# Probabilistic Central Bell Polynomials

Rongrong Xu  School of Science, Xi’an Technological University, Xi’an, 710021,
Shaanxi, P. R. China<EMAIL_ADDRESS>,  Yuankui Ma  School of Science,
Xi’an Technological University, Xi’an, 710021, Shaanxi, P. R. China
<EMAIL_ADDRESS>,  Taekyun Kim  School of Science, Xi’an Technological
University, Xi’an 710021, Shaanxi, China; Department of Mathematics, Kwangwoon
University, Seoul 139-701, Republic of Korea<EMAIL_ADDRESS>,  Dae San Kim
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
<EMAIL_ADDRESS>and  Salah Boulaaras  Department of Mathematics, Qassim
University, Buraydah Al-Qassim, 51452, Saudi Arabia
<EMAIL_ADDRESS>
###### Abstract.

Let $Y$ be a random variable whose moment generating function exists in a
neighborhood of the origin. In this paper, we study the probabilistic central
Bell polynomials associated with random variable $Y$, as probabilistic
extension of the central Bell polynomials. In addition, we investigate the
probabilistic central factorial numbers of the second kind associated with $Y$
and the probabilistic central Fubini polynomials associated with $Y$. The aim
of this paper is to derive some properties, explicit expressions, certain
identities and recurrence relations for those polynomials and numbers.

###### Key words and phrases:

probabilistic central factorial numbers of the second kind associated with
$Y$; probabilistic central Bell polynomials associated with $Y$; probabilistic
central Fubini polynomials associated with $Y$

###### 2010 Mathematics Subject Classification:

11B73; 11B83

∗ Corresponding authors

##  1\. Introduction

Assume that $Y$ is a random variable satisfying the moment condition (see
(16)). The aim of this paper is to study, as probabilistic extensions of
central Bell polynomials, the probabilistic central Bell polynomials
associated with $Y$, along with the probabilistic central factorial numbers of
the second kind associated with $Y$ and the probabilistic central Fubini
polynomials associated with $Y$. We derive some properties, explicit
expressions, certain identities and recurrence relations for those polynomials
and numbers. In addition, we consider the case that $Y$ is the Poisson random
variable with parameters $\alpha>0$.

The outline of this paper is as follows. In Section 1, we recall the central
factorials, the central factorial numbers of the second and the central Bell
polynomials. We remind the reader of the Stirling numbers of the second kind,
the Bell polynomials, the partial Bell polynomials and the complete Bell
polynomials. Assume that $Y$ is a random variable such that the moment
generating function of $Y$,
$E[e^{tY}]=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}E[Y^{n}],\quad(|t|<r)$, exists
for some $r>0$. Let $(Y_{j})_{j\geq 1}$ be a sequence of mutually independent
copies of the random variable $Y$, and let
$S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\,\,(k\geq 1)$, with $S_{0}=0$. Then we recall
the probabilistic Stirling numbers of the second kind associated with $Y$,
${n\brace k}_{Y}$ and the probabilistic Bell polynomials. Section 2 is the
main results of this paper. Let $(Y_{j})_{j\geq
1},\,\,S_{k},\,\,(k=0,1,\dots)$ be as in the above. We define the
probabilistic central factorial numbers of the second kind associated with
$Y$, $T^{Y}(n,k)$. We derive an explicit expression for $T^{Y}(n,k)$ in
Theorem 2.1. We define the probabilistic central Bell polynomials associated
$Y$, $B_{n}^{(c,Y)}(x)$. The generating function of $B_{n}^{(c,Y)}(x)$ is
found in Theorem 2.2. Explicit expressions for $B_{n}^{(c,Y)}(x)$ are deduced
in Theorems 2.3 and 2.4. An additional explicit expression for
$B_{n}^{(c,Y)}(x)$ is derived in Theorem 2.15 when $Y$ is the Poisson random
variable with parameter $\alpha$. We define the probabilistic central Fubini
polynomials associated with $Y$, $F_{n}^{(c,Y)}(x)$. We obtain explicit
expressions for $F_{n}^{(c,Y)}(x)$ in Theorems 2.5 and 2.6. As to
$B_{n}^{(c,Y)}(x)$, several facts are found, namely expressions in terms of
partial Bell polynomials in Theorems 2.7 and 2.10, a recurrence relation in
Theorem 2.8, convolution formula in Theorem 2.9 and higher-order derivatives
in Theorem 2.13. We derive some identities involving $T^{Y}(n,k)$ and the
partial Bell polynomials in Theorem 2.11 and 2.12. In Theorem 2.14, we find an
expression of $T^{Y}(2n,2k)$ in terms of the partial Bell polynomial. In the
rest of this section, we recall the facts that are needed throughout this
paper.

For $n\in\mathbb{N}\cup\\{0\\}$, the central factorials $x^{[n]}$ are defined
by

(1) $\displaystyle x^{[0]}=1,\quad x^{[n]}$
$\displaystyle=x\bigg{(}x+\frac{n}{2}-1\bigg{)}\bigg{(}x+\frac{n}{2}-2\bigg{)}\cdots\bigg{(}x+\frac{n}{2}-n+1\bigg{)}$
$\displaystyle=x\binom{x+\frac{n}{2}-1}{n-1}(n-1)!,\quad(n\geq
1),\quad(\mathrm{see}\ [7,10-12,17,18]).$

From (1), we note that

(2)
$\bigg{(}\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+1}\bigg{)}^{2x}=\sum_{n=0}^{\infty}x^{[n]}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[10-12,17,18]).$

The central factorial numbers of the second kind are defined by

(3) $x^{n}=\sum_{k=0}^{n}T(n,k)x^{[k]},\quad(n\geq 0),\quad(\mathrm{see}\
[11]).$

From (2) and (3), we have

(4)
$\frac{1}{k!}\Big{(}e^{\frac{t}{2}}-e^{-\frac{t}{2}}\Big{)}^{k}=\sum_{n=k}^{\infty}T(n,k)\frac{t^{n}}{n!},\quad(k\geq
0),\quad(\mathrm{see}\ [7,11]).$

The central Bell polynomials are defined by

(5) $B_{n}^{(c)}(x)=\sum_{k=0}^{n}T(n,k)x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [12]).$

When $x=1$, $B_{n}^{(c)}=B_{n}^{(c)}(1)$ are called the central Bell numbers.
From (5), we note that

(6)
$e^{x(e^{\frac{t}{2}}-e^{-\frac{t}{2}})}=\sum_{n=0}^{\infty}B_{n}^{(c)}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[12,17,18]).$

From (6), we have

(7)
$B_{n}^{(c)}(x)=\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\binom{l+j}{l}(-1)^{j}\bigg{(}\frac{l}{2}-\frac{j}{2}\bigg{)}^{n}\frac{x^{l+j}}{(l+j)!},\quad(n\geq
0),\quad(\mathrm{see}\ [12]).$

For $n\geq 0$, the Stirling numbers of the second kind are defined as

(8) $x^{n}=\sum_{k=0}^{n}{n\brace k}(x)_{k},\quad(n\geq 0),\quad(\mathrm{see}\
[1-24]),$

where $(x)_{0}=1,\ (x)_{n}=x(x-1)\cdots(x-n+1),\ (n\geq 1)$.
The Bell polynomials are defined by

(9) $\phi_{n}(x)=\sum_{k=0}^{n}{n\brace k}x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [7,15,16,19]).$

When $x=1$, $\phi_{n}=\phi_{n}(1)$ are called the Bell numbers.

For any integer $k\geq 0$, the partial Bell polynomials are defined by

(10)
$\frac{1}{k!}\bigg{(}\sum_{m=1}^{\infty}x_{m}\frac{t^{m}}{m!}\bigg{)}^{k}=\sum_{n=k}^{\infty}B_{n,k}\big{(}x_{1},x_{2},\dots,x_{n-k+1}\big{)}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[7,13,17]).$

Thus, by (10), we get

(11) $\displaystyle B_{n,k}(x_{1},x_{2},\dots,x_{n-k+1})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1}+\cdots+l_{n-k+1}=k\\\
l_{1}+2l_{2}+\cdots+(n-k+1)l_{n-k+1}=n\end{subarray}}\frac{n!}{l_{1}!l_{2}!\cdots
l_{n-k+1}!}\bigg{(}\frac{x_{1}}{1!}\bigg{)}^{1!}\bigg{(}\frac{x_{2}}{2!}\bigg{)}^{l_{2}}\cdots\bigg{(}\frac{x_{n-k+1}}{(n-k+1)!}\bigg{)}^{l_{n-k+1}}.$

Thus, by (10), we get

(12) $B_{n,k}(1,1,\dots,1)={n\brace k},\quad(n\geq k\geq 0).$

The complete Bell polynomials are given by

(13)
$\exp\bigg{(}\sum_{i=1}^{\infty}x_{i}\frac{t^{i}}{i!}\bigg{)}=\sum_{n=0}^{\infty}B_{n}(x_{1},x_{2},\dots,x_{n})\frac{t^{n}}{n!},\quad(\mathrm{see}\
[7,13,17]).$

From (10) and (13), we note that

(14)
$B_{n}(x_{1},x_{2},\dots,x_{n})=\sum_{k=0}^{n}B_{n,k}(x_{1},x_{2},\dots,x_{n-k+1}),\quad(n\geq
0).$

By (9) and (14), we get

(15) $B_{n}(x,x,\dots,x)=\phi_{n}(x),\quad(n\geq 0).$

Assume that $Y$ is a random variable such that the moment generating function
of $Y$,

