frequencies (For HRS122, 163 and 204, we do
not display these because of the lack of high-quality data at
$1500\;[\mbox{MHz}]$ data). On the bottom right, we show the SDSS-RGB stacking
image for each galaxy.