# Polynomial identities that involve binomial coefficients, double and rising
factorials and their probabilistic interpretations and proofs.

Paweł J. Szabłowski  Emeritus in Department of Mathematics and Information
Sciences,
Warsaw University of Technology
ul Koszykowa 75, 00-662 Warsaw, Poland
<EMAIL_ADDRESS>
(Date: May 2022)

###### Abstract.

We formulate several polynomial identities. One side of these identities has a
nice simple form. Whereas the other has a form of a polynomial whose
coefficients contain binomial coefficients double factorials or (and) rising
factorials. The origins and the proofs of these identities are probabilistic.
However, their form suggests universal applications in simplifying
expressions. Many useful simplifying formulae are presented in the sequel.

###### Key words and phrases:

Binomial coefficient. Pochhammer symbol, bivariate normal distribution ,
bivariate gamma distribution, Hermite polynomials, Laguerre polynomials,
Fibonacci Numbers, Lucas numbers

###### 2020 Mathematics Subject Classification:

Primary 05A10, 05D40; Secondary 11B65, 11B73

##  1\. Introduction

In this note, we will present some identities involving factorials, binomial
coefficients, and the so-called rising factorials (sometimes called Pochhammer
symbols). Many of them have the forms of polynomials whose coefficients often
have the form of binomial coefficients and (or) of rising factorials of some
additional variables. These variables appear on both sides of the identity.
The domains of these variables can be extended to all complex numbers. All
these identities have a common origin, except for the sometimes similar form.
Namely, the origins of all are a few, rather deep probabilistic
interpretations and sometimes following non-trivial computations.

The paper is organized as follows. The next section is dedicated to the
presentation of the identities and some of the particular, interesting
particular cases. The next section is devoted to the presentation of the
probabilistic background of the results presented in the previous section and
then, finally, the presentation of the calculations leading to the identities.

##  2\. Identities

Let us set $(-1)!!\allowbreak=\allowbreak(0)!!\allowbreak=\allowbreak 1$ and
$\binom{n}{k}\allowbreak=\allowbreak 0$ when $n<k$. $i$ would always denote
imaginary unit , i.e.,
$i\allowbreak=\allowbreak\sqrt{-1}\allowbreak=\allowbreak\exp(i\pi/2)$. In
order to simplify notation let us introduce the so-called rising factorial
(sometimes called also Pochhammer symbol), which is the following function:

$(x)^{(n)}=x(x+1)\ldots(x+n-1),$

defined for all complex $x.$ Notice that we have for all $x\neq 0:$

$(x)^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)},$

where $\Gamma(x)$ denotes the Euler’s gamma function. To learn more about
Pochhammer symbols or binomial coefficients, see, e.g., [6]. Notice only that,
e.g., $\left(1\right)^{\left(n\right)}\allowbreak=\allowbreak n!$ or
$\left(1/2\right)^{\left(n\right)}\allowbreak=\allowbreak(2n-1)!!/2^{n}$. We
will use these values below.

######  Theorem 1.

For all natural $k$ and all complex $\rho$ and $\beta$ we have:

i)

(2.1)
$\frac{(2k)!}{k!}(1+\rho)^{k}=\sum_{j=0}^{k}\binom{2k}{2j}(1+\rho)^{2k-2j}(1-\rho^{2})^{j}(2j-1)!!(2k-2j-1)!!.$

ii)

(2.2) $(1+\rho)^{k}=\frac{1}{2^{k}}\sum_{j=0}^{2k}\sum_{m=0}^{\left\lfloor
j/2\right\rfloor}(2\rho)^{j-2m}\binom{k}{j-2m}\binom{k-j+2m}{m}.$

iii) For all integer $m$ we have

(2.3)
$\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\frac{\left(\beta\right)^{\left(j+m\right)}}{\left(\beta\right)^{\left(j\right)}}=(-1)^{n}n!\binom{m}{n}\frac{\left(\beta\right)^{\left(m\right)}}{\left(\beta\right)^{\left(n\right)}}.$

iv)

(2.4)
$\displaystyle\sum_{m=0}^{n}\left(-1\right)^{m}\binom{n}{m}\left(\beta\right)^{\left(m\right)}\left(\beta\right)^{\left(n-m\right)}\sum_{k=0}^{m}\binom{m}{k}\binom{n-m}{k}\frac{\rho^{k}k!}{\left(\beta\right)^{\left(k\right)}}$
(2.7) $\displaystyle=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{ is
odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}(1-\rho)^{n/2}&\text{if}&n\text{
is even}\end{array}\right..$

v)

(2.8)
$\displaystyle\sum_{m=0}^{n}(-1)^{n-m}\binom{n}{m}\sum_{j=0}^{m}\binom{m}{j}(1-\rho)^{j}\rho^{m-j}\left(\beta\right)^{\left(n-j\right)}\left(\beta+m-j\right)^{\left(j\right)}$
(2.11) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{ is odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}(1-\rho)^{n/2}&\text{if}&n\text{
is even}\end{array}\right..$

Below we present some, believed to be important, particular cases, remarks and
corollaries.

######  Remark 1.

Taking first $\rho\allowbreak=\allowbreak\frac{2}{3}$ then
$\rho\allowbreak=\allowbreak\frac{1}{3}$ and finally
$\rho\allowbreak=\allowbreak\frac{4}{3}$ in (2.1) we get:

$\displaystyle\sum_{j=0}^{k}\binom{2k}{2j}5^{j}(2j-1)!!(2k-2j-1)!!$
$\displaystyle=$ $\displaystyle\frac{(2k)!3^{k}}{k!},$
$\displaystyle\sum_{j=0}^{k}\binom{2k}{2j}2^{j}(2j-1)!!(2k-2j-1)!!$
$\displaystyle=$ $\displaystyle\frac{(2k)!3^{k}}{k!2^{k}},$
$\displaystyle\sum_{j=0}^{k}\binom{2k}{2j}(-7)^{j}(2j-1)!!(2k-2j-1)!!$
$\displaystyle=$ $\displaystyle(-1)^{k}\frac{(2k)!3^{k}}{k!}.$

######  Remark 2.

Let us take $\rho\allowbreak=\allowbreak\sqrt{5}$ in (2.1) and recall that
$(1+\sqrt{5})^{n}\allowbreak=\allowbreak
2^{n}(\frac{1+\sqrt{5}}{2})^{n}\allowbreak=\allowbreak
2^{n-1}(L_{n}+\sqrt{5}F_{n}),$ where $L_{n}$ and $F_{n}$ are respectively
$n-$th Lucas and Fibonacci numbers. Now, equating integer and irrational parts
we end up with the following two identities valid for $k\geq 0:$

$\displaystyle\frac{(2k)!}{k!}L_{k}$ $\displaystyle=$ $\displaystyle
2^{k}\sum_{j=0}^{k}\binom{2k}{2j}(-1)^{j}L_{2k-2j}(2j-1)!!(2k-2j-1)!!,$
$\displaystyle\frac{(2k)!}{k!}F_{k}$ $\displaystyle=$ $\displaystyle
2^{k}\sum_{j=0}^{k}\binom{2k}{2j}(-1)^{j}F_{2k-2j}(2j-1)!!(2k-2j-1)!!$

######  Remark 3.

Taking $\rho\allowbreak=\allowbreak i$ in (2.1), further noticing that
$(1+i)^{k}\allowbreak=\allowbreak 2^{k/2}\exp(ik\pi/4)$ and taking firstly
$k\allowbreak=\allowbreak 4n$ and cancelling out $(-4)^{n}$ on both sides, we
get for all $n\geq 0:$

$\frac{(8n)!}{(4n)!}=\sum_{j=0}^{4n}\binom{8n}{2j}(2j-1)!!(8n-2j-1)!!,$

and the taking $k\allowbreak=\allowbreak 4n+1$ and equating the real and
imaginary parts we get for all $n\geq 0:$

$\displaystyle\frac{(8n+2)!}{(4n+1)!}(-4)^{n}$ $\displaystyle=$ $\displaystyle
2^{4n+1}\sum_{m=0}^{2n}(-1)^{m}\binom{8n+2}{4m+2}(4m+1)!!(8n-4m-1)!!,$
$\displaystyle\frac{(8n+2)!}{(4n+1)!}(-4)^{n}$ $\displaystyle=$ $\displaystyle
2^{4n+1}\sum_{m=0}^{2n}(-1)^{m}\binom{8n+2}{4m}(4m-1)!!(8n-4m+1)!!.$

######  Remark 4.

Taking $\rho\allowbreak=\allowbreak 1/2$ in (2.2) we get the following
identity:

$1=\sum_{j=0}^{2k}(-1)^{j}\sum_{m=0}^{\left\lfloor
j/2\right\rfloor}\binom{k}{j-2m}\binom{k-j+2m}{m}.$

######  Remark 5.

Taking $\rho\allowbreak=\allowbreak i$ and $k\allowbreak=\allowbreak 4n$ in
(2.2), further noticing that $(1+i)^{4n}\allowbreak=\allowbreak(-4)^{n}$ and
$(2i)^{j-2m}\allowbreak=\allowbreak i^{j}(-1)^{m}$ and finally splitting the
right-hand side into real and imaginary parts we get for all $n\geq 0:$

$\displaystyle 4^{n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}}{16^{n}}\sum_{j=0}^{4n}(-1)^{j}\sum_{m=0}^{j}(-1)^{m}4^{j-m}\binom{4n}{2j-2m}\binom{4n-2j+2m}{m},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{4n}(-1)^{j}\sum_{m=0}^{j}(-1)^{m}4^{j-m}\binom{4n}{2j-2m+1}\binom{4n-2j-1+2m}{m}.$

######  Remark 6.

i) Taking $\beta\allowbreak=\allowbreak 1$ and integer $k$ in (2.3) we get

$\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\binom{j+k}{j}=(-1)^{n}\binom{k}{n}.$

ii) Taking $\beta\allowbreak=\allowbreak 1/2$ and integer $k$ in (2.3) we get:

$\sum_{j=0}^{n}(-1)^{j}\frac{(2n-1)!!(2j+2k-1)!!}{j!(n-j)!(2j-1)!!}=(-1)^{n}2^{n}\binom{k}{n}(2k-1)!!.$