(16)
$E\big{[}e^{tY}\big{]}=\sum_{n=0}^{\infty}E[Y^{n}]\frac{t^{n}}{n!},\quad(|t|<r),\
\ \textrm{exists for some $r>0$}.$

Let $(Y_{j})_{j\geq 1}$ be a sequence of mutually independent copies of the
random variable $Y$, and let

$S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\quad(k\in\mathbb{N}),\ \mathrm{with}\quad
S_{0}=0.$

The probabilistic Stirling numbers of the second kind are given by

(17) $\frac{1}{k!}\Big{(}E[e^{tY}]-1\Big{)}^{k}=\sum_{n=k}^{\infty}{n\brace
k}_{Y}\frac{t^{n}}{n!},\quad(k\geq 0),\quad(\mathrm{see}\ [3,13]).$

In view of (9), the probabilistic Bell polynomials are defined by

(18) $\phi_{n}^{Y}(x)=\sum_{k=0}^{n}{n\brace k}_{Y}x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [13]).$

From (17), we have

${n\brace
k}_{Y}=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}E\big{[}S_{j}^{n}\big{]},\quad(n\geq
k\geq 0).$

When $Y=1$, ${n\brace k}_{Y}={n\brace k},\quad(n\geq k\geq 0)$.
By (18), we get

(19) $\sum_{n=0}^{\infty}\phi_{n}^{Y}(x)\frac{t^{n}}{n!}=e^{x(E[e^{tY}]-1)}.$

When $Y=1$, $\phi^{Y}(x)=\phi_{n}(x),\quad(n\geq 0)$.

##  2\. Probabilistic Central Bell polynomials

Let $Y$ be a random variable and let $(Y_{j})_{j\geq 1}$ be a sequence of
mutually independent copies of the random variable $Y$ with

$S_{0}=0,\quad S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\quad(k\geq 1).$

In view of (4), we define the probabilistic central factorial numbers of the
second kind associated with $Y$ by

(20)
$\frac{1}{k!}\Big{(}E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}]\Big{)}^{k}=\sum_{n=k}^{\infty}T^{Y}(n,k)\frac{t^{n}}{n!},\quad(k\geq
0).$

When $Y=1$, we have $T^{Y}(n,k)=T(n,k),\ (n\geq k\geq 0)$. From (20), we have

(21) $\displaystyle\sum_{n=k}^{\infty}T^{Y}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}\Big{)}^{k-j}(-1)^{j}\Big{(}E\big{[}e^{-\frac{Y}{2}t}\big{]}\Big{)}^{j}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}E\Big{[}e^{-\frac{t}{2}(Y_{1}+Y_{2}+\cdots+Y_{j})}\Big{]}E\Big{[}e^{\frac{t}{2}(Y_{1}+Y_{2}+\cdots+Y_{k-j})}\Big{]}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}E\Big{[}e^{-\frac{t}{2}S_{j}}\Big{]}E\Big{[}e^{\frac{t}{2}S_{k-j}}\Big{]}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{k!}\sum_{j=0}^{k}\sum_{l=0}^{n}\binom{n}{l}\binom{k}{j}\bigg{(}\frac{1}{2}\bigg{)}^{n}(-1)^{n-l-j}E\big{[}S_{k-j}^{l}\big{]}E\big{[}S_{j}^{n-l}\big{]}\frac{t^{n}}{n!}.$

Thus, by comparing the coefficients on both sides of (21), we obtain the
following theorem.

######  Theorem 2.1.

For $n\geq k\geq 0$, we have

$2^{n}T^{Y}(n,k)=\frac{1}{k!}\sum_{j=0}^{k}\sum_{l=0}^{n}\binom{n}{l}\binom{k}{j}(-1)^{n-l-j}E\big{[}S_{k-j}^{l}\big{]}E\big{[}S_{j}^{n-l}\big{]}.$

In view of (5), we define the probabilistic central Bell polynomials
associated with $Y$ by

(22) $B_{n}^{(c,Y)}(x)=\sum_{k=0}^{n}T^{Y}(n,k)x^{k},\quad(n\geq 0).$

When $Y=1$, $B_{n}^{(c,Y)}(x)=B_{n}^{(c)}(x),\ (n\geq 0)$. In particular, for
$x=1$, $B_{n}^{(c,Y)}=B_{n}^{(c,Y)}(1)$ are called the probabilistic central
Bell numbers.
From (22), we note that

(23) $\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}x^{k}T^{Y}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\sum_{n=k}^{\infty}T^{Y}(n,k)\frac{t^{n}}{n!}\bigg{)}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}\frac{1}{k!}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-E\big{[}e^{-\frac{Y}{2}t}\big{]}\Big{)}^{k}$
$\displaystyle=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}.$

Therefore, by (23), we obtain the following theorem.

######  Theorem 2.2.

The generating function of the probabilistic central Bell polynomials
associated with $Y$ is given by

(24)
$e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}=\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}.$

From (24), we have

(25)
$\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=e^{x(E[e^{\frac{Y}{2}t}])}e^{-x(E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\sum_{l=0}^{\infty}\frac{x^{l}}{l!}E\big{[}e^{\frac{t}{2}S_{l}}\big{]}\sum_{j=0}^{\infty}\frac{(-1)^{j}}{j!}x^{j}E\big{[}e^{-\frac{t}{2}S_{j}}\big{]}$
$\displaystyle=\sum_{l=0}^{\infty}\frac{x^{l}}{l!}\sum_{m=0}^{\infty}E[S_{l}^{m}]\frac{1}{m!}\bigg{(}\frac{t}{2}\bigg{)}^{m}\sum_{j=0}^{\infty}\frac{(-1)^{j}}{j!}x^{j}\sum_{p=0}^{\infty}E\big{[}S_{j}^{p}\big{]}\frac{(-1)^{p}}{p!}\bigg{(}\frac{t}{2}\bigg{)}^{p}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\frac{x^{l+j}}{l!j!}(-1)^{j}\bigg{(}\frac{1}{2}\bigg{)}^{n}\sum_{m=0}^{n}\binom{n}{m}(-1)^{n-m}E\big{[}S_{l}^{m}\big{]}E\big{[}S_{j}^{n-m}\big{]}\frac{t^{n}}{n!}.$

Thus, by comparing the coefficients on both sides of (25), we obtain the
following theorem.

######  Theorem 2.3.

For $n\geq 0$, we have

$B_{n}^{(c,Y)}(x)=\frac{1}{2^{n}}\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\frac{x^{l+j}}{l!j!}(-1)^{j}\sum_{m=0}^{n}\binom{n}{m}(-1)^{n-m}E\big{[}S_{l}^{m}\big{]}E\big{[}S_{j}^{n-m}\big{]}.$

When $Y=1$, we get the following result in (7):

$B_{n}^{(c)}(x)=\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\binom{l+j}{l}(-1)^{j}\frac{x^{l+j}}{(l+j)!}\bigg{(}\frac{l}{2}-\frac{j}{2}\bigg{)}^{n}.$

Now, we observe that

(26)
$\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\sum_{l=0}^{\infty}x^{l}\frac{1}{l!}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-1\Big{)}^{l}\sum_{j=0}^{\infty}(-1)^{j}x^{j}\frac{1}{j!}\Big{(}E\big{[}e^{-\frac{Y}{2}t}\big{]}-1\Big{)}^{j}$
$\displaystyle=\sum_{l=0}^{\infty}x^{l}\sum_{i=l}^{\infty}{i\brace
l}_{Y}\bigg{(}\frac{1}{2}\bigg{)}^{i}\frac{t^{i}}{i!}\sum_{j=0}^{\infty}(-1)^{j}x^{j}\sum_{k=j}^{\infty}{k\brace
j}_{Y}(-1)^{k}\bigg{(}\frac{1}{2}\bigg{)}^{k}\frac{t^{k}}{k!}$
$\displaystyle=\sum_{i=0}^{\infty}\sum_{l=0}^{i}x^{l}{i\brace
l}_{Y}\bigg{(}\frac{1}{2}\bigg{)}^{i}\frac{t^{i}}{i!}\sum_{k=0}^{\infty}\sum_{j=0}^{k}(-1)^{j}x^{j}{k\brace
j}_{Y}(-1)^{k}\bigg{(}\frac{1}{2}\bigg{)}^{k}\frac{t^{k}}{k!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\frac{1}{2}\bigg{)}^{n}\sum_{k=0}^{n}\sum_{j=0}^{k}\sum_{l=0}^{n-k}x^{j+l}{n-k\brace
l}_{Y}\binom{n}{k}{k\brace j}_{Y}(-1)^{j+k}\frac{t^{n}}{n!}.$

Therefore, by (26), we obtain the following theorem.

######  Theorem 2.4.

For $n\geq 0$, we have

$B_{n}^{(c,Y)}(x)=\bigg{(}\frac{1}{2}\bigg{)}^{n}\sum_{k=0}^{n}\sum_{j=0}^{k}\sum_{l=0}^{n-k}(-1)^{j+k}\binom{n}{k}{n-k\brace
l}_{Y}{k\brace j}_{Y}x^{j+l}.$

Now, we define the probabilistic central Fubini polynomials associated with
$Y$ by

(27)
$\sum_{n=0}^{\infty}F_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=\frac{1}{1-x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}.$

In particular, $F_{n}^{(c,Y)}=F_{n}^{(c,Y)}(1),\ (n\geq 0)$, are called the
probabilistic central Fubini numbers. By (20) and (27), we get

(28) $\displaystyle\sum_{n=0}^{\infty}F_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}k!\frac{1}{k!}\Big{(}E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}]\Big{)}^{k}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}k!\sum_{n=k}^{\infty}T_{n}^{Y}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}x^{k}k!T_{n}^{Y}(n,k)\frac{t^{n}}{n!}.$

Therefore, by (28), we obtain the following theorem.

######  Theorem 2.5.

For $n\geq 0$, we have

$F_{n}^{(c,Y)}(x)=\sum_{k=0}^{n}k!T_{n}^{Y}(n,k)x^{k}.$

From (27), we note that

(29)
$\displaystyle\sum_{n=0}^{\infty}F_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=\frac{1}{1-x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}\sum_{j=0}^{k}\binom{k}{j}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-1\Big{)}^{k-j}(-1)^{j}\Big{(}E\big{[}e^{-\frac{Y}{2}t}\big{]}-1\Big{)}^{j}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}{k!}\sum_{j=0}^{k}\frac{1}{(k-j)!}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-1\Big{)}^{k-j}(-1)^{j}\frac{1}{j!}\Big{(}E\big{[}e^{-\frac{Y}{2}t}\big{]}-1\Big{)}^{j}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}{k!}\sum_{j=0}^{k}(-1)^{j}\sum_{n=k}^{\infty}\sum_{i=k-j}^{n-j}{i\brace
k-j}_{Y}\bigg{(}\frac{1}{2}\bigg{)}^{n}{n-i\brace
j}_{Y}(-1)^{n-i}\binom{n}{i}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{2^{n}}\sum_{k=0}^{n}\sum_{j=0}^{k}\sum_{i=k-j}^{n-j}x^{k}k!{i\brace
k-j}_{Y}{n-i\brace j}_{Y}\binom{n}{i}(-1)^{n-i-j}\frac{t^{n}}{n!}.$

Thus, by comparing the coefficients on both sides of (29), we obtain the
following theorem.

######  Theorem 2.6.

For $n\geq 0$, we have

$F_{n}^{(c,Y)}(x)=\frac{1}{2^{n}}\sum_{k=0}^{n}\sum_{j=0}^{k}\sum_{i=k-j}^{n-j}(-1)^{n-i-j}k!{i\brace
k-j}_{Y}{n-i\brace j}_{Y}\binom{n}{i}x^{k}.$

By (24), we get

(30) $\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}\frac{t^{n}}{n!}$
$\displaystyle=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\bigg{(}x\sum_{j=1}^{\infty}\frac{1-(-1)^{j}}{2^{j}}E[Y^{j}]\frac{t^{j}}{j!}\bigg{)}^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{n=k}^{\infty}B_{n,k}\Big{(}xE[Y],0,\frac{x}{2^{2}}E[Y^{3}],0,\dots,\frac{x(1-(-1)^{n-k+1})}{2^{n-k+1}}E[Y^{n-k+1}]\Big{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}B_{n,k}\Big{(}xE[Y],0,\frac{x}{2^{2}}E[Y^{3}],0,\dots,\frac{x(1-(-1)^{n-k+1})}{2^{n-k+1}}E[Y^{n-k+1}]\Big{)}\frac{t^{n}}{n!}.$

Therefore, by (30), we obtain the following theorem.

######  Theorem 2.7.

For $n\geq 0$, we have

$B_{n}^{(c,Y)}(x)=\sum_{k=0}^{n}B_{n,k}\Big{(}xE[Y],0,\frac{x}{2^{2}}E[Y^{3}],0,\dots,\frac{x(1-(-1)^{n-k+1})}{2^{n-k+1}}E[Y^{n-k+1}]\Big{)}.$

From (24), we note that

(31)
$\displaystyle\sum_{n=0}^{\infty}B_{n+1}^{(c,Y)}(x)\frac{t^{n}}{n!}=\frac{d}{dt}\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=\frac{d}{dt}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=x\bigg{(}E\bigg{[}\frac{Y}{2}e^{\frac{Y}{2}t}\bigg{]}+E\bigg{[}\frac{Y}{2}e^{-\frac{Y}{2}t}\bigg{]}\bigg{)}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=x\sum_{l=0}^{\infty}\bigg{(}\frac{1}{2}\bigg{)}^{2l}E\big{[}Y^{2l+1}\big{]}\frac{t^{2l}}{(2l)!}\sum_{m=0}^{\infty}B_{m}^{(c,Y)}(x)\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=0}^{\infty}x\sum_{l=0}^{[\frac{n}{2}]}\binom{n}{2l}\bigg{(}\frac{1}{2}\bigg{)}^{2l}E\big{[}Y^{2l+1}\big{]}B_{n-2l}^{(c,Y)}(x)\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (29), we obtain the
following theorem.

######  Theorem 2.8.

For $n\geq 0$, we have

$B_{n+1}^{(c,Y)}(x)=x\sum_{l=0}^{[\frac{n}{2}]}\binom{n}{2l}\bigg{(}\frac{1}{2}\bigg{)}^{2l}E\big{[}Y^{2l+1}\big{]}B_{n-2l}^{(c,Y)}(x).$

We observe that

(32) $\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x+y)\frac{t^{n}}{n!}$
$\displaystyle=e^{(x+y)(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}e^{y(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\sum_{k=0}^{\infty}B_{k}^{(c,Y)}(x)\frac{t^{k}}{k!}\sum_{m=0}^{\infty}B_{m}^{(c,Y)}(y)\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}B_{k}^{(c,Y)}(x)B_{n-k}^{(c,Y)}(y)\frac{t^{n}}{n!}.$

Therefore, we obtain the following convolution formula.

######  Theorem 2.9 (Convolution formula).

For $n\geq 0$, we have

$B_{n}^{(c,Y)}(x+y)=\sum_{k=0}^{n}\binom{n}{k}B_{k}^{(c,Y)}(x)B_{n-k}^{(c,Y)}(y).$

By using (24), we have

(33)
$\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\Big{(}e^{(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}-1+1\Big{)}^{x}=\sum_{k=0}^{\infty}\binom{x}{k}k!\frac{1}{k!}\bigg{(}\sum_{j=1}^{\infty}B_{j}^{(c,Y)}\frac{t^{j}}{j!}\bigg{)}^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\binom{x}{k}k!\sum_{n=k}^{\infty}B_{n,k}\Big{(}B_{1}^{(c,Y)},B_{2}^{(c,Y)},\dots,B_{n-k+1}^{(c,Y)}\Big{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{x}{k}k!B_{n,k}\Big{(}B_{1}^{(c,Y)},B_{2}^{(c,Y)},\dots,B_{n-k+1}^{(c,Y)}\Big{)}\frac{t^{n}}{n!}.$

Therefore, by (33), we obtain the following theorem.

######  Theorem 2.10.

For $n\geq 0$, we have

$B_{n}^{(c,Y)}(x)=\sum_{k=0}^{n}\binom{x}{k}k!B_{n,k}\Big{(}B_{1}^{(c,Y)},B_{2}^{(c,Y)},\dots,B_{n-k+1}^{(c,Y)}\Big{)}.$

It is easy to show that

(34)
$te^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}=t\sum_{j=0}^{\infty}B_{j}^{(c,Y)}(x)\frac{t^{j}}{j!}=\sum_{j=1}^{\infty}B_{j-1}^{(c,Y)}(x)j\frac{t^{j}}{j!}.$

Thus, by (34), we get

(35)
$\displaystyle\bigg{(}\sum_{j=1}^{\infty}jB_{j-1}^{(c,Y)}(x)\frac{t^{j}}{j!}\bigg{)}^{k}=t^{k}\Big{(}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}\Big{)}^{k}$
$\displaystyle=t^{k}\sum_{j=0}^{\infty}k^{j}x^{j}\frac{1}{j!}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-E\big{[}e^{-\frac{Y}{2}t}\big{]}\Big{)}^{j}$
$\displaystyle=t^{k}\sum_{j=0}^{\infty}k^{j}x^{j}\sum_{n=j}^{\infty}T^{Y}(n,j)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\sum_{j=0}^{n}k^{j}x^{j}T^{Y}(n,j)\frac{t^{n+k}}{n!}$
$\displaystyle=\sum_{n=k}^{\infty}k!\sum_{j=0}^{n-k}\binom{n}{k}k^{j}x^{j}T^{Y}(n-k,j)\frac{t^{n}}{n!}.$

From (35), we note that

(36)
$\displaystyle\sum_{n=k}^{\infty}\sum_{j=0}^{n-k}\binom{n}{k}k^{j}x^{j}T^{Y}(n-k,j)\frac{t^{n}}{n!}=\frac{1}{k!}\bigg{(}\sum_{j=1}^{\infty}jB_{j-1}^{(c,Y)}(x)\frac{t^{j}}{j!}\bigg{)}^{k}$
$\displaystyle=\sum_{n=k}^{\infty}B_{n,k}\Big{(}B_{0}^{(c,Y)}(x),2B_{1}^{(c,Y)}(x),3B_{2}^{(c,Y)}(x),\dots,(n-k+1)B_{n-k}^{(c,Y)}(x)\Big{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (36), we obtain the
following theorem.