######  Remark 7.

i) Taking $\beta\allowbreak=\allowbreak 1/2$ and and integer $k$ in (2.4) we
get:

$\sum_{j=0}^{n}(-1)^{n-j}\binom{n}{j}\frac{(2n+2k+2j-1)!!}{(2j-1)!!}=2^{n}\frac{(n+k)!(2n+2k-1)!!}{k!(2n-1)!!}.$

ii) Taking $\beta\allowbreak=\allowbreak 1$ in (2.4) we get for all integer
$n$ and $k.$

(2.12)
$\sum_{m=0}^{n}\left(-1\right)^{m}\sum_{k=0}^{m}\binom{m}{k}\binom{n-m}{k}\rho^{k}=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{
is odd}\\\ (1-\rho)^{n/2}&\text{if}&n\text{ is even}\end{array}\right..$

iii) Changing the order of summation in (2.12) and comparing the coefficients
in expansions in powers of $\rho$ we get:

$\sum_{m=k}^{n}(-1)^{m-k}\binom{m}{k}\binom{n-m}{k}=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{
is odd}\\\ \binom{n/2}{k}&\text{if}&n\text{ is even}\end{array}\right..$

iv) Taking additionally $\rho\allowbreak=\allowbreak 1$ in we get, for all,
$n\geq 0:$

$\sum_{m=0}^{n}\left(-1\right)^{m}\sum_{k=0}^{m}\binom{m}{k}\binom{n-m}{k}=0.$

v) Taking $\beta\allowbreak=\allowbreak 1/2$ in (2.4) we get after denoting
$2\rho\allowbreak=\allowbreak x$ and canceling out $n!:$

$\displaystyle\sum_{m=0}^{n}\left(-1\right)^{m}(2m-1)!!(2n-2m-1)!!\sum_{k=0}^{m}\frac{1}{k!(m-k)!(n-m-k)!}\frac{x^{k}}{(2k-1)!!}$
$\displaystyle=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{ is odd}\\\
\frac{(n-1)!!}{(n/2)!}(2-x)^{n/2}&\text{if}&n\text{ is
even}\end{array}\right..$

vi) By taking additionally in the above-mentioned identity
$x\allowbreak=\allowbreak 0$ we get:

$\sum_{m=0}^{n}\left(-1\right)^{m}\binom{n}{m}(2m-1)!!(2n-2m-1)!!=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{
is odd}\\\ \frac{n!(n-1)!!}{(n/2)!}2^{n/2}&\text{if}&n\text{ is
even}\end{array}\right..$

vii) Taking $\rho\allowbreak=\allowbreak 1$ in (2.4) we get for all complex
$\beta\neq 0$ and $n\geq 0$:

$\sum_{m=0}^{n}\left(-1\right)^{m}\binom{n}{m}\left(\beta\right)^{\left(m\right)}\left(\beta\right)^{\left(n-m\right)}\sum_{k=0}^{m}\binom{m}{k}\binom{n-m}{k}\frac{k!}{\left(\beta\right)^{\left(k\right)}}=0.$

######  Remark 8.

i) Taking $\rho\allowbreak=\allowbreak 0$ in (2.8) we get

$\sum_{m=0}^{n}(-1)^{n-m}\binom{n}{m}\left(\beta\right)^{\left(n-m\right)}\left(\beta\right)^{\left(m\right)}=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{
is odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}&\text{if}&n\text{ is
even}\end{array}\right.,$

which is related, but not a direct generalization of a well-known (also from
umbral calculus) equality:

$\sum_{m=0}^{n}\binom{n}{m}\left(\beta\right)^{\left(n-m\right)}\left(\alpha\right)^{\left(m\right)}=\left(\alpha+\beta\right)^{\left(n\right)}.$

ii) Taking $\rho\allowbreak=\allowbreak 1/2$ in (2.8), we get for all complex
$\beta\neq 0:$

$\displaystyle\sum_{m=0}^{n}(-1)^{n-m}\binom{n}{m}2^{-m}\sum_{j=0}^{m}\binom{m}{j}\left(\beta\right)^{\left(n-j\right)}\left(\beta+m-j\right)^{\left(j\right)}$
$\displaystyle=\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{ is odd}\\\
\frac{n!}{2^{n/2}(n/2)!}\left(\beta\right)^{\left(n/2\right)}&\text{if}&n\text{
is even}\end{array}\right..$

iii) Changing the order of summation in (2.8), comparing the coefficients in
expansions in powers of $\rho$, multiplying both sides by
$\left(\beta\right)^{\left(k\right)}(n-k)!$ and cancelling out $n!$ we get for
all $\beta$ complex and $0\leq k\leq n$:

$\displaystyle\sum_{m=k}^{n}(-1)^{m-k}\frac{(n-k)!}{(m-k)!(n-m-k)!}\left(\beta\right)^{\left(m\right)}\left(\beta\right)^{\left(n-m\right)}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ccc}0&\text{if}&n\text{
is odd }\\\
(n/2)!\binom{n-k}{n/2}\left(\beta\right)^{\left(k\right)}\left(\beta\right)^{\left(n/2\right)}&\text{if}&n\text{
is even}\end{array}\right..$

We have also the following relationships between Fibonacci
$\left\\{F_{n}\right\\}_{n\geq 0}$ and Lucas $\left\\{L_{n}\right\\}_{n\geq
0}$ numbers. For Fibonacci and Lucas numbers see, e.g.: [3], [7] or [4]. For
other relationships of Fibonacci and Lucas numbers with the so-called Catalan
triangles (expressed by binomial coefficients) see, e.g., [12].

######  Corollary 1.

Let us denote by $\phi$ the following number $\left(\sqrt{5}-1\right)/2$ that
is called ”golden ratio”. Take $\rho\allowbreak=\allowbreak-\phi$ in (2.1). We
get then

$\displaystyle 2\frac{\left(2k\right)!}{k!}L_{k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}(-1)^{j}\binom{2k}{2j}(2j-1)!!(2k-2j-1)!!(L_{2k-2j}L_{j}-5F_{2k-2j}F_{j}),$
$\displaystyle 2\frac{\left(2k\right)!}{k!}F_{k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}(-1)^{j}\binom{2k}{2j}(2j-1)!!(2k-2j-1)!!(F_{2k-2j}L_{j}-L_{2k-2j}F_{j}).$

###### Proof.

We know that for all $n\geq 0:$

$(1+\phi)^{n}=((1+\sqrt{5})/2)^{n}=L_{n}/2+F_{n}\sqrt{5}/2.$

Now notice also that we have $\phi\allowbreak=\allowbreak 1/(1+\phi)$ and that

$L_{n}^{2}-5F_{n}^{2}=(-1)^{n}4.$

Hence we have:

$\phi^{n}=(-1)^{n}(L_{n}/2-F_{n}\sqrt{5}/2).$

Now we take $\rho\allowbreak=\allowbreak-\phi$ and insert it in (2.1). Notice
that $1-\rho\allowbreak=\allowbreak(1+\phi)$, hence
$(1-\rho)^{2k-2j}\allowbreak=\allowbreak L_{2k-2j}/2\allowbreak+\allowbreak
F_{2k-2j}\sqrt{5}/2$ and $1-\rho^{2}\allowbreak=\allowbreak\phi$, consequently
$(1-\rho^{2})^{j}\allowbreak=\allowbreak(-1)^{j}(L_{j}/2\allowbreak-\allowbreak
F_{j}\sqrt{5}/2)$. Now it remains to perform multiplication and separation of
terms with and without $\sqrt{5}$.

##  3\. Probabilistic background and the proofs

###  3.1. Probabilistic background

All identities presented in Theorem 1 stem from calculating the following
moments $E(X-Y)^{n},$ $n\geq 0,$ where $X$ and $Y$ are two normalized random
variables. The joint distribution of these random variables has one parameter
and is either bivariate Normal (Gaussian) or bivariate Gamma distribution.
More precisely, in the first case, the joint distribution of $(X,Y)$ has the
following well-known density, valid for all $x,y\in\mathbb{R}$ and
$\left|\rho\right|<1:$

(3.1) $f_{N}(x,y|\rho)=\frac{1}{2\pi(1-\rho^{2})}\exp\left(-\frac{x^{2}-2\rho
xy+y^{2}}{2(1-\rho^{2})}\right),$

while in the second case valid for all $x,y,\beta>0$ and
$\left|\rho\right|<1:$

$f_{G}(x,y|\rho)=f_{g}(x|\beta)f_{g}(y|\beta)\frac{\exp\left(-\frac{\rho\left(x+y\right)}{1-\rho}\right)}{\left(1-\rho\right)\left(xy\rho\right)^{(\beta-1)/2}}I_{\beta-1}\left(\frac{2\sqrt{xy\rho}}{1-\rho}\right),$

where

$f_{g}(x|\beta)\allowbreak=\allowbreak x^{\beta-1}\exp(-x)/\Gamma(\beta),$

for $x>0$ and $0$ otherwise. $I_{\alpha}$ denotes the modified Bessel function
of the first kind. The most important property of these joint densities is
that they allow the so-called Lancaster expansions. To learn more about
Lancaster expansions, their probabilistic interpretations and convergence
problems associated with them, see, e.g., [1], [8], [9], [13], [12], [11]. The
orthogonal polynomials that we are using are well presented, for example, in
two well-known monographs [5], and [13].

Namely, in the first case, we have the so-called Poisson-Mehler expansion

(3.2) $f_{N}(x,y|\rho)=f_{N}(x)f_{N}(x)\sum_{n\geq
0}\rho^{n}h_{n}(x)h_{n}(y),$

where we denoted for simplicity

$f_{N}(x)\allowbreak=\allowbreak\exp(-x^{2}/2)/\sqrt{2\pi},$

and

$h_{n}(x)\allowbreak=\allowbreak H_{n}(x)/\sqrt{n!}.$

$h_{n}(x)$ is the orthonormal modification of the so-called probabilistic
Hermite polynomials $\left\\{H_{n}(x)\right\\}$, i.e. polynomials defined by
the following three-term recurrence :

$H_{n+1}(x)\allowbreak=\allowbreak xH_{n}(x)-nH_{n-1}(x),$

with $H_{-1}(x)\allowbreak=\allowbreak 0$ and $H_{0}(x)\allowbreak=\allowbreak
1.$ It is also known, that

(3.3)
$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}h_{n}(x)h_{m}(x)\exp(-x^{2}/2)dx=\delta_{nm},$

where $\delta_{nm}$ denotes Kronecker’s delta.