######  Theorem 2.11.

For $n\geq k\geq 0$, we have

$\sum_{j=0}^{n-k}\binom{n}{k}k^{j}x^{j}T^{Y}(n-k,j)=B_{n,k}\Big{(}B_{0}^{(c,Y)}(x),2B_{1}^{(c,Y)}(x),3B_{2}^{(c,Y)}(x),\dots,(n-k+1)B_{n-k}^{(c,Y)}(x)\Big{)}.$

By making use of (24), we note that

(37)
$\displaystyle\sum_{n=k}^{\infty}B_{n,k}\Big{(}B_{1}^{(c,Y)}(x),B_{2}^{(c,Y)}(x),\dots,B_{n-k+1}^{(c,Y)}(x)\Big{)}\frac{t^{n}}{n!}=\frac{1}{k!}\bigg{(}\sum_{j=1}^{\infty}B_{j}^{(c,Y)}(x)\frac{t^{j}}{j!}\bigg{)}^{k}$
$\displaystyle=\frac{1}{k!}\Big{(}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}-1\Big{)}^{k}=\sum_{j=k}^{\infty}{j\brace
k}x^{j}\frac{1}{j!}\Big{(}E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}]\Big{)}^{j}$
$\displaystyle=\sum_{j=k}^{\infty}{j\brace
k}x^{j}\sum_{n=j}^{\infty}T^{Y}(n,j)\frac{t^{n}}{n!}=\sum_{n=k}^{\infty}\bigg{(}\sum_{j=k}^{n}{j\brace
k}T^{Y}(n,j)x^{j}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (37), we obtain the
following theorem.

######  Theorem 2.12.

For $n\geq k\geq 0$, we have

$B_{n,k}\Big{(}B_{1}^{(c,Y)}(x),B_{2}^{(c,Y)}(x),\dots,B_{n-k+1}^{(c,Y)}(x)\Big{)}=\sum_{j=k}^{n}{j\brace
k}T^{Y}(n,j)x^{j}.$

From (24), we note that

(38)
$\displaystyle\sum_{n=0}^{\infty}\bigg{(}\frac{d}{dx}\bigg{)}^{k}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=\bigg{(}\frac{d}{dx}\bigg{)}^{k}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=\frac{1}{k!}k!\Big{(}E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}]\Big{)}^{k}e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=k!\sum_{l=k}^{\infty}T^{Y}(l,k)\frac{t^{l}}{l!}\sum_{j=0}^{\infty}B_{j}^{(c,Y)}(x)\frac{t^{j}}{j!}=\sum_{n=k}^{\infty}k!\sum_{j=0}^{n-k}\binom{n}{j}B_{j}^{(c,Y)}(x)T^{Y}(n-j,k)\frac{t^{n}}{n!}.$

Therefore, by (38), we obtain the following theorem.

######  Theorem 2.13.

For $k\geq 1$, we have

$\bigg{(}\frac{d}{dx}\bigg{)}^{k}B_{n}^{(c,Y)}(x)=k!\sum_{j=0}^{n-k}\binom{n}{j}B_{j}^{(c,Y)}(x)T^{Y}(n-j,k).$

From (20), we get

(39) $\displaystyle\sum_{n=k}^{\infty}T^{Y}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{k!}\Big{(}E\big{[}e^{\frac{Y}{2}t}\big{]}-E\big{[}e^{-\frac{Y}{2}t}\big{]}\Big{)}^{k}$
$\displaystyle=\frac{1}{k!}\bigg{(}\sum_{j=1}^{\infty}E[Y^{j}]\bigg{(}\frac{1}{2}\bigg{)}^{j}\big{(}1-(-1)^{j}\big{)}\frac{t^{j}}{j!}\bigg{)}^{k}$
$\displaystyle=\sum_{n=k}^{\infty}B_{n,k}\bigg{(}E[Y],0,\bigg{(}\frac{1}{2}\bigg{)}^{2}E[Y^{3}],0,\dots,\bigg{(}\frac{1}{2}\bigg{)}^{n-k+1}E\big{[}Y^{n-k+1}\big{]}\big{(}1-(-1)^{n-k+1}\big{)}\bigg{)}\frac{t^{n}}{n!}.$

Thus, by (39), we get

(40) $\displaystyle T^{Y}(n,k)$
$\displaystyle=B_{n,k}\bigg{(}E[Y],0,\bigg{(}\frac{1}{2}\bigg{)}^{2}E[Y^{3}],0,\dots,\bigg{(}\frac{1}{2}\bigg{)}^{n-k+1}E\big{[}Y^{n-k+1}\big{]}\big{(}1-(-1)^{n-k+1}\big{)}\bigg{)},$

where $n\geq k\geq 0$.
From (40), we obtain the following theorem.

######  Theorem 2.14.

For $n\geq k\geq 0$, we have

$\displaystyle T^{Y}(2n,2k)$
$\displaystyle=B_{2n,2k}\bigg{(}E[Y],0,\bigg{(}\frac{1}{2}\bigg{)}^{2}E[Y^{3}],0,\dots,\bigg{(}\frac{1}{2}\bigg{)}^{2n-2k}E\big{[}Y^{2n-2k+1}\big{]}\bigg{)}.$

Let $Y$ be the Poisson random variable with parameter $\alpha>0$. Then we have

(41)
$E\big{[}e^{\frac{Y}{2}t}\big{]}=\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}e^{-\alpha}e^{\frac{n}{2}t}=e^{\alpha(e^{\frac{t}{2}}-1)},$

and

$E\big{[}e^{-\frac{Y}{2}t}\big{]}=\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}e^{-\alpha}e^{-\frac{n}{2}t}=e^{\alpha(e^{-\frac{t}{2}}-1)}.$

Thus, by (24) and (41), we get

(42)
$\displaystyle\sum_{n=0}^{\infty}B_{n}^{(c,Y)}(x)\frac{t^{n}}{n!}=e^{x(E[e^{\frac{Y}{2}t}]-E[e^{-\frac{Y}{2}t}])}$
$\displaystyle=e^{x(e^{\alpha(e^{\frac{t}{2}}-1)}-e^{\alpha(e^{-\frac{t}{2}}-1)})}$
$\displaystyle=e^{x(e^{\alpha(e^{\frac{t}{2}}-1)}-1)}e^{-x(e^{\alpha(e^{-\frac{t}{2}}-1)}-1)}$
$\displaystyle=\sum_{j=0}^{\infty}\phi_{j}(x)\frac{\alpha^{j}}{j!}\big{(}e^{\frac{t}{2}}-1\big{)}^{j}\sum_{k=0}^{\infty}\phi_{k}(-x)\frac{\alpha^{k}}{k!}\big{(}e^{-\frac{t}{2}}-1\big{)}^{k}$
$\displaystyle=\sum_{j=0}^{\infty}\phi_{j}(x)\alpha^{j}\sum_{i=j}^{\infty}{i\brace
j}\bigg{(}\frac{1}{2}\bigg{)}^{i}\frac{t^{i}}{i!}\sum_{k=0}^{\infty}\phi_{k}(-x)\alpha^{k}\sum_{l=k}^{\infty}{l\brace
k}(-1)^{l}\bigg{(}\frac{1}{2}\bigg{)}^{l}\frac{t^{l}}{l!}$
$\displaystyle=\sum_{i=0}^{\infty}\sum_{j=0}^{i}\phi_{j}(x)\alpha^{j}{i\brace
j}\bigg{(}\frac{1}{2}\bigg{)}^{i}\frac{t^{i}}{i!}\sum_{l=0}^{\infty}\sum_{k=0}^{l}\phi_{k}(-x)\alpha^{k}{l\brace
k}(-1)^{l}\bigg{(}\frac{1}{2}\bigg{)}^{l}\frac{t^{l}}{l!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{l=0}^{n}\sum_{k=0}^{l}\sum_{j=0}^{n-l}\phi_{j}(x)\alpha^{j+k}\binom{n-l}{j}\bigg{(}\frac{1}{2}\bigg{)}^{n}\phi_{k}(-x){l\brace
k}(-1)^{l}\binom{n}{l}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (42), we obtain the following theorem.

######  Theorem 2.15.

Let $Y$ be the Poisson random variable with parameter $\alpha>0$. For $n\geq
0$, we have

$B_{n}^{(c,Y)}(x)=\frac{1}{2^{n}}\sum_{l=0}^{n}\sum_{k=0}^{l}\sum_{j=0}^{n-l}(-1)^{l}\binom{n}{l}\binom{n-l}{j}{l\brace
k}\alpha^{j+k}\phi_{j}(x)\phi_{k}(-x).$

Let $Y$ be the Bernoulli random variable with probability of success $p$. Then
we have

(43)
$E\big{[}Y^{n}\big{]}=\sum_{i=0}^{1}i^{n}p(i)=0^{n}p(0)+1^{n}p(1)=p,\quad(n\in\mathbb{N}).$

By Theorem 2.14 and (43), we get

$\displaystyle T^{Y}(2n,2k)$
$\displaystyle=B_{2n,2k}\bigg{(}p,0,\bigg{(}\frac{1}{2}\bigg{)}^{2}p,0,\dots,\bigg{(}\frac{1}{2}\bigg{)}^{2n-2k}p\bigg{)}$
$\displaystyle=p^{2k}T(2n,2k).$

##  3\. Conclusion

Assume that $Y$ is a random variable such that the moment generating function
of $Y$ exists in a neighborhood of the origin. In this paper, we studied by
using generating functions probabilistic extensions of several special
polynomials and numbers, namely the probabilistic central factorial numbers of
the second kind associated with $Y$, the probabilistic central Bell
polynomials associated with $Y$ and the probabilistic central Fubini
polynomials associated with $Y$.

In more detail, we obtained an explicit expression for $T^{Y}(n,k)$ in Theorem
2.1 and a representation in terms of $B_{2n,2k}$ for $T^{Y}(2n,2k)$ in Theorem
2.14. A generating function of $B_{n}^{(c,Y)}(x)$ was derived in Theorem 2.2.
We deduced explicit expressions for $B_{n}^{(c,Y)}(x)$ in Theorems 2.3 and
2.4, and in Theorem 2.15 for the special case when $Y$ is the Poisson random
variable with parameter $\alpha>0$. We found explicit expressions for
$F_{n}^{(c,Y)}(x)$ in Theorems 2.5 and 2.6. We deduced several facts about
$B_{n}^{(c,Y)}(x)$, namely representions in terms of $B_{n,k}$ in Theorems 2.7
and 2.10, a recursive formula in Theorem 2.8, a convolution formula in Theorem
2.9 and higher-order derivatives in Theorem 2.13. We obtained identities
involving $T^{Y}(n,k)$ and $B_{n,k}$ in Theorems 2.11 and 2.12.

It is one of our future projects to continue to study probabilistic versions
of many special polynomials and numbers and to find their applications to
physics, science and engineering as well as to mathematics.

## Funding

The third author of this research has been conductedby the Research Grant of
Kwangwoon Universityin 2024. And this research was funded by the National
Natural Science Foundation of China (No. 12271320), Key Research and
Development Program of Shaanxi (No. 2023-ZDLGY-02).

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# Cross-cultural Inspiration Detection and Analysis
in Real and LLM-generated Social Media Data

Oana Ignat1 Gayathri Ganesh Lakshmy${}^{2}{{}^{*}}$ Rada Mihalcea1
1University of Michigan - Ann Arbor, USA
2Sri Sivasubramaniya Nadar College of Engineering - Chennai, Tamil Nadu, India
{oignat<EMAIL_ADDRESS><EMAIL_ADDRESS>
Oana Ignat and Gayathri Ganesh Lakshmy contributed equally to the manuscript

###### Abstract

Inspiration is linked to various positive outcomes, such as increased
creativity, productivity, and happiness. Although inspiration has great
potential, there has been limited effort toward identifying content that is
inspiring, as opposed to just engaging or positive. Additionally, most
research has concentrated on Western data, with little attention paid to other
cultures. This work is the first to study cross-cultural inspiration through
machine learning methods. We aim to identify and analyze real and AI-generated
cross-cultural inspiring posts. To this end, we compile and make publicly
available the InspAIred dataset, which consists of 2,000 real inspiring posts,
2,000 real non-inspiring posts, and 2,000 generated inspiring posts evenly
distributed across India and the UK. The real posts are sourced from Reddit,
while the generated posts are created using the GPT-4 model. Using this
dataset, we conduct extensive computational linguistic analyses to (1) compare
inspiring content across cultures, (2) compare AI-generated inspiring posts to
real inspiring posts, and (3) determine if detection models can accurately
distinguish between inspiring content across cultures and data sources.

Cross-cultural Inspiration Detection and Analysis
in Real and LLM-generated Social Media Data

Oana Ignat1††thanks: Oana Ignat and Gayathri Ganesh Lakshmy contributed
equally to the manuscript Gayathri Ganesh Lakshmy${}^{2}{{}^{*}}$ Rada
Mihalcea1 1University of Michigan - Ann Arbor, USA 2Sri Sivasubramaniya Nadar
College of Engineering - Chennai, Tamil Nadu, India {oignat,
<EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

Inspiration has been a part of our world for millennia, starting with ancient
Greece, where Muses were responsible for delivering divine knowledge by
whispering in a poet’s ear Leavitt and Leavitt (1997), all the way to today’s
creativity domain, where it is still common for artists and scientists to
attribute their best ideas to a higher power, independent of their own
control.

According to Thrash and Elliot (2003, 2004), inspiration is a general
construct that consists of three core characteristics: evocation,
transcendence, and approach motivation. Evocation refers to the process of
being triggered by a stimulus, either from within (such as a creative idea
that comes from the subconscious) or from outside (such as a person, object,
music, or nature). Transcendence allows one to perceive something beyond their
usual concerns (Milyavskaya et al., 2012). Finally, an inspired person is
motivated to express, transmit, or act on their inspiration (Elliot and
Thrash, 2002).

Inspiration is an area of study with promising cross-disciplinary applications
in creative fields (e.g., advertisement, storytelling), education, therapy,
mentorship, coaching, or social media. For instance, social network
recommendation systems can mitigate potential harms by showing more positive
and inspiring content to users (Ignat et al., 2021). Access to inspiring
content can have a positive impact on people’s lives by offering them a fresh
perspective and motivating them to take action, particularly during periods of
uncertainty and concern (Oleynick et al., 2014). Moreover, inspiration
facilitates progress towards goals Milyavskaya et al. (2012) and increases
overall well-being Thrash et al. (2010).

Figure 1: We compare AI-generated and human-written inspiring Reddit content
across India and the UK. Although it is challenging for a person to
distinguish between them, we find significant linguistic cross-cultural
differences between generated and real inspiring posts.

Despite the compelling motivation, little research has been done on the
automatic identification of content that is inspiring, as opposed to simply
positive Ignat et al. (2021), and no studies have been conducted on how
inspiration varies across different cultures, and whether it can be
automatically generated. At the same time, the impressive generative ability
of LLMs can create new opportunities for automatically generating inspiring
content.

In this work, we aim to address these research gaps by focusing on three main
research questions. (RQ1:) How do inspiring posts compare across cultures?
(RQ2:) How do AI-generated inspiring posts compare to real inspiring posts
across cultures? And (RQ3:) Can detection models effectively differentiate
inspiring posts across diverse cultures and data sources?

We summarize our contributions as follows. First, we share a novel dataset
comprised of cross-cultural real and generated inspiring and non-inspiring
posts, for a total of 2,000 real inspiring posts, 2,000 non-inspiring posts,
and 2,000 LLM-generated posts, balanced across India and the UK. Second, we
make use of the dataset to conduct extensive computational linguistic analyses
to compare inspiring content across cultures and data sources, i.e., India vs.
the UK and AI-generated vs. human-written. Finally, we explore the
effectiveness of machine learning models for detecting inspiration across
diverse data sources and cultures.

##  2 Related Work

##### Automatic Inspiration Detection.

There have been a limited number of research studies conducted on inspiration,
with the majority of them being carried out by the psychology and sociology
communities. These studies, such as the ones by Thrash and Elliot (2003, 2004)
and Elliot and Thrash (2002), have established the fundamental characteristics
of inspiration. Additionally, they have developed a scale to measure the
frequency with which people experience inspiration in their daily lives. These
studies found that individuals who are inspired tend to be more open to new
experiences and show greater absorption in their tasks. They are also more
intrinsically motivated, have a strong drive to master their work, and are
less competitive.

The work most similar to ours is from Ignat et al. (2021), who are the first
to study inspiration through machine learning methods. To facilitate research
in this domain, they release a weakly labeled dataset consisting of inspiring
and non-inspiring posts collected from Reddit and annotate these posts with
their effect on the reader and the emotions they transmit. They also provide a
RoBERTa (Liu et al., 2019) classifier fine-tuned on human labels to provide a
strong baseline for determining whether a post is inspiring. Finally, they
perform extensive data analyses to gain insights into which topics inspire
readers and how they influence them. Our work builds on Ignat et al. (2021) by
extending it to other cultures, India and the UK, and by collecting AI-
generated inspiring posts in order to compare them to human-written posts.

##### Human vs. LLM-generated cross-cultural text.

With the rapid development of LLMs, these models demonstrate remarkable
proficiency in generating human-like text across multiple languages and styles
Wu et al. (2023); Tang et al. (2023).

In particular, LLMs excel at creative writing, such as story generation (Yuan
et al., 2022), advertising slogan creation (Murakami et al., 2023), and news
composition (Yanagi et al., 2020). Tools like Yuan et al. (2022) can help
users in their creative pursuits. By generating inspiring content, our work
provides a more indirect approach, with the same final goal of helping the
user express, transmit, and act on their insight.

More similar to our work, LLMs have also started to be applied to motivate
people Cox et al. (2023). For example, Karinshak et al. (2023) used GPT-3 to
generate messages to persuade people to receive the Covid-19 vaccine. To the
best of our knowledge, we are the first to generate inspiring content using
LLMs. Our work is also part of the emerging work on modeling cultural factors
in LLMs (Huang and Yang, 2023; Fung et al., 2022; Ramezani and Xu, 2023).
Inspiration varies across cultures. Therefore, we test the cultural knowledge
of LLMs about inspiration in India and the UK, and compare it to inspiring
Reddit posts from users in these countries.

##### Computational Linguistics for Social Media Analysis.

The advent of computational linguistics techniques has enabled researchers to
analyze vast amounts of social media data for various purposes, including
sentiment analysis, topic modeling, and linguistic variation across cultures
(Pennebaker et al., 2007; Pang et al., 2008; Imran et al., 2020). These works
have facilitated the extraction of meaningful insights from diverse linguistic
contexts, paving the way for studies on cross-cultural communication in online
environments.

Social media is a key source of inspiring content for younger audiences (Raney
et al., 2018). Features such as hope and appreciation of beauty and excellence
trigger self-transcendent emotions in videos tagged with “inspiration” on
YouTube (Dale et al., 2017), as well as in #inspiring and #meaningful Tumblr
memes and Facebook posts (Rieger and Klimmt, 2019; Dale et al., 2020).
Similarly, we collect Reddit posts from subreddits related to inspiration
across UK and Indian cultures and analyze them using computational linguistic
tools such as Pennebaker et al. (2007).