In the second case we have

(3.4) $f_{G}(x,y|\rho)=f_{g}(x|\beta)f_{g}(y|\beta)\sum_{j\geq
0}\rho^{n}l_{n}(x|\beta)l_{n}(y|\beta),$

where
$l_{n}(x|\beta)\allowbreak=\allowbreak\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}L_{n}(x|\beta)$
is the orthonormal modification of the so-called generalized Laguerre
polynomials $\left\\{L_{n}(x|\beta)\right\\}$, defined by the following
expansions:

(3.5)
$L_{n}(x|\beta)=\sum_{k=0}^{n}(-1)^{k}\frac{(\beta)^{(n)}}{(n-k)!(\beta)^{(k)}}x^{k}/k!.$

Let us remark that the expansion (3.4) is known under the name Hardy-Hille
(see, e.g. [13], p.102) formula. One knows that :

(3.6)
$\int_{0}^{\infty}l_{n}(x|\beta)l_{m}(x|\beta)f_{g}(x|\beta)dx=\delta_{n,m}.$

Our aim is to calculate $\left\\{E(X-Y)^{n}\right\\}_{n\geq 0}.$ We will do it
in two ways.

The first way is to calculate the generating function of the set of these
numbers i.e. the function

$g(t)=\sum_{n\geq 0}t^{n}E(X-Y)^{n}/n!.$

It can be calculated in the following way since it is known that all moments
exist, hence one can exchange integration and summation. Namely, we have

$g(t)=\int\int\exp\left(tx-ty\right)f(x,y|\rho)dxdy.$

Obviously here and below the integration is depending on the case either over
whole $\mathbb{R}^{2}$ or over $\mathbb{R}^{+}\times\mathbb{R}^{+}$. The
second way is to calculate the number using an expansion:

$E(-X+Y)^{n}=\sum_{m=0}^{n}(-1)^{m}EX^{m}Y^{n-m}.$

Now

$EX^{m}Y^{n-m}=\int\int x^{m}y^{n-m}f(x,y|\rho)dxdy.$

where $f$ is either $f_{N}$ or $f_{G}$ presented above. Using one of the
expansions (3.2) or (3.4) we will find these moments using the numbers

$H_{m,j}=EX^{m}k_{j}(X),$

where $k_{i}(x)$ is either $h_{j}(x)$ if we consider the Normal case or
$l_{j}(x)$ if we consider Gamma case. Since both expansions (3.2) and (3.4)
have similar structure we have:

(3.7) $EX^{m}Y^{n-m}=\sum_{j=0}^{\min(m,n-m)}\rho^{j}H_{m,j}H_{n-m,j}.$

This is so since $X$ and $Y$ have the same distributions and since
$H_{m,j}\allowbreak=\allowbreak 0$ if $j>m$ because of the orthogonality of
polynomials $h$ or $l.$ We will calculate this function assuming either
expansion (3.2) or (3.4).

######  Case 1.

So let us start with expansion (3.2). First let us calculate the auxiliary
numbers $H_{j,n}$. We have:

(3.8)
$H_{j,n}=EX^{j}h_{n}(X)=\frac{1}{\sqrt{n!}}EX^{j}H_{n}(X)=\left\\{\begin{array}[]{ccc}0&\text{if
}&n>j\text{ or }j-n\text{ odd}\\\
\frac{j!}{2^{(j-n)/2}((j-n)/2)!\sqrt{n!}}&\text{if}&j-n\text{ is
even}\end{array}\text{.}\right.$

This is so since polynomials $h_{n}$ are orthogonal and also since it is
common knowledge, that $\forall n\geq 0:$

$x^{j}\allowbreak=\allowbreak j!\sum_{m=0}^{\left\lfloor
j/2\right\rfloor}\frac{1}{2^{m}m!(j-2m)!}H_{j-2m}(x).$

Secondly, let us calculate the following auxiliary function that will simplify
many further calculations.

$\displaystyle m_{n}(t)\allowbreak$ $\displaystyle=$ $\displaystyle\allowbreak
Eh_{n}(X)\exp(tX)\allowbreak=\allowbreak\frac{1}{\sqrt{2\pi
n!}}\int_{-\infty}^{\infty}H_{n}(x)\exp(tx)\exp(-x^{2}/2)dx$ $\displaystyle=$
$\displaystyle\frac{\exp(t^{2}/2)}{\sqrt{2\pi
n!}}\int_{-\infty}^{\infty}H_{n}(x)\exp(-(x-t)^{2}/2)dx.$

Now, let us change the variable under the integral by setting
$y\allowbreak=\allowbreak(x-t).$ Then, we get:

$m_{n}(t)\allowbreak=\allowbreak\frac{\exp(t^{2}/2)}{\sqrt{2\pi
n!}}\int_{-\infty}^{\infty}H_{n}(y+t)\exp(-y^{2}/2)dy.$

Next, we utilize the following the well-known expansion

$H_{n}(x+y)\allowbreak=\allowbreak\sum_{j=0}^{n}\binom{n}{j}H_{j}(x)y^{n-j}.$

Now, since we have (3.3), we see that $m_{n}(t)\allowbreak=\allowbreak
t^{n}\frac{\exp(t^{2}/2)}{\sqrt{n!}}$ and we get:

$\displaystyle E\exp\left(tX-tY\right)\allowbreak=\allowbreak\sum_{n\geq
0}E\exp\left(tX-tY\right)$ $\displaystyle=\frac{1}{2\pi}\sum_{n\geq
0}\rho^{n}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\left(tx-
ty\right)\exp\left(-\left(x^{2}+y^{2}\right)/2\right)h_{n}(x)h_{n}(y)dxdy$
$\displaystyle\sum_{n\geq 0}\rho^{n}m_{n}(t)m_{n}(-t)=\exp(t^{2})\sum_{n\geq
0}(-\rho)^{n}t^{2n}/n!=\exp(t^{2}(1-\rho))$ $\displaystyle\sum_{n\geq
0}t^{2n}(1-\rho)^{n}/n!.$

Thus, we deduce that:

(3.9) $E(X-Y)^{n}=\left\\{\begin{array}[]{ccc}0&\text{if }&n\text{ is odd}\\\
\frac{n!}{(n/2)!}(1-\rho)^{n/2}&\text{if }&n\text{ is
even}\end{array}\right..$

######  Case 2.

Now let us consider expansion (3.4). We need to recall some simple, well-known
facts: The Gamma distribution with rate parameter zero and shape parameter
$\beta>0,$ is the distribution with the density $f_{g}$ presented above. It is
easy to see, recalling the definition of the Euler’s Gamma function that:

(3.10)
$EX^{n}=\int_{0}^{\infty}x^{n}f_{g}(x|\beta)dx=\left(\beta\right)^{\left(n\right)}.$

As before, let us calculate the set of auxiliary quantities and functions: We
start with the numbers:

$H_{j,n}=EX^{j}l_{n}(X).$

We have:

(3.11)
$H_{j,n}=\frac{\sqrt{n!}}{\Gamma(\beta)\sqrt{\left(\beta\right)^{\left(n\right)}}}\int_{0}^{\infty}x^{j}L_{n}^{(\beta)}(x)x^{\beta-1}\exp(-x)dx.$

Now, we use the well-known expansion of $x^{j}$ in powers of Laguerre
polynomials:

(3.12) $x^{j}\allowbreak=\allowbreak
j!\sum_{k=0}^{j}(-1)^{k}\frac{(\beta)^{(j)}}{(j-k)!(\beta)^{(k)}}L_{k}^{(\beta)}(x),$

and then we use the orthogonality of the Laguerre polynomials. Hence we have:

(3.13)
$H_{j,n}=(-1)^{n}\binom{j}{n}(\beta)^{(j)}\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}.$

Let us calculate now, the following auxiliary functions:

$\displaystyle m_{n}(t)$ $\displaystyle=$ $\displaystyle
El_{n}(X)\exp(tX)=\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}E\exp(tX)L_{n}(X|\beta)\allowbreak$
$\displaystyle=$
$\displaystyle\allowbreak\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{1}{\Gamma(\beta)}\int_{0}^{\infty}\exp(xt)L_{n}(x|\beta)x^{\beta-1}\exp(-x)dx\allowbreak$
$\displaystyle=$
$\displaystyle\allowbreak\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{1}{\Gamma(\beta)}\int_{0}^{\infty}L_{n}(x|\beta)x^{\beta-1}\exp(-x(1-t))dx\allowbreak.$

Now, let us change variables under the integral by considering
$y\allowbreak=\allowbreak x(1-t).$ Then we get:

$m_{n}(t)=\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{1}{1-t}\frac{1}{\Gamma(\beta)}\int_{0}^{\infty}L_{n}(\frac{y}{1-t}|\beta)\left(\frac{y}{1-t}\right)^{\beta-1}\exp(-y)dy\allowbreak.$

We will now utilize the following well-known recurrence relation for Laguerre
polynomials (see, e.g., [2]) :

$L_{n}(y|\beta)=\sum_{j=0}^{n}L_{n-j}(x|\beta+j)(y-x)^{j}/j!.$

So we get:

$\displaystyle
m_{n}(t)=\allowbreak\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{1}{(1-t)^{\beta}}\frac{1}{\Gamma(\beta)}\times$
$\displaystyle\int_{0}^{\infty}\left(\sum_{j=0}^{n}\left(\frac{-ty}{1-t}\right)^{j}L_{n-j}(x|(\beta+j))/j!\right)y^{\beta-1}\exp(-y)dy.$

Further, we have:

$\displaystyle
m_{n}(t)=\allowbreak\allowbreak\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{1}{(1-t)^{\beta}}\frac{1}{\Gamma(\beta)}\times$
$\displaystyle\sum_{j=0}^{n}\int_{0}^{\infty}(-1)^{j}\frac{t^{j}}{(1-t)^{j}}L_{n-j}(y\left|\left(\beta+j\right)\right|)y^{j+\beta-1}\exp(-y)dy\allowbreak$
$\displaystyle=\allowbreak(-1)^{n}\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{t^{n}\Gamma(\beta+n)}{\Gamma(\beta)n!(1-t)^{\beta+n}}\allowbreak=\allowbreak(-1)^{n}\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}\frac{t^{n}\left(\beta\right)^{\left(n\right)}}{n!(1-t)^{\beta+n}}.\allowbreak$

Hence we have now:

$\displaystyle\allowbreak g(t)=E\exp(tX-tY)=\sum_{n\geq
0}\rho^{n}\int_{0}^{\infty}\exp(tx-
ty)l_{n}(x|\beta)l_{n}(y|\beta)f_{g}(x|\beta)f_{g}(y|\beta)dxdy\allowbreak\allowbreak$
$\displaystyle=\allowbreak\sum_{n\geq
0}\rho^{n}m_{n}(t)m_{n}(-t)=\allowbreak\allowbreak\sum_{n\geq
0}\rho^{n}\frac{n!(-1)^{n}t^{2n}\left(\beta\right)^{\left(n\right)}\left(\beta\right)^{\left(n\right)}}{\left(\beta\right)^{\left(n\right)}n!n!(1-t^{2})^{n+\beta}}$
$\displaystyle=\frac{1}{(1-t^{2})^{\beta}}\sum_{n\geq
0}\rho^{n}\frac{(-1)^{n}t^{2n}\left(\beta\right)^{\left(n\right)}}{n!(1-t^{2})^{n}}$
$\displaystyle=\frac{1}{(1-t^{2})^{\beta}}\sum_{n\geq
0}(-1)^{n}\left(\frac{\rho
t^{2}}{1-t^{2}}\right)^{n}\left(\beta\right)^{\left(n\right)}/n!,$

and further we have:

$\displaystyle
g(t)\allowbreak=\allowbreak\frac{1}{(1-t^{2})^{\beta}}\left(1+\frac{\rho
t^{2}}{1-t^{2}}\right)^{-\beta}\allowbreak=$
$\displaystyle=\allowbreak\frac{1}{(1-t^{2}(1-\rho))^{\beta}}=\sum_{n\geq
0}t^{2n}(1-\rho)^{n}\left(\beta\right)^{\left(n\right)}/n!.$

We use here twice the well-known formula for the so-called binomial series.
Thus we deduce

(3.14)
$E(X-Y)^{n}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}0&\text{if
}&n\text{ is odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}(1-\rho)^{n/2}&\text{if
}&n\text{ is even}\end{array}\right..$

###  3.2. Proofs

All proofs are based on considering different ways of calculating numbers
$\left\\{E(X-Y)^{n}\right\\}_{n\geq 1}$ when assuming one of the discussed
above joint distributions.

Proof of assertion i) of Theorem 1:

Let $(X,Y)$ have bivariate Gaussian distribution given by (3.1). Then it is
well known that every linear transformation of such vriable has also gaussin
distribution. In particular we deduce that $X-Y\sim N(0,(1-\rho)^{2}).$ hence,
keeping in mind that
$EZ^{j}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}0&\text{if}&j\text{
is odd}\\\ 1&\text{if}&j=0\\\ \sigma^{j}(2j-1)!!&\text{if}&j\text{ is
even}\end{array}\right.$ if only $Z\sim N(0,\sigma^{2})$. Thus the left hand
side of assertion i) is equal to
$(2k-1)!!(1-\rho)^{k}\allowbreak=\allowbreak\frac{(2k)!}{k!}(1-\rho)^{k}.$ Now
it is enough to change $\rho$ to $-\rho$.

In order to get right hand side we calculate:

$\displaystyle E(X-Y)^{2k}\allowbreak=\allowbreak E(X+\rho
Y-(1+\rho)Y)^{2k}\allowbreak=$
$\displaystyle\allowbreak\sum_{j=0}^{2k}\binom{2k}{j}(-1)^{j}(1+\rho)^{2k-j}E(X+\rho
Y)^{j}Y^{2k-j}\allowbreak=$
$\displaystyle\allowbreak\sum_{j=0}^{2k}\binom{2k}{j}(-1)^{j}(1+\rho)^{2k-j}E(E(X+\rho
Y)^{j}|Y)Y^{2k-j}\allowbreak.$

Now, we use the fact know that random variables $(X+\rho Y)$ and $Y$ are
independent, hence $(E(X+\rho
Y)^{j}|Y)\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}0&\text{if}&j\text{
is odd}\\\ 1&\text{if}&j=0\\\ (2j-1)!!(1-\rho^{2})^{j/2}&\text{if}&j\text{ is
even}\end{array}\right..$ Thus we get

$E(X-Y)^{2k}=\sum_{j=0}^{k}\binom{2k}{2j}(1-\rho)^{2k-2j}(1-\rho^{2})^{j}(2j-1)!!(2k-2j-1)!!.$

Proof of assertion ii) of Theorem 1:

We use (3.7) with (3.8) getting:

$\frac{(2k)!}{k!}(1-\rho)^{k}=2\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}\sum_{n=0}^{j}\frac{\rho^{n}}{n!}H_{j,n}H_{2k-j,n}+(-1)^{k}\binom{2k}{k}\sum_{n=0}^{k}\frac{\rho^{n}}{n!}H_{k,n}^{2},$

where $H_{j,n}=\left\\{\begin{array}[]{ccc}0&\text{if }&n>j\text{ or
}j-n\text{ odd}\\\ \frac{j!}{2^{(j-n)/2}((j-n)/2)!}&\text{if}&j-n\text{ is
even}\end{array}\text{.}\right.$ Notice that if $j-n$ is even then
$\binom{2k}{j}H_{j,n}H_{2k-j,n}\allowbreak=\allowbreak
2^{n}\binom{2k}{k}\binom{k}{n}\binom{k-n}{(j-n)/2}/2^{k}$ and when $k-n$ is
even $H_{k,n}^{2}\allowbreak=\allowbreak
2^{n}\binom{k}{n}\binom{k-n}{(k-n)/2}/2^{k},$ hence (2.2) can be reformulated
in the following way:

$\frac{1}{2^{k-1}}\sum_{j=0}^{k-1}(-1)^{j}\sum_{n=0-}^{j}(2\rho)^{n}\binom{k}{n}mh_{j,n,k}+\frac{(-1)^{k}}{2^{k}}\sum_{n=0}^{k}(2\rho)^{n}\binom{k}{n}mh_{k,n,k},=(1-\rho)^{k},$

where
$mh_{j,n,k}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}0&\text{if}&j>k-1\text{
or }j-n\text{ is odd}\\\ \binom{k-n}{(j-n)/2}&\text{if }&j-n\text{ is
even}\end{array}.\right.$ Now it remains to change the order of summation in
the internal sums and $\rho$ to $-\rho$.

Proof of assertion iii) of Theorem 1:

Let us recall (3.5) and (3.11) and let us calculate $H_{j,n}$ directly,
getting:

$\displaystyle
H_{j,n}=\frac{\sqrt{n!}}{\Gamma(\beta)\sqrt{\Gamma(n+\beta)}}\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\frac{(\beta)^{(n)}}{(n-k)!(\beta)^{(k)}}\int_{0}^{\infty}x^{k+j}x^{\beta-1}\exp(-x)dx$
$\displaystyle=\frac{\sqrt{n!}}{\sqrt{\Gamma(n+\beta)}}\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\frac{(\beta)^{(n)}(\beta)^{(k+j)}}{(n-k)!(\beta)^{(k)}}.$

Now, we compare it with the calculated already $H_{j,n}$ for Gamma
distribution i.e.,
$(-1)^{n}\binom{j}{n}(\beta)^{(j)}\sqrt{\frac{n!}{\left(\beta\right)^{\left(n\right)}}}.$

Proof of assertion iv) of Theorem 1:

Recall (3.13). We have:

$\displaystyle\left\\{\begin{array}[]{ccc}0&\text{if }&n\text{ is odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}(1-\rho)^{n/2}&\text{if
}&n\text{ is even}\end{array}\right.=E(-X+Y)^{n}\allowbreak$
$\displaystyle=\sum_{m=0}^{n}(-1)^{m}\binom{n}{m}EX^{m}Y^{n-m}.\allowbreak$

Now we use (3.7) with $H_{j,m}$ given by (3.13). By (3.14) we have

$\displaystyle
E(-X+Y)^{n}=\sum_{m=0}^{n}(-1)^{m}\binom{n}{m}\sum_{j=0}^{\min(m,n-m)}\rho^{j}H_{m,j}H_{n-m,j}$
$\displaystyle=\sum_{m=0}^{n}(-1)^{n-m}\binom{n}{m}\sum_{j=0}^{\min(m,n-m)}\rho^{j}\binom{m}{j}\binom{n-m}{j}\left(\beta\right)^{\left(m\right)}\left(\beta\right)^{\left(n-m\right)}\frac{j!}{\left(\beta\right)^{\left(j\right)}}$
$\displaystyle=\sum_{m=0}^{n}(-1)^{n-m}\binom{n}{m}\left(\beta\right)^{\left(m\right)}\left(\beta\right)^{\left(n-m\right)}\sum_{j=0}^{m}\binom{m}{j}\binom{n-m}{j}\frac{j!\rho^{j}}{\left(\beta\right)^{\left(j\right)}}$
$\displaystyle=\left\\{\begin{array}[]{ccc}0&\text{if }&n\text{ is odd}\\\
\frac{n!}{(n/2)!}\left(\beta\right)^{\left(n/2\right)}(1-\rho)^{n/2}&\text{if
}&n\text{ is even}\end{array}\right..$

This is so, since $\binom{m}{j}\binom{n-m}{j}$ is zero whenever $m>j$ or
$n-m>j.$

Proof of assertion iv) of Theorem 1:

One can easily notice, basing on (3.4), that
$E(l_{n}(X)|Y)\allowbreak=\allowbreak\rho^{n}l_{n}(Y).$ Let us hence calculate
first, the conditional moments $\eta_{n}(y|\beta,\rho)\allowbreak=\allowbreak
E(X^{n}|Y=y)$. We have by (3.12)