##  3 The InspAIred Dataset

To answer our research questions, we compile a novel dataset, which we refer
to as InspAIred \- AI-generated Inspiring Reddit Content. Our dataset contains
inspiring and non-inspiring posts from India and the UK, from two different
sources: (1) crawled from Reddit and (2) generated by an LLM.

###  3.1  Real Inspiring Content

We collect 2,000 weakly labeled inspiring posts and 2,000 weakly labeled non-
inspiring posts, balanced across India and the UK. We describe our data
collection and annotation process below.

##### Data Collection.

We scrape around 5,300 posts from Reddit, a popular online platform,
specifically focusing on culturally inspiring content. Following the data
collection process from Ignat et al. (2021), we conduct searches within
culture-related or discussion-related flairs of the subreddits using keywords,
such as “inspiration” and “motivation”, to identify the relevant data in the
form of both posts and comments.

More specifically, for Indian data, we primarily target the regions of Kerala,
Karnataka, Maharashtra, and Tamil Nadu. Besides the general “r/india”, we also
explore subreddits at the state level, such as “r/Kerala”, “r/Karnataka”,
“r/Maharashtra”, and “r/TamilNadu”, which serve as hubs for focused
discussions on regional culture, traditions, and social issues. Finally, we
also examine subreddits from specific cities, including capital cities like
Chennai and Bangalore, to capture more local perspectives and experiences. For
the UK data, we follow a similar strategy to collect Reddit posts, targeting
both state level subreddits, such as “r/UnitedKingdom”, as well as regional
subreddits representing areas and capital cities, such as “r/London”.

We group the collected posts into two main categories: inspiring posts from
India and inspiring posts from the UK. Initially, we attempted to create a
more fine-grained split at the region or city level, but we faced difficulties
in finding annotators from those specific regions. However, we encourage
future research to explore this direction and investigate how inspiration
varies within a country or within a region, along with exploring other
demographic information such as language, age, gender, or income.

Figure 2: Potato Pei et al. (2022) annotation platform that provides workers
with guidelines to annotate whether a given text is inspiring.

##### Data Filtering.

The collected posts are further filtered with an inspiration classifier model.
We use the XLM-RoBERTa base Conneau et al. (2019) from
HuggingFace.111https://huggingface.co/FacebookAI/xlm-roberta-base The model is
a multilingual version of RoBERTa Liu et al. (2019), and is pre-trained on
2.5TB of filtered CommonCrawl Wenzek et al. (2020)222We choose a multilingual
model, as the Indian data contains code-mix data.. We fine-tune the model for
five epochs using the dataset described below, with a learning rate of $2e-5$
and a batch size of $8$. More implementation details can be found in the
Section A.1.

For fine-tuning, we use the data from Ignat et al. (2021). The data contains
around 12,000 annotated “general”, i.e., no target culture, balanced Reddit
inspiring and non-inspiring posts. More details about the fine-tuning data can
be found in Ignat et al. (2021). Next, we evaluate the quality of the fine-
tuned model predictions by annotating a subset of the posts and comparing them
with the model predictions.

##### Data Annotation.

The posts are annotated by crowd-sourced workers from India and the UK. For
each country, we select a sample of 200/ 1,000 posts equally sampled from
inspiring and non-inspiring predicted posts, to be labeled by three annotators
from that country.

We use Potato Pei et al. (2022) to create the user interface and store the
data. Following the guidelines from Ignat et al. (2021), the user interface
contains a definition of inspiration and examples of inspiring and non-
inspiring posts, as seen in Figure 2. To find and hire UK annotators, we
connect Potato to Prolific.333www.prolific.com We select annotators who have
the following qualifications: an approval rate >98%, living in the UK, and a
right to vote (as a proxy for age). The annotators independently label 200
posts each, categorizing the posts based on their subjective judgment of
whether they found the content inspiring or not. We conduct a similar
annotation process with three reliable Indian annotators to label the data
from India. To ensure consistency amongst the annotators, they are provided
with the same interface as used for the UK annotators and shown in Figure 2.

Finally, we compute the agreement score between the annotators using the
Fleiss Kappa measure Fleiss and Cohen (1973). We obtain a score of $0.24$ for
UK data and $0.29$ for Indian data, indicating a fair agreement, expected for
a subjective task as inspiration detection.

Given that each post is annotated by three annotators, following Ignat et al.
(2021), we mark a post as inspiring if at least one annotator labeled it as
inspiring.

| # non-inspiring ✗ | # inspiring
---|---|---
Annotated |  Weakly-Labeled |  Annotated |  Weakly-Labeled
[length=0.5cm, width=0.3cm]IN |  100 |  900 |  100 |  900
[length=0.5cm, width=0.3cm]GB |  100 |  900 |  100 |  900

Table 1: Final number of inspiring () and non-inspiring (✗) posts across India
and the UK.

We find that the predictions made by the fine-tuned model are quite similar to
human annotations. For Indian data, the accuracy rate is $72.9\%$, and the F1
score is $75.5\%$. Similarly, for UK data, the accuracy rate is $73.5\%$, and
the F1 score is $80.1\%$. Therefore, we decide to not annotate more data and
instead use the model predictions. Additionally, we further fine-tune the
model on the annotated data and use it to collect more weakly labeled data.
This approach aims to leverage the insights gained from the initial fine-
tuning process and apply them to the user-annotated subset, thereby refining
the model to better capture the nuances present in this specific dataset. We
use the fine-tuned model predictions and the annotated subset as our final
labeled data.

##### Quality Assurance.

We remove posts that are classified as toxic or hate speech using two fine-
tuned XLM-RoBERTa classifiers from
HuggingFace.444https://huggingface.co/textdetox/xlmr-large-toxicity-
classifier555https://huggingface.co/facebook/roberta-hate-speech-
dynabench-r4-target We also remove profanity from posts using a profanity
classifier.666https://github.com/snguyenthanh/better_profanity We acknowledge
that automatic detection of profanity, toxicity, and hate speech are active
research areas that have yet to be solved (Dale et al., 2021; Vidgen et al.,
2021). Finally, for each country, we manually inspect 100 random posts to
ensure syntactic and semantic correctness and find the remaining toxic or hate
speech posts. We find that around 99% of the posts are of good quality.

##### Data Statistics.

The final data statistics are shown in Table 1.

###  3.2  LLM-Generated Inspiring Content

We generate 2,000 inspiring posts with
GPT-4,777https://platform.openai.com/docs/guides/text-generation/chat-
completions-api balanced across India and the UK. Our study can be conducted
with any LLM. However, we chose GPT-4 because it is one of the largest LLMs
available and has been shown to effectively emulate human texts. Achiam et al.
(2023)

####  3.2.1 Prompt Design and Robustness

GPT-4 takes as input a list of message objects, and returns an inspiring
Reddit post. We use messages, which are more interactive and dynamic compared
to the classical prompt style. Specifically, we use messages with three
different roles: system, user or
assistant.888https://help.openai.com/en/articles/7042661-chatgpt-api-
transition-guide

The prompt is first formatted with a system role, which sets the behavior of
the model. This is followed by a conversation between the user and assistant,
in a few-shot prompting fashion. Prior work found that LLMs function better
with few-shot prompts (i.e., instructions alongside example output) rather
than using zero-shot prompts (with no examples). (Brown et al., 2020)

##### System Prompt.

We find that we can obtain high-quality responses with additional context in
our prompts. Therefore, we instruct the model to be a Reddit user from either
the UK or India. To ensure that the generated data is diverse and reliable, we
collect five versions of system prompts with different phrasing, with one
example shown below.

> Imagine you’re a person from {location} and use Reddit regularly.

##### User-Assistant Prompts.

We design two rounds of conversations between a user and an assistant, where
the user asks for a Reddit post or comment, and the assistant responds to the
request. We use few-shot prompting by providing the assistant in the first
round of conversation with an inspiring post, which is randomly extracted from
the real posts annotated as inspiring by the majority of annotators. Finally,
the answer to the second user is automatically generated by the assistant and
used to collect the GPT-4 generated inspiring post. A user message is shown
below.

> Write a Reddit post or comment of maximum 100 tokens about what inspires
> you.

##### Quality Assurance.

To ensure the quality of our generated data, we conduct sanity checks to
review approximately 200 inspiring posts balanced across the UK and India. The
posts are checked for cultural knowledge, factuality, semantic and syntax
errors, and style. Based on the feedback, we find that the posts are
semantically and syntactically accurate, possess cultural knowledge, and are
often more complex than real posts.

|  Real Post  |  GPT4-generated Post
---|---|---
[length=0.5cm, width=0.3cm]IN |  The youngest freedom fighter martyr of the country, the brave boy who refused to take the British cops across the river and was shot to death. His name was Baji Rout and he was 12 at the time of his death. |  Dr. APJ Abdul Kalam - His humble beginnings, insatiable thirst for knowledge, and absolute dedication to his country have been my greatest inspiration. It pushed me to work harder, dream big, and contribute to society.
✗ What is a common meal there? I’ve always thought Indian palates to be quite spicy and rich. |  ✗ N/A
[length=0.5cm, width=0.3cm]GB |  Dr. Helen Sharman. I’m very pleased to come from a country whose first astronaut isn’t a man by default. Equality shouldn’t be about women catching up, it should be about women being first 50% of the time. |  Absolutely love Sir David Attenborough’s documentaries. His passion and commitment to preserving the environment is truly inspiring in these challenging times.
✗ Mate all he does is play football and misses pens. Not that special |  ✗ N/A

Table 2: Random samples of inspiring () and non-inspiring (✗) posts from India
and UK.