$\displaystyle\eta_{j}(y|\beta,\rho)$ $\displaystyle=$ $\displaystyle
E(X^{j}|Y=y)\allowbreak=\allowbreak
j!\sum_{k=0}^{j}(-1)^{k}\frac{(\beta)^{(j)}}{(j-k)!(\beta)^{(k)}}\rho^{k}L_{k}^{(\beta)}(y)$
$\displaystyle=$ $\displaystyle
j!\sum_{k=0}^{j}(-1)^{k}\frac{(\beta)^{(j)}}{(j-k)!(\beta)^{(k)}}\rho^{k}\sum_{m=0}^{k}(-1)^{m}\frac{(\beta)^{(k)}}{(k-m)!(\beta)^{(m)}}y^{m}/m!$
$\displaystyle=$ $\displaystyle
j!\sum_{m=0}^{j}\frac{(\beta)^{(j)}}{(\beta)^{(m)}(j-m)!m!}(\rho
y)^{m}\sum_{k=m}^{j}(-1)^{k-m}\frac{(j-m)!}{(k-m)!(j-k)!}\rho^{k-m}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{j}\binom{j}{m}\frac{(\beta)^{(j)}}{(\beta)^{(m)}}(\rho
y)^{m}(1-\rho)^{j-m}=j!(1-\rho)^{j}L_{j}(-\frac{\rho y}{1-\rho}).$

Further we get:

$\displaystyle
E\left((X-Y)^{n}|Y=y\right)=\sum_{j=0}^{n}(-1)^{n-j}\binom{n}{j}\eta_{j}(y|\beta,\rho)y^{n-j}$
$\displaystyle=\sum_{j=0}^{n}(-1)^{n-j}y^{n-j}\binom{n}{j}\sum_{m=0}^{j}\binom{j}{m}\frac{(\beta)^{(j)}}{(\beta)^{(m)}}(\rho
y)^{m}(1-\rho)^{j-m}$
$\displaystyle=\sum_{m=0}^{n}\frac{n!}{m!(n-m)!}\frac{(\rho
y)^{m}}{(\beta)^{(m)}}\sum_{j=m}^{n}(-1)^{n-j}\frac{(n-m)!(\beta)^{(j)}y^{n-j}(1-\rho)^{j-m}}{(n-j)!(j-m)!}$
$\displaystyle\sum_{m=0}^{n}\frac{n!}{m!(n-m)!}(\rho
y)^{m}\sum_{s=0}^{n-m}(-1)^{n-m-s}\binom{n-m}{s}y^{n-m-s}(1-\rho)^{s}(\beta+m)^{(s)}$
$\displaystyle=\sum_{t=0}^{n}(-1)^{n-t}y^{n-t}\frac{n!}{t!(n-t)!}\sum_{s=0}^{t}\binom{t}{s}(\rho
y)^{t-s}(1-\rho)^{s}(\beta+t-s)^{(s)}$
$\displaystyle=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho)^{s}\sum_{t=s}^{n}(-1)^{n-t}\frac{(n-s)!}{(t-s)!t!}\rho^{t-s}(\beta+t-s)^{(s)}$
$\displaystyle=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho)^{s}\sum_{m=0}^{n-s}(-1)^{n-m-s}\binom{n-s}{m}\rho^{m}(\beta+m)^{(s)}.$

Hence we get using (3.10) we have;

$\displaystyle
E(X-Y)^{n}=\int_{0}^{\infty}(\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho)^{s}\times$
$\displaystyle\sum_{m=0}^{n-s}(-1)^{n-m-s}\binom{n-s}{m}\rho^{m}(\beta+m)^{(s)})f_{g}(x|\beta)dx$
$\displaystyle=\sum_{s=0}^{n}\binom{n}{s}(\beta)^{(n-s)}(1-\rho)^{s}\sum_{m=0}^{n-s}(-1)^{n-m-s}\binom{n-s}{m}\rho^{m}(\beta+m)^{(s)}$
$\displaystyle=\sum_{t=0}^{n}(-1)^{n-t}\frac{n!(\beta)^{(n-t)}}{t!(n-t)!}\sum_{s=0}^{t}\frac{t!}{s!(t-s)!}(1-\rho)^{s}\rho^{t-s}(\beta+n-t)^{(t-s)}(\beta+t-s)^{(s)}.$

In the last line, we have changed the order of summation. To get the assertion
we use the formula:

$\left(\beta\right)^{\left(n\right)}\left(\beta+n\right)^{\left(m\right)}=\left(\beta\right)^{\left(n+m\right)},$

true for all complex $\beta$.

## References

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# Understanding and Detecting Hateful Content using Contrastive Learning

Felipe González-Pizarro${}^{\vardiamondsuit}$ , and Savvas
Zannettou${}^{\spadesuit\blacktriangle}$
${}^{\vardiamondsuit}$University of British Columbia, ♠TU Delft,
${}^{\blacktriangle}$Max Planck Institute for Informatics
<EMAIL_ADDRESS><EMAIL_ADDRESS>Work done during an internship at
Max Planck Institute for Informatics.

###### Abstract

The spread of hate speech and hateful imagery on the Web is a significant
problem that needs to be mitigated to improve our Web experience. This work
contributes to research efforts to detect and understand hateful content on
the Web by undertaking a multimodal analysis of Antisemitism and Islamophobia
on 4chan’s /pol/ using OpenAI’s CLIP. This large pre-trained model uses the
Contrastive Learning paradigm. We devise a methodology to identify a set of
Antisemitic and Islamophobic hateful textual phrases using Google’s
Perspective API and manual annotations. Then, we use OpenAI’s CLIP to identify
images that are highly similar to our Antisemitic/Islamophobic textual
phrases. By running our methodology on a dataset that includes 66M posts and
5.8M images shared on 4chan’s /pol/ for 18 months, we detect 573,513 posts
containing 92K Antisemitic/Islamophobic images and 246K posts that include 420
hateful phrases. Among other things, we find that we can use OpenAI’s CLIP
model to detect hateful content with an accuracy score of 0.84 (F1 score =
0.58). Also, we find that Antisemitic/Islamophobic imagery is shared in 2x
more posts on 4chan’s /pol/ compared to Antisemitic/Islamophobic textual
phrases, highlighting the need to design more tools for detecting hateful
imagery. Finally, we make publicly available a dataset of 420
Antisemitic/Islamophobic phrases and 92K images that can assist researchers in
further understanding Antisemitism/Islamophobia and developing more accurate
hate speech detection models.

##  1 Introduction

The spread of hateful content on the Web is an everlasting and vital issue
that adversely affects society. The problem of hateful content is longstanding
on the Web for various reasons. First, there is no scientific consensus on
what constitutes hateful content (i.e., no definition of what hate speech is)
[56]. Second, the problem is complex since hateful content can spread across
various modalities (e.g., text, images, videos, etc.), and we still lack
automated techniques to detect hateful content with acceptable and
generalizable performance [4]. Third, we lack moderation tools to proactively
prevent the spread of hateful content on the Web [35]. This work focuses on
assisting the community in addressing the issue of the lack of tools to detect
hateful content across multiple modalities.

Most of the research efforts in the space of detecting hateful content focus
on designing and training machine learning models that are specifically
tailored towards detecting specific instances of hateful content (e.g., hate
speech on text or particular cases of hateful imagery). Some examples of such
efforts include Google’s Perspective API [46] and the HateSonar classifier
[17] that aim to detect toxic and offensive text. Other methods aim to detect
instances of hateful imagery like Antisemitic images [71] or hateful memes
[33, 69]. These efforts and tools are essential and valuable, however, they
rely on human-annotated datasets that are expensive to create, and therefore
they are also small. At the same time, these datasets focus on specific
modalities (i.e., text or images in isolation). All these drawbacks limit
their broad applicability.

The lack of large-scale annotated datasets for solving problems like hate
speech motivated the research community to start developing techniques that do
not rely on annotated datasets (a paradigm known as _unsupervised learning_).
Over the past years, the research community released large-scale models that
depend on huge unlabeled datasets such as OpenAI’s GPT-3 [6], Google’s BERT
[18], OpenAI’s CLIP [48], etc. These models are trained on large-scale
datasets and usually can capture general knowledge extracted from the datasets
that can be valuable for performing classification tasks that the model was
not explicitly trained on.

Motivated by these recent advancements on large-scale pre-trained machine
learning models, in this work, we investigate how we can use such models to
detect hateful content on the Web across multiple modalities (i.e., text and
images). Specifically, we focus on OpenAI’s CLIP model because it helps us
capture content similarity across modalities (i.e., measure similarity between
text and images). To achieve this, CLIP leverages a paradigm known as
Contrastive Learning; the main idea is that the model maps text and images to
a high-dimensional vector space and is trained in such a way that similar
text/images are mapped closer to this vector space (for more details see
Section 2).

Focus & Research Questions. This work focuses on understanding the spread of
Antisemitic/Islamophobic content on 4chan’s /pol/ board. We concentrate on
hateful content targeted towards these two demographics mainly because
previous work indicates that 4chan’s /pol/ is known for disseminating
Antisemitic/Islamophobic content [71, 47]. Specifically, we focus on shedding
light on the following research questions:

  * •

RQ1: Can large pre-trained models that leverage the Contrastive Learning
paradigm, like OpenAI’s CLIP, identify hateful content with acceptable
performance?

  * •

RQ2: How prevalent is Antisemitic/Islamophobic imagery and textual hate speech
on 4chan’s /pol/?

To answer these research questions, we obtain all the posts and images shared
on 4chan’s /pol/ between July 1, 2016, and December 31, 2017, ultimately
collecting 66M textual posts and 5.8M images. Then, we leverage the
Perspective API and manual annotations to construct a dataset of 420
Antisemitic and Islamophobic textual phrases. We retrieve 246K posts that
include any of our 420 hateful phrases. Finally, we use OpenAI’s CLIP to
detect Antisemitic/Islamophobic images when provided as input the above-
mentioned hateful phrases and all images shared on 4chan’s /pol/; we find 92K
images.

Contributions & Main Findings. Our work makes the following contributions/main
findings:

  * •

We investigate whether large pre-trained models based on Contrastive Learning
can assist in detecting hateful imagery. We find that large pre-trained models
like OpenAI’s CLIP [48] can detect Antisemitic/Islamophobic imagery with 0.84,
0.47, 0.80, 0.58, accuracy, precision, recall, and F1 score, respectively
(RQ1).