##  4 Cross-Cultural Inspiration Analysis across Real and LLM-generated posts

In line with previous work (Jakesch et al., 2023), we also find during manual
inspection that it is challenging to differentiate between LLM-generated texts
and those written by humans (see Table 2). Therefore, we perform computational
linguistic analyses to compare real and LLM-generated text across cultures.
This section addresses our first two research questions: RQ1: How do inspiring
posts compare across cultures? and RQ2: How do AI-generated inspiring posts
compare to real inspiring posts across cultures?.

###  4.1 Stylistic and Structural Features

We assess the linguistic style and structure of real and LLM-generated
inspiring posts across India and the UK in terms of (1) analytic writing, (2)
descriptiveness, and (3) readability.

##### Analytic writing

index measures the complexity and sophistication of the writing, which can be
an indicator of advanced thinking. The formula for analytic writing is
$[articles+prepositions-pronouns-auxiliaryverbs-adverb-conjunctions-
negations]$ from LIWC scores (Jordan et al., 2019; Pennebaker et al., 2014)
More information about LIWC can be found in Section 4.2. We display the scores
in Table 3. The low complexity scores of Reddit posts are primarily negative
due to their high usage of pronouns and lack of articles. We find that LLM-
generated inspiring posts from the UK and India are more complex than real
posts, which aligns with our initial observations from data quality checks. At
the same time, there is no significant difference in text complexity between
real inspiring posts from India and those from the UK.

##### Descriptiveness

can be measured by the frequency of adjectives used in language patterns.
Texts with high rates of adjectives tend to be more elaborate and narrative-
like compared to texts with low rates of adjectives. (Chung and Pennebaker,
2008) In Table 3, we find that LLM-generated inspiring posts from the UK and
India are more descriptive than real posts. Additionally, real inspiring posts
from India are more descriptive than those from the UK.

##### Readability

considers not only word count, but also word complexity. For instance, longer
words are more complex than shorter ones. We use the Flesch Reading Ease
metric (Flesch, 1948), which counts the number of words per sentence and
syllables per word. In Table 3, we find that LLM-generated inspiring posts
from the UK and India are less readable than real posts. Additionally, real
inspiring posts from India are less readable than those from the UK.
Furthermore, when measuring post length, LLM-generated UK posts are longer
than real posts and UK posts are shorter than Indian posts.

|  [length=0.5cm, width=0.3cm]IN |  [length=0.5cm, width=0.3cm]GB
---|---|---
|  |  |  |
Analytic | -16.7 $\pm$ 18.0 | -6.9 $\pm$ 10.0 | -17.1 $\pm$ 19.9 | -7.0 $\pm$ 11.4
Descriptive | 8.2 $\pm$ 7.8 | 8.9 $\pm$ 3.6 | 6.6 $\pm$ 5.9 | 9.1 $\pm$ 3.9
Readable | 36.3 $\pm$ 63.1 | 12.1 $\pm$ 20.6 | 53.2 $\pm$ 52.2 | 29.6 $\pm$ 18.0
Word Count | 61.1 $\pm$ 59.8 | 66.9 $\pm$ 20.3 | 43.9 $\pm$ 49.3 | 49.9 $\pm$ 16.8

Table 3: To what degree is the LLM-generated text () stylistically and
structurally different from the real text ()? We compute the mean and standard
deviation for the inspiring posts, across cultures. The differences are
statistically significant, based on the Student t-test (Student, 1908),
p-value < 0.05.

###  4.2 Semantic and Psycholinguistic Features

We assess the semantic and psycholinguistic differences between LLM-generated
and real inspiring posts across India and the UK, using topic modeling and
LIWC psycholinguistic markers.

##### Data Pre-processing.

We use spaCy library999https://spacy.io/models to pre-process the data:
tokenize each post, lowercase tokens, remove stop words, remove numbers,
symbols, emojis, links, and lemmatize tokens.

##### Topic Modeling.

We use Scattertext Kessler (2017), a tool used to create interactive
visualizations of linguistic patterns. After initial pre-processing, we
structure the text data for a better representation of information. Next, we
use the n-gram representation of the text data to conduct topic extraction
through sentence-level clustering. We utilize methods like Non-negative Matrix
Factorization (NMF) to decompose the TF-IDF matrix to identify the latent
thematic structures within the corpus Ramos et al. (2003).

Results. We analyze the n-gram and topic distributions by various dimensions:
inspiring vs. non-inspiring, Indian vs. UK, and real vs. generated. We display
the topic distribution across real and generated UK inspiring posts in Figure
3. Further topic and n-gram distributions are displayed in Section A.2.

Figure 3: Visualization of topics used in the real and generated ( vs. )
inspiring posts from the UK. Points are colored red or blue based on the
association of their corresponding terms with UK Real inspiring posts or UK
LLM-Generated inspiring posts. The most associated topics are listed under Top
Generated and Top Real headings. Interactive version:
https://anonymous.4open.science/r/cross_inspiration.

vs. ✗: Comparing inspiring to non-inspiring real posts, we find that, in the
Indian data (Fig. 5, 7), the topic of people is frequently discussed in an
inspiring context. Moreover, the most commonly occurring theme in non-
inspiring posts is bot, encompassing textual content regarding Reddit rules
and moderators.

In the UK data (Fig. 6, 8), the topic of life is amongst the most commonly
occurring themes in inspiring posts, with discussions surrounding career,
luck, and pension. On the other hand, the non-inspiring posts contain themes
like dark, including conversations about rain and winter.

vs. : The LLM-generated Indian data (Fig. 10, 11) often places a significant
emphasis on the topic of inspiring, within which common words include
dedication and motivational. There is also a significant number of posts
surrounding isro (Indian Space Research Organisation), featuring terms such as
space, mars, and mission. In contrast, the real data from India contains
mentions of housing, summer, and living, grouped under the category live. The
topic of movie is also popular, containing words such as hype, stardom, and
socialize, hinting at the culture surrounding the film industry.

In the LLM-generated UK data (Fig. 3, top left), the topic nhs is emphasized,
where common words include pandemic, heroes, and staff. Additionally,
discussions often revolve around adversity, with mentions of resilience,
determination, and spirit. In contrast, in the real data from the UK (Fig. 3,
bottom right), discussions related to job dominate, with mentions of salary,
savings, and time. Moreover, there is also a significant focus on exercise,
featuring words like discipline, and motivate.

##### LIWC psycholinguistic markers.

We use Linguistic Inquiry and Word Count (LIWC), a gold-standard text analysis
tool (Pennebaker et al., 2007, 2015), to obtain the words related to human
cognitive processes from each post. Specifically, we use the LIWC2015
dictionary, which contains 6,400 words and word stems, each related to a
cognitive category. As an example, the word “mother” is assigned the following
cognitive categories: female, family, social.

Results. We display the top 24 categories and their LIWC scores, with the most
significant differences across dimensions in Table 4. Across all data, we find
that male words are more frequent than female, the most common pronoun is I,
the most frequent tense is present, and the most common emotion is positive.

vs. ✗: Comparing inspiring to non-inspiring real posts, we find that, in the
Indian data, inspiring posts are more likely to be related to socializing,
insight, feelings, perception, affection, and contain more positive emotions.
Additionally, Indian inspiring posts tend to have more comparisons and use
more words related to achievement, health, reward, and work.

UK real inspiring posts contain more words related to affection, comparisons,
feelings, achievement, health, home, leisure, money, reward, and work than
non-inspiring posts. Furthermore, compared to Indian inspiring posts, UK
inspiring posts have fewer words related to family, socializing, affection,
perception, religion, less positive emotions and more words related to
achievement, health, home, leisure, money, rewards, and work.

vs. : Comparing real to LLM-generated inspiring posts, we find that real
Indian posts tend to include more words related to family, social
interactions, comparisons, feelings, perceptions as well as home, leisure and
rewards. Conversely, real Indian posts contain fewer words related to
affection, insight, achievement, health, and religion.

UK real inspiring posts contain more words related to comparisons, feelings,
health, home, leisure, money, rewards and work than LLM-generated UK posts.
Conversely, real UK posts contain fewer words related to socializing,
affection, insight, perception, achievement and religion than LLM-generated UK
posts. Furthermore, compared to LLM-generated Indian posts, LLM-generated UK
posts have fewer words related to socializing, achievement, leisure, money,
religion, work and more words related to affection, perception and more
positive emotions.

|  [length=0.5cm, width=0.3cm]IN |  [length=0.5cm, width=0.3cm]GB
---|---|---
LIWC |  |  ✗ |  |  |  ✗ |
Social Processes
FAMILY | 0.4 | 0.4 | 0.2 | 0.2 | 0.1 | 0.2
FRIEND | 0.3 | 0.4 | 0.2 | 0.2 | 0.4 | 0.2
FEMALE | 0.4 | 0.3 | 0.2 | 0.4 | 0.4 | 0.2
MALE | 1.1 | 1.0 | 0.9 | 0.9 | 1.3 | 1.7
SOCIAL | 9.3 | 7.7 | 8.6 | 7.4 | 8.1 | 8.1
Affective Processes
AFFECT | 6.1 | 4.3 | 8.4 | 5.4 | 4.8 | 9.4
NEGEMO | 1.4 | 1.3 | 1.0 | 1.3 | 1.6 | 0.9
POSEMO | 4.5 | 3.0 | 7.3 | 4.1 | 3.1 | 8.3
Cognitive Processes
COMPARE | 3.0 | 2.4 | 1.8 | 3.0 | 2.7 | 1.8
INSIGHT | 2.3 | 2.0 | 4.8 | 2.2 | 2.1 | 4.9
Perceptual Processes
FEEL | 0.6 | 0.4 | 0.4 | 0.7 | 0.3 | 0.4
PERCEPT | 2.4 | 2.0 | 2.1 | 2.1 | 2.1 | 2.4
Personal Concerns
ACHIEV | 1.9 | 1.0 | 3.5 | 2.4 | 1.6 | 2.6
HEALTH | 0.7 | 0.4 | 0.9 | 1.0 | 0.8 | 0.8
HOME | 0.3 | 0.2 | 0.1 | 0.7 | 0.4 | 0.2
LEISURE | 1.5 | 1.4 | 1.2 | 1.7 | 1.4 | 0.9
MONEY | 0.7 | 0.8 | 0.6 | 1.2 | 0.8 | 0.4
RELIG | 0.4 | 0.5 | 1.1 | 0.1 | 0.1 | 0.6
REWARD | 2.0 | 1.1 | 1.0 | 2.2 | 1.6 | 1.1
WORK | 3.0 | 1.9 | 3.0 | 3.9 | 2.7 | 1.9

Table 4: Comparing LIWC scores across culture (India and the UK) in inspiring
vs. non-inspiring ( vs. ✗) and real vs. LLM-generated posts ( vs. ). We
present the top 24 categories with the most significant differences across
these dimensions.

##  5 Cross-Cultural Inspiration Detection across Real and LLM-generated posts

To answer our last research question – RQ3: Can detection models effectively
differentiate inspiring posts across diverse cultures and data sources? – we
fine-tune a multi-label classification model to identify if a post is
inspiring, represents India or the UK, and whether it is real or LLM-
generated.

##### Implementation Details.

We use the XLM-RoBERTa base model Conneau et al. (2019), pre-trained on 2.5TB
of filtered CommonCrawl Wenzek et al. (2020). We fine-tune the model on our
dataset for five epochs with a learning rate of $2e-5$ and a batch size of
$8$, while monitoring the validation performance and selecting the best model
checkpoint based on the F1 metric.

##### Model Training Setup.

We experiment with two setups to split the data into training, validation, and
test sets: a default train-val-test split of 64-16-20% and a few-shot split of
8-2-90%.

##### Results.

Since the dataset is evenly distributed across classes, a random baseline
results in a 50% accuracy score.

The XLM-RoBERTa model with few-shot setup achieves an accuracy of $75.0\%$ and
F1 score of $91.1\%$ across all labels, while the default setup achieves an
accuracy of $89.1\%$ and F1 score of $96.3\%$. Figure 4 displays the
disaggregated results for each label. We find that even with very few training
data (600 posts), in the few-shot setup, the XLM-RoBERTa model learns to
accurately distinguish inspiring content across cultures (India, UK) and data
sources (real and generated). Furthermore, we do not find significant
differences in performance across cultures and data sources. Finally, the
model can accurately distinguish between real and generated inspiring posts in
both few-shot and default training setups.

Figure 4: Multi-label classification test results with the few-shot and
default setups with the XLM-RoBERTa base Conneau et al. (2019) model.

##  6 Conclusion

In this paper, we introduced the task of cross-cultural inspiration detection
and generation in social media data. To facilitate research in this domain, we
released InspAIred, a dataset of 2,000 real inspiring posts, 2,000 real non-
inspiring posts, and 2,000 LLM-generated inspiring posts, evenly distributed
across India and the UK. We performed extensive linguistic data analyses to
gain insight into what topics inspire each culture and compare AI-generated
inspiring posts to real inspiring posts across various linguistic dimensions.
Despite the difficulty humans have in distinguishing between real posts and
those generated by LLMs, we found that these posts have noticeable differences
in style, structure, and semantics and that, even with little data, fine-tuned
models accurately distinguish inspiring content across cultures and data
sources.

We hope our work will enable the exploration of various applications to
improve creativity and motivation, including storytelling, advertising, and
social media, as well as therapy and coaching. Our dataset can be used to
test, fine-tune, and analyze other models, and it is publicly available at
https://github.com/MichiganNLP/cross_inspiration.

## Limitations

### 1\. A more fine-grained data split

Our posts are divided into two main categories - inspiring posts from India
and inspiring posts from the UK. Initially, we tried to create a more detailed
classification at the region or city level, but we faced difficulties in
finding annotators from those specific regions. However, we encourage future
research to explore this direction and investigate how inspiration varies
within a country or region, as well as explore other demographic information,
such as language, age, gender, or income.

##### Study case for India.

The extensive discussions on Indian subreddits, like r/AskIndia and r/India,
offer straightforward access to text-based motivational narratives.
Additionally, region-based subReddits, like r/Chennai and r/Mumbai, provide
diverse insights into localized experiences and discussions. The nature of
posts on this topic is mostly anecdotal. Reddit users contribute by sharing
personal experiences that have inspired them.

### 2\. Limited number of cross-cultural inspiring posts

It is crucial to recognize that inspiration is not always explicitly
articulated through words like “inspiring” or “motivation”, especially within
cultures that aren’t as open about talking about such topics. In many
instances, it manifests more subtly, embedded within narratives, imagery, or
cultural expressions. This implicit nature of inspiration adds another layer
of complexity to the data collection process.

The language barrier also introduces an additional barrier, as it requires
linguistic fluency and cultural understanding to interpret and analyze
inspiring content effectively. Posts in languages other than English may
contain nuances and cultural references that are not easily translatable.
Moreover, it becomes more challenging to collect inspiring content when data
collection is restricted to certain countries. That is why we collected fewer
posts compared to Ignat et al.’s “general” inspiring posts.

### 3\. Relevance of LLM-based data to current times

In our reliance on LLM-based data, it’s imperative to recognize the temporal
limitations inherent in its training corpus. While these models offer
remarkable capabilities in understanding and generating text, they might not
fully capture the current cultural and societal context in their outputs. This
limitation can be primarily attributed to the fact that these models are
trained on data only up to a specific cutoff date, i.e., GPT-4 only learned
from data dated up to September 2021.

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##  Appendix A Appendix

###  A.1 Fine-tuning Process

##### XLM-RoBERTa Model.

Implementation Details. We experiment with three different fine-tuned
classification models for the purpose of weakly labeling the presence of
inspiration in the dataset. The first attempt involved fine-tuning the model
on the general inspiration dataset introduced in Ignat et al. (2021). In each
of the experiments, the data is split into three subsets: training,
validation, and test sets, using an 80:10:10 ratio. The fine-tuning process
involved training the model for 5 epochs with a learning rate of $2e-5$ and a
batch size of $8$. We monitored the model performance on a separate validation
set and selected the best model checkpoint based on accuracy metric.

In the second experiment, we focused on a subset of the dataset used in
annotation, consisting of 200 posts. These posts were labeled as inspiring if
at least one user considered them to be so. We fine-tuned the base XLM-RoBERTa
model using this subset, maintaining the same training configurations as in
the initial experiment. In the last experiment, we leveraged the user-
annotated posts to further refine the model trained on the general inspiration
dataset from the first experiment. This approach aimed to leverage the
insights gained from the initial fine-tuning process and apply them to the
user-annotated subset, thereby refining the model to better capture the
nuances present in this specific dataset.

###  A.2 Topic Analysis

Figure 5: Scattertext visualization of unigrams used in the real inspiring and
non-inspiring ( vs. ✗) Reddit posts from India. Points are colored in red or
blue based on the association of their corresponding terms with Indian Non-
inspiring posts or Indian inspiring posts. The most associated terms are
listed under “Top inspiring” and “Top Non-inspiring” headings. Figure 6:
Scattertext visualization of unigrams used in the real inspiring and non-
inspiring ( vs. ✗) Reddit posts from the UK. Points are colored in red or blue
based on the association of their corresponding terms with UK Non-inspiring
posts or UK inspiring posts. The most associated terms are listed under “Top
inspiring” and “Top Non-inspiring” headings.  Figure 7: Scattertext
visualization of topics used in the real inspiring and non-inspiring ( vs. ✗)
Reddit posts from India. Points are colored in red or blue based on the
association of their corresponding terms with India Non-inspiring posts or
India inspiring posts. The most associated topics are listed under “Top
inspiring” and “Top Non-inspiring” headings.  Figure 8: Scattertext
visualization of topics used in the real inspiring and non-inspiring ( vs. ✗)
Reddit posts from the UK. Points are colored red or blue based on the
association of their corresponding terms with UK Non-inspiring posts or UK
inspiring posts. The most associated topics are listed under “Top inspiring”
and “Top Non-inspiring” headings.  Figure 9: Scattertext visualization of
unigrams used in the real and generated ( vs. ) inspiring posts from the UK.
Points are colored red or blue based on the association of their corresponding
terms with the UK Real inspiring posts or the UK LLM-Generated inspiring
posts. The most associated topics are listed under Top Generated and Top Real
headings.  Figure 10: Scattertext visualization of unigrams used in the real
and generated ( vs. ) inspiring posts from India. Points are colored red or
blue based on the association of their corresponding terms with India Real
inspiring posts or India LLM-Generated inspiring posts. The most associated
topics are listed under Top Generated and Top Real headings.  Figure 11:
Scattertext visualization of topics used in the real and generated ( vs. )
inspiring posts from India. Points are colored red or blue based on the
association of their corresponding terms with India Real inspiring posts or
India LLM-Generated inspiring posts. The most associated topics are listed
under Top Generated and Top Real headings.