  * •

We find that on 4chan’s /pol/ Antisemitic/Islamophobic imagery appears in 2x
more posts compared to Antisemitic/Islamophobic textual hateful content. This
finding highlights the need for the development and use of multimodal hate
speech detectors for understanding and mitigating the problem (RQ2).

  * •

We make publicly available (upon acceptance of the manuscript) a large dataset
of Antisemitic/Islamophobic posts, phrases, and images shared on 4chan’s
/pol/. The released dataset can assist researchers in future work focusing on
detecting and understanding the spread of hateful content on the Web across
multiple modalities (i.e., text and images).

Ethical Considerations. We emphasize that we rely entirely on publicly
available and anonymous data shared on 4chan’s /pol/. The authors of this work
did all manual annotations performed in this study, hence there are no other
individuals exposed to hateful content. Also, we follow standard ethical
guidelines [51] like reporting our results on aggregate and not attempting to
deanonymize users.

Disclaimer. This manuscript contains Antisemitic and Islamophobic textual and
graphic elements that are offensive and are likely to disturb the reader.

##  2 Background

This section provides background information on Contrastive Learning and
OpenAI’s CLIP model, as well as details on Google’s Perspective API.

Contrastive Learning. To understand Contrastive Learning, it is essential to
grasp its differences compared to traditional Machine/Deep Learning (ML/DL)
classifiers. Traditional ML/DL classifiers take as an input a set of unlabeled
data, each accompanied with a class, and aim to predict the class from the
unlabeled data, a paradigm known as supervised learning [7]. On the other
hand, Contrastive Learning is an unsupervised technique, meaning that there is
no need to have labeled data (i.e., classes). The main idea behind Contrastive
Learning is that you train a model that relies entirely on unlabeled data, and
the model learns general features from the dataset by teaching it which input
samples are similar/different to each other [28]. In other words, Contrastive
Learning relies on a set of unlabeled data samples with additional information
on which of these samples are similar to each other. Contrastive Learning is
becoming increasingly popular in the research community with several
applications on visual representations [12, 34], textual representations [23,
65, 22], graph representations [66, 30], and multimodal (i.e., text/images,
images/videos, etc.) representations [48, 19, 67, 72].

Figure 1: Example of text queries and images that are similar to the text
queries on 4chan (i.e., cosine similarity between the text CLIP-representation
and the image CLIP-representation equals to 0.3 or more). On the left side, we
show a benign text query (“cute cat sleeping”), while on the right we show the
results for a toxic and antisemitic query (“gas the jews”).

OpenAI’s CLIP. OpenAI recently released a model called Contrastive Language-
Image Pre-training (CLIP) [48] that leverages Contrastive Learning to generate
representations across text and images. The model relies on a text encoder and
an image encoder that maps text and images to a high-dimensional vector space.
Subsequently, the model is trained to minimize the cosine distance between
similar text/image pairs. To train CLIP, OpenAI created a huge dataset that
consists of 400M pairs of text/images collected from various Web sources and
covers an extensive set of visual concepts.111The exact methodology for
creating this dataset was not made publicly available by OpenAI. By training
CLIP with this vast dataset, the model learns general visual representations
and how these representations are described using natural language, which
results in the model obtaining general knowledge in various topics (e.g.,
identify persons, objects, etc.).

In this work, we leverage the CLIP model to extract representations for our
4chan textual/image datasets and assess the similarity between the text and
image representations. The main idea is that by providing to the pre-trained
CLIP model a set of hateful text phrases, we will be able to identify a set of
hateful images that are highly similar to the hateful text query. To
demonstrate CLIP’s potential in discovering hateful imagery from hateful text-
based queries, we show an example from our 4chan dataset in Fig. 1. On the
left side, we show examples of images that are highly similar to a benign text
query like “cute cat sleeping,” while on the right, we show examples of images
that are similar to an antisemitic and toxic phrase (“gas the jews”)222In this
work, we treat an image as similar to the text phrase if it has a cosine
similarity of 0.3 or higher (see Section 4.2).. For each image, we report the
cosine similarity between the representation obtained from the text query and
the representation of the image in our dataset. This example shows that the
CLIP model can detect objects in images (i.e., cats) and provide relevant
images to the queries (i.e., the cats are indeed sleeping according to the
query). Furthermore, by looking at the images for the toxic query, we observe
that CLIP can identify harmful images based on the query and can link
historical persons to it (e.g., the textual input does not mention Adolf
Hitler; however, the model knows that Hitler was responsible for the
holocaust). Also, CLIP can detect images that share hateful ideology by adding
text on memes (i.e., CLIP also performs Optical Character Recognition and can
correlate that text with the text-based query). Overall, this example shows
the predictive power of the CLIP model in detecting hateful imagery from
hateful text phrases.

Google’s Perspective API. As a first step towards identifying hateful phrases,
we use Google’s Perspective API [46], which provides a set of Machine Learning
models for identifying how rude/aggressive/hateful a comment is. We use the
Perspective API for identifying hateful text mainly because it outperforms
other publicly available hate speech classifiers like HateSonar [17, 70]. This
work focuses on the SEVERE_TOXICITY model available from Perspective API
because it is more robust to positive uses of curse words, and it is a
production-ready model. The SEVERE_TOXICITY model returns a score between 0
and 1, which can be interpreted as the probability of the text being rude and
toxic.

##  3 Dataset

Our work focuses on 4chan, particularly the Politically Incorrect board
(/pol/). /pol/ is the main board for discussing world events and politics and
is infamous for the spread of conspiracy theories [68, 58] and racist/hateful
content [31, 71]. We collect the data about posts on /pol/ using the publicly
available dataset released by Papasavva et al. [45]; the dataset includes
textual data about 134.5M posts shared on /pol/ between June 2016 and November
2019. Our work focuses on the period between July 1, 2016, and December 31,
2017 (to match the time period of the image dataset mentioned below),
including 66,383,955 posts We complement the above dataset with the image
dataset collected by Zannettou et al. [71]. The dataset includes 5,859,439
images shared alongside /pol/ posts between July 1, 2016, and December 31,
2017.

Overall, our dataset comprises all textual and image activity on /pol/ between
July 1, 2016, and December 31, 2017, including 66M posts and 5.8M images.

##  4 Methodology

This section describes our methodology for detecting hateful text phrases and
hateful imagery, focusing on Antisemitic and Islamophobic content.

###  4.1 Identifying Antisemitic and Islamophobic phrases

Here, our goal is to identify a set of phrases that are
Antisemitic/Islamophobic. To do this, we follow a multi-step semi-automated
methodology. First, we use the SEVERE_TOXICITY scores from the Perspective API
to identify posts that are toxic/offensive without considering the target
(e.g., if it is antisemitic). Specifically, we consider all posts that have a
score of 0.8 or more as toxic, following the methodology by Ribeiro et al.
[50]. Out of the 66M posts in our dataset, we find 4.5M (6.7%) toxic posts.

Having extracted a set of toxic posts from 4chan’s /pol/, we then aim to
identify the main targets of hate speech on /pol/ by extracting the top
keywords. To do this, we preprocess the data to remove HTML tags, stop words,
and URLs, and then we create a term frequency-inverse document frequency array
(TF-IDF). Next, we manually inspect the top 200 words based on their TF-IDF
values and identify the words related to Jews or Muslims. As a result, we find
seven keywords: “jews,” “kike,” “jew,” “kikes,” “jewish,” “muslims,” and
“muslim.” Then, based on these keywords, we filter the toxic posts obtained
from the previous step, hence getting a set of 336K posts with a
SEVERE_TOXICITY score of 0.8 and include at least one of the seven keywords.

Since our goal is to create a set of Antisemitic/Islamophobic phrases, we need
to break down the toxic 4chan posts into sentences and then identify the ones
that are Antisemitic/Islamophobic. To do this, we apply a sentence tokenizer
[41] on the 336K posts, obtaining 976K sentences. To identify common phrases
used on 4chan’s /pol/, we apply WordNet lemmatization [42], excluding all
sentences that appear less than five times. We obtain 4,582 unique common
phrases; not all of these sentences are Antisemitic/Islamophobic.

Identifying whether a phrase is Antisemitic/Islamophobic is not a
straightforward task and can not be easily automated. Therefore, we use manual
annotation on the 4,582 common phrases to annotate the common phrases as
Antisemitic/Islamophobic or irrelevant. Two authors of this paper
independently annotated the 4.5K common phrases. We discard long phrases (over
seven words) during the annotation since our preliminary experiments showed
that OpenAI’s CLIP returns a considerable amount of false positives when
provided with long text queries. We also consider as irrelevant phrases that
target multiple demographic groups (e.g., hateful towards Muslims and Jews
like “fuck jews and muslims” or hateful towards African Americans and Jews
like “fuck niggers and jews”). The two annotators agreed on 91% of the
annotations with a Cohen’s Kappa score of 0.69, which indicates a substantial
agreement [36]. After the independent annotations, the two annotators
discussed the disagreements to come up with a final annotation on whether a
phrase is Antisemitic/Islamophobic or irrelevant. After our annotation, we
find 326 Antisemitic and 94 Islamophobic phrases. The list of the
Antisemitic/Islamophobic phrases is publicly available [3].

Finally, we search for these Antisemitic/Islamophobic phrases on the entire
dataset. We extract all posts that include any of the Antisemitic/Islamophobic
phrases (_Textual_ dataset), finding 247K posts. Note that we remove 864 posts
that contain both Antisemitic and Islamophobic phrases. Overall, we find 209K
Antisemitic posts and 37K Islamophobic posts (see Table 1).

Dataset | Textual | Visual
---|---|---
| # Phrases | # Posts | # Images | # Posts
Antisemitism | 326 | 209,224 | 69,610 | 472,048
Islamophobia | 94 | 37,354 | 22,519 | 101,465
Total | 420 | 246,578 | 92,129 | 573,513
Table 1: Overview of our Antisemitism/Islamophobia Textual and Visual
datasets. The number of phrases is based on lemmatized versions, and the
number of images is based on the unique pHash values.

###  4.2 Identifying Antisemitic and Islamophobic images

Our goal is to identify Antisemitic and Islamophobic imagery using the pre-
trained CLIP model [48]. To do this, we encode all images in our dataset using
the image encoder on the CLIP model, hence obtaining a high-dimensional vector
for each image. Also, we encode all the Antisemitic/Islamophobic phrases
(extracted from the previous step), using the text encoder on the CLIP model,
obtaining a vector for each phrase. Then, we calculate all the cosine
similarities between the image and text vectors, which allows us to assess the
similarity between the phrases and the images. The main idea is that by
comparing a hateful phrase to all the images, images with a high cosine
similarity score will also be hateful. To identify a suitable cosine
similarity threshold where we treat a text and an image similarly, we perform
a manual annotation process.

Figure 2: Performance of the CLIP model in identifying
Antisemitic/Islamophobic imagery for varying cosine similarity thresholds. The
lines refer to the average metric for ten random phrases (2K images), while
the area refers to the standard deviation across the ten phrases.

First, we extract a random sample of ten Antisemitic/Islamophobic phrases
(eight Antisemitic and two Islamophobic to match the percentage of
Antisemitic/Islamophobic phrases in our dataset). Then, we extract a random
sample of 200 images for each phrase while ensuring that the images cover the
whole spectrum of cosine similarity scores. Specifically, we extract 50 random
images with cosine similarity scores for each of the following ranges: [0.0,
0.20), [0.2, 0.25), [0.25, 0.3), [0.3, 0.4]. To select these ranges, we plot
the CDF of all cosine similarity scores obtained by comparing the ten randomly
selected phrases and all the images in our dataset (we omit the figure due to
space constraints). We find that 40% of the scores are below 0.2, and we
expect these images to be entirely irrelevant for the phrase. To verify this,
we select the [0.0, 0.20) range. Additionally, we select the [0.2, 0.25]
because it has a considerable percentage of the scores (50%), and we expect
that the images will not be very similar again. Finally, we select the [0.25,
0.3) and [0.3-0.4] ranges because we expect that the ideal threshold is
somewhere in these two ranges, and devoting half of the selected images in
these ranges will help us identify a suitable threshold.

Then, two authors of this paper independently annotated the 2K images to
identify which are Antisemitic/Islamophobic or irrelevant. The annotators
agreed on 94% of the annotations with a Cohen’s Kappa score of 0.75, which
indicates a substantial agreement. Again, the two annotators solved the
disagreements by discussing the images and deciding a final annotation on
whether the image is Antisemitic/Islamophobic or irrelevant.

Figure 3: CDF of the number of Antisemitic/Islamophobic posts containing each
phrase.

Having constructed a ground truth dataset of Antisemitic and Islamophobic
imagery, we then find the best performing cosine similarity threshold. We vary
the cosine similarity threshold, and we treat each image as
Antisemitic/Islamophobic (depending on the phrase used for the comparison) if
the cosine similarity between the phrase and the image is above the threshold.
Then, we calculate the accuracy, precision, recall, and F1 score, for each of
the ten phrases. We report the average performance across all phrases and the
standard deviation (as the area) in Fig. 2. We observe that the model performs
best with a cosine similarity threshold of 0.3 as we achieve a 0.84, 0.47,
0.80, 0.58 for accuracy, precision, recall, and f1 score, respectively.
Indeed, the 0.3 threshold is also used by previous work by Schuhmann et al.
[55] that inspected CLIP’s cosine similarities between text and images and
determined that 0.3 is a suitable threshold.

Finally, to construct our Antisemitic/Islamophobic image dataset (_Visual_
dataset), we extract all images that have a cosine similarity of 0.3 or higher
with any of the Antisemitic/Islamophobic text phrases. We label each image as
Antisemitic or Islamophobic depending on whether the textual phrase is
Antisemitic or Islamophobic. To identify unique images, we use the Perceptual
Hashing (pHash) algorithm [39] that calculates a fingerprint for each image in
such a way that any two images that look similar to the human eye map have
minor differences in their hashes. Similar to the Textual dataset, we remove
all images labeled as both Antisemitic and Islamophobic (3,325 images), mainly
because our manual inspections indicate that most of them are noise. Overall,
we find 69,610 Antisemitic and 22,519 Islamophobic images that are shared in
472,048 and 101,465 posts, respectively (see Table 1).

Antisemitic phrases | Islamophobic phrases
---|---
Phrase | # Posts | Phrase | # Posts
a kike | 51,216 | fuck muslim | 7,993
fuck kike | 23,604 | kill muslim | 4,639
fuck jew | 20,241 | fuck islam | 3,464
gas the kike | 13,353 | kill all muslim | 1,672
fuck off kike | 11,108 | muslim be terrorist | 1,445
kike shill | 10,134 | i hate muslim | 1,379
gas the kike race war now | 6,105 | muslim shithole | 1,208
kill jew | 5,815 | muslim shit | 1,085
you fuck kike | 5,007 | all muslim be terrorist | 1,039
filthy kike | 4,111 | muslim be bad | 1032
jew fuck | 3,557 | ban all muslim | 958
kike faggot | 3,540 | fuck mudslimes | 951
kike on a stick | 3,537 | muslim cunt | 907
gas the jew | 3,081 | i hate islam | 881
faggot kike | 2,905 | fuck sandniggers | 876

Table 2: Top 15 phrases (lemmatized versions), in terms of the number of
posts, in our Antisemitic and Islamophobic Textual dataset. For each phrase,
we report the number of posts that contain it. Figure 4: CDF of the number of
Antisemitic/Islamophobic posts containing each image.

##  5 Results

This section presents our results from analyzing the Antisemitic/Islamophobic
Textual and Visual datasets.

Figure 5: Top 15 images, in terms of the number of posts, in our Antisemitic
Visual datasets. We report the number of posts that the image appears and the
most related Antisemitic phrase that matches each image.

###  5.1 Popular phrases in Textual dataset

We start our analysis by looking into the most popular phrases in our
Antisemitic/Islamophobic textual datasets. Fig. 3 shows the Cumulative
Distribution Function (CDF) of the number of posts per each
Antisemitic/Islamophobic phrase. We observe that these hateful phrases tend to
appear in a considerable amount of posts. For instance, 90.4% and 92.47% of
the Antisemitic and Islamophobic phrases appear in at least ten posts.
Furthermore, we identify that the percentage of Antisemitic phrases (41.8%)
that appear in at least 100 posts is slightly lower than the percentage of
Islamophobic phrases (49.46%). At the same time, we observe that a small
percentage of phrases (11.54%) is shared in more than 1000 posts on the
Antisemitic/Islamophobic textual datasets combined.

We also report the top 15 phrases, in terms of the number of posts, in our
Antisemitic and Islamophobic Textual dataset (see Table 2). In the first
dataset, we observe that 12 out of the 15 most frequent phrases contain the
term “kike”, a derogatory term to denote Jews. We also identify three phrases
related to the extermination procedure in the gas chambers during the
holocaust. Indeed, 16,433 (7.85%) of the Antisemitic posts contain at least
one of these phrases: “gas the kike”, “gas the jew”, or “gas the kike race war
now”. Phrases accusing jews of being accomplices (“kike shill”) or alluding to
a supposed good social-economic status (“filthy kike”) are also trendy,
appearing in 10,134 and 4,111 posts, respectively.

We also show the top 15 most popular Islamophobic phrases in Table 2. Here, we
observe many posts with phrases calling terrorists to Muslims. For instance,
“Muslims be terrorist” and “All Muslim be terrorist” appear in 1,445 and 1,039
posts, respectively. The second and fourth most popular phrases are calls for
attacks targeting Muslims; “Kill Muslim” and “Kill all Muslim” appear in
approximately 4.6K and 1.6K posts. We also find phrases against Islam; “Fuck
Islam” appears in 3.4K posts and ”I hate Islam” in 881 posts. Finally, we also
identify phrases containing the terms “mudslimes” [59] and “sandniggers” [60],
which are derogatory names to refer to Muslims and Arabs.

Figure 6: Top 15 images, in terms of the number of posts, in our Islamophobic
Visual dataset. We report the number of posts that the image appeared and the
most related Islamophobic phrase for each image.

###  5.2 Popular Images in Visual Dataset

We also look into the popularity of images in our Antisemitic/Islamophobic
datasets (in terms of the number of posts they shared). Fig. 4 shows the CDF
of the number of posts for each Antisemitic/Islamophobic image. We observe
that Antisemitism and Islamophobia imagery is a diverse problem, with 58.64%
and 62.58% of the images appearing only in one post for Antisemitism and
Islamophobia, respectively. At the same time, we have a small percentage of
images that are shared many times on 4chan’s /pol/; 8.94% of all the
Antisemitic/Islamophobic imagery are shared at least ten times. Overall, we
observe a similar pattern in the distribution of the number of posts per image
for Antisemitism and Islamophobia.

Next, we look into the most popular images in our Antisemitic and Islamophobic
visual datasets. Fig. 5 shows the top 15 Antisemitic images; for each image,
we report the number of posts it appears and the most relevant matching
textual phrase. We observe that the Happy Merchant meme appears in a lot of
the most popular Antisemitic memes on 4chan’s /pol/ (images I, II, III, IV, X,
XI, XIII, and XV in Fig. 5). The most popular image is particularly
interesting as it likely aims to disseminate the idea that Canadians are
behind every shitpost on /pol/ (note the antisemitic connotation in the leaf
as it has the Happy Merchant’s face). We also observe some false positives
among the most popular images (image V, VI, and IX in Fig. 5). These images
are not Antisemitic; however, by looking into the matching phrase, we can
understand why the CLIP model treats it as similar. For instance, image V in
Fig. 5 is likely returned because of the word “worshiper” in the query. This
specific example highlights the sensitivity of the CLIP model to the input
queries and how it can result in the generation of false positives when
considering the task of hateful content detection. Other interesting examples
of popular images include image VIII and image X in Fig. 5; both hinting that
members of the Jewish are allegedly the masterminds of lousy stuff happening
or conspiracy theories (i.e., shut down the Jewish plan or Rabbi painting Nazi
symbols).

Figure 7: Number of Antisemitic posts per day in our Textual/Visual datasets.
Figure 8: Number of Islamophobic posts per day in our Textual/Visual datasets.

We also show the top 15 most popular Islamophobic images in Fig. 6. Here, we
observe some images that are pretty graphic; for example, image V shows a pig
having sex with a Muslim, or image VI shows a Muslim eating his excreta. Other
popular images include ones that show Pepe the frog as a Muslim (these images
are not directly Islamophobic, but CLIP considers them as related to the
query), images that include sarcasm and link Muslims to terrorism (image VII
and XIII in Fig. 6), as well as images linking Muslims to the Happy Merchant
meme (e.g., image XIV in Fig. 6). Again, similarly to the most popular images
in the Antisemitic dataset, we have some false positives like image III
(likely because the CLIP model does not know the slur “sandniggers”) and image
XV in Fig. 6. Finally, the second most popular image in our Islamophobic
dataset (Fig. 6) likely highlights that the CLIP model has some biases; e.g.,
it links the phrase “all muslims must hang” with an image showing Pepe the
Frog dressed as a terrorist, likely indicating that the model thinks that
Muslims are terrorists.

###  5.3 Antisemitic/Islamophobic content over time

This section presents our temporal analysis that shows the distribution of
Antisemitic/Islamophobic content over time. Fig. 7 shows the number of hateful
posts per day in the Antisemitic Textual/Visual datasets. We run Kendall’s
tau-b correlation to determine the relationship between the number of posts in
the Antisemitic Textual and Visual dataset. We find a strong, positive, and
statistically significant correlation ($\tau=.582,p<.001)$, indicating that
hateful content is spread both using text and images in a similar fashion. We
also observe the highest volume of textual and image content between April 6,
2017, and April 9, 2017, with 4,132 (1.97% of the dataset) posts in the
Textual dataset and 7,134 (1.51%) posts in the Visual dataset. This finding
confirms previous findings from Zannettou et al. [71] that identified a spike
in the spread of the Happy Merchant memes on April 7, 2017.

By inspecting the top 15 most frequent images during that period (we omit the
figure due to space constraints), we identified that those images are related
to the decision of Donald Trump to remove Steve Bannon from the National
Security Council Post on April 5, 2017 [15] and a missile attack in Syria on
April 7, 2017 [53]. According to newspapers [5, 27], Jared Kushner, the Jewish
Trump’s son-in-law, seemed to be acting as a shadow secretary of state
visiting and taking Middle East portfolios after that event. This political
decision spread a volume of image content with the face of Jared Kushner.
Also, there are some references to Donald Trump that indicate that he is
controlled by Israel ( e.g., the most popular image is associated with the
phrase “fuck Drumpf and fuck Jewish people”). The term “Drumpf” is a German
surname most commonly known as the likely predecessor to the family name of
Donald Trump.

We also evaluate the distribution of Islamophobic posts over time. Fig. 8
shows the number of Islamophobic posts per day in our Textual/Visual datasets.
We also find a statistically significant, strong, and positive correlation
($\tau=.470,p<.001)$. In both datasets, we find a peak of activity on May 23,
2017, with 482 and 937 posts in the Textual and Visual datasets, respectively.
By manually inspecting the top 15 images shared that day, we identify that the
high volume of posts is related to the Manchester Bombing; on May 22, 2017, a
British man detonated a suicide bomb in the foyer of the Manchester Arena as
people were leaving a concert by pop singer Ariana Grande. On May 23, ISIS
claimed responsibility for the attack. [14]. This event raised hateful online
narratives defining Muslims as terrorists [20]. We find images that contain
explicit references to this attack and images questioning whether Islam is a
religion of peace. Overall, our findings highlight that both textual and
visual hateful content is likely influenced by real-world events, with peaks
of hateful activity observed during important real-world events that are
related to the demographic groups we study.

##  6 Related Work

Hate Speech detection. Hate speech has recently received much research
attention, with several works focusing on detecting hate speech in online
social media. Initial research on hate speech analysis is typically oriented
towards monolingual and single classification tasks due to the complexity of
the task. They used simple methods such as dictionary lookup [26], bag of
words [26], or SVM classifiers [37, 57]. Recent efforts are proposing
multilingual and multitask learning by using deep learning models [63, 43, 24,
62]. While previous approaches to characterize and identify hate speech focus
purely on the content posted in social media, some research efforts shift the
focus towards detecting hateful users by exploiting other contextual data [49,
1, 11, 64]. Furthermore, other research efforts investigate to what extend the
models trained to detect general abusive language generalize between different
datasets labeled with different abusive language types [32, 38, 52, 54, 40].
While less explored, some work focus on multimodal settings, formed by text
and images [16, 33]. Gomez et al. [25] build a large dataset for multimodal
hate speech detection retrieved from Twitter using specific hateful seed
keywords, finding that multimodal models do not outperform the unimodal text
ones.

Antisemitism. Antisemitism have grown and proliferated rapidly online and have
done so mostly unchecked; Zannettou et al. [71] call for new techniques to
understand it better and combat it. Ozalp et al. [44] train a scalable
supervised machine learning classifier to identify antisemitic content on
Twitter. Chandra et al. [9] propose a multimodal system that uses text,
images, and OCR to detect the presence of Antisemitic textual and visual
content. They apply their model on Twitter and Gab, finding that multiple
screenshots, multi-column text, and texts expressing irony, sarcasm posed
problems for the classifiers. To characterize antisemitism, Enstad [21]
propose an analytical framework composed of three indicators: antisemitic
attitudes, incidents targeting Jews, and Jew’s exposure to antisemitism. Their
results show that attitudes vary by geographic and cultural region and among
population sub-groups.

Islamophobia. Surveys show that Islamophobia is rising on Web communities
[29]. Vidgen and Yasseri [61] build an SVM classifier to distinguish between
tweets non-Islamophobic, weak Islamophobic, and strong Islamophobic with a
balanced accuracy of 83%. Cervi [8] use clause-based semantic text analysis to
identify the presence of Islamophobia in electoral discourses of political
parties from Spain and Italy. Chandra et al. [10] apply topic modeling and
temporal analysis over tweets from the #coronajihad to identify the existence
of Islamophobic rhetoric around COVID-19 in India. Civila et al. [13] apply
content analysis over 474 images and texts in from Instagram posts under the
hashtag #StopIslam. Alietti and Padovan [2] conduct telephone surveys on 1.5K
Italians on Antisemitic and Islamophobic attitudes, finding an overlap of
ideology for both types of hate speech.

##  7 Discussion & Conclusion

In this work, we explored the problem of Antisemitism/Islamophobia on 4chan’s
/pol/ using OpenAI’s CLIP model. We devised a methodology to identify
Antisemitic/Islamophobic textual phrases using Google’s Perspective API and
manual annotations and then used the CLIP model to identify hateful imagery
based on the phrases. We found that the CLIP can play a role in detecting
hateful content; using our methods, the CLIP can detect hateful content with
an accuracy of 84%. Also, we found that Antisemitic/Islamophobic imagery
exists in 2x more posts when compared to Antisemitic/Islamophobic speech on
4chan’s /pol/. Additionally, our work contributes to research efforts focusing
on detecting hateful content by making a dataset of 420
Antisemitic/Islamophobic phrases, 246K textual posts, and 92K images publicly
available. Below, we discuss the implications of our findings for researchers
focusing on detecting hate speech and for researchers working on large pre-
trained models like OpenAI’s CLIP.

Prevalence of Antisemitic/Islamophobic Imagery. Our findings show that images
play a significant role in the spread of hateful content, and on 4chan’s /pol/
they even overshadow hateful content spread via text. This is likely because
4chan is an imageboard and a fringe Web community; hence, a large volume of
hateful content is disseminated via images. Nevertheless, the problem of
hateful imagery exists on other mainstream platforms (e.g., Twitter), hence it
is of paramount importance to develop better and more accurate systems for the
detection of hateful content across multiple modalities. For instance, we
argue that the spread of hateful content via videos is an unexplored problem,
and there is a need to develop models across text, images, and videos.

Performance and Sensitivity of CLIP model. Our experiments indicate that
large-pre-trained models like CLIP are pretty powerful and have general
knowledge that can be used for various tasks. When considering the hateful
content detection task, the CLIP model should be used with caution. This is
because the CLIP model highly depends on how the input text query is written,
influencing the number of false positives returned. When CLIP is used for
moderation purposes, we argue that it is essential to have humans in the loop
to ensure that the automated model works as expected. Additionally, we
observed that the CLIP model performs worse when considering input text
queries that comprise many words. This indicates that we need more powerful
text encoders that can capture the primary meaning of textual phrases,
irrespectively of how long they are.

Biases on CLIP model. Large pre-trained models like OpenAI’s CLIP are trained
on large-scale datasets from the Web, and these datasets might include biases,
hence some of the bias is transferred to the trained model. From our
experiments and manual annotations, we observed some instances of such biases;
e.g., the CLIP model identifying an image showing a terrorist as similar to a
text phrase talking about Muslims (i.e., the model is biased towards Muslims,
thinking they are terrorists). When considering that these models can be used
for moderation purposes (e.g., detecting and removing hateful content), such
biases can result in false positives biased towards specific demographics.
This can cause users to lose trust in the platform and its moderation systems
and may cause them to stop using the platform. Overall, given the increasing
use of such models in real-world applications, there is a pressing need to
develop techniques and tools to diminish such biases from large pre-trained
models.

Limitations. Our work has several limitations. First, we rely on Google’s
Perspective API to initially identify hateful text, which has its limitations
(e.g., might not understand specific slurs posted on 4chan) and biases when
detecting hateful text. Second, our analysis focuses only on short textual
phrases (at most seven words), mainly because our preliminary results showed
that CLIP does not perform well in detecting hateful imagery when considering
long phrases. Therefore, it is likely to miss some Antisemitic/Islamophobic
text and imagery. Third, we rely entirely on a pre-trained CLIP model; this is
not ideal since the CLIP model is trained on a public dataset obtained from
multiple Web resources and is not specific to our platform of interest (i.e.,
4chan). This might result in the model not recognizing 4chan slurs or slang
language. As part of our future work, we intend to explore the possibility of
fine-tuning the CLIP model with datasets obtained from fringe Web communities
like 4chan.

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