60

I will stay away from the academic politics of hiring "professors of data science", but if I interpret the question more specifically as "does data science offer problems of mathematical interest", I might refer to Bandeira's list of 42 Open Problems in Mathematics of Data Science.
(The full list from 2016 is here, and Bandeira's home page links to ...

39

Fundamentally a lot of what a modern data scientist does is very similar to what in previous generations would have been the responsibility of a statistician, and it shouldn't surprise you that there are professors of statistics. Mathematically there are quite a few interesting things that come up in a lot of modern data science, but first let me make a non-...

33

The Mathematics of Data may go some way towards answering your question. As one example of a mathematically interesting topic that is motivated by data science, you might want to look at the concept of persistent homology.

19

To begin, there is a family of results which are sometimes referred to as "No free lunch" theorems. Each of these results, in their own way, asserts that any optimization algorithm is just as good as any other if you average over the space of all optimization problems. On the other hand, we know that in specific domains some algorithms vastly outperform ...

12

Maybe relevant:
Yu Chen, Jin Cheng, Yu Jiang, Keji Liu, A Time Delay Dynamical Model for Outbreak of 2019-nCoV and the Parameter Identification, https://arxiv.org/abs/2002.00418

10

$\newcommand\bar\overline$
Letting $t:=\ln p$, we see that the limit in question is the limit of
$$d(t):=\frac1t\Big(\sum_1^n x_j e^{tx_j}\Big/\sum_1^n e^{tx_j}-m_{e^t}\Big)$$
as $t\to0$.
Next, letting
$\bar x:=\frac1n\,\sum_1^n x_j$, $\bar{x^2}:=\frac1n\,\sum_1^n x_j^2$, and $s^2=\bar{x^2}-\bar x^2$, we have
$$\sum_1^n x_j e^{tx_j}=\sum_1^n x_j (1+tx_j+o(t))...

9

The following paper is a little strange, since it dates back to 2015, but has some valuable data:
A SARS-like cluster of circulating bat coronaviruses shows potential for human emergence, Nature Medicine, 2015.

8

The book by Gábor Csárdi, Tamás Nepusz, Edoardo Airoldi,
Statistical Network Analysis with igraph
Based around popular software library igraph, Wikipedia link
contains whole chapter with source codes (in R) on Epidemics on networks in particular 6.5 Vaccination strategies
Let me quote the content of the chapter:
6 Epidemics on networks
6.2 Branching ...

7

Suppose that $k$ is a fixed natural number, $n\to\infty$, and
\begin{equation}
a_i=\frac nk+o(n^{2/3})
\end{equation}
for each $i=1,\dots,k$.
Let
\begin{equation}
h_i:=\frac kn\,a_i-1=o(n^{-1/3}).
\end{equation}
By Stirling's formula:
\begin{equation}
m!\sim\sqrt{2\pi m}(m/e)^m
\end{equation}
as $m\to\infty$,
\begin{equation}
\binom{n}{a_1,\ldots,...

7

The $\tau$-expectile, say $E_\tau X$, of a random variable (r.v.) $X$ is the root $t$ of the equation
$$r_X(t)=\rho(\tau),$$
where
$$r_X(t):=\frac{E(X-t)_+}{E(t-X)_+}, \quad \rho(\tau):=\frac{1-\tau}\tau.$$
For any real $a>0$, we have $r_{aX}(at)=r_X(t)$, whence
$$E_\tau(aX)=a\,E_\tau X.$$
For any real $a<0$, we have $r_{aX}(at)=1/r_X(t)$, whence
$...

7

The lecture notes by Frank Nielsen are succinct and fairly self-contained and maybe good to get a first idea. The books [1,2] by Amari can serve for a more in-depth study and contain a fair bit of differential geometry background. In order to get a flavour for some of the applications, I would suggest [3].
EDIT: Some additional remarks on the differential ...

7

What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.
Technology transfer from the general linear model (not generalized!), that is with gaussian errors and identity link. A lot of what ...

6

This is known as the truncated Stieltjes moment problem, and there is a necessary and sufficient condition taking the form of a semidefinite program. See Section 5 of the classic paper by Curto and Fialkow.

6

Here is a recent paper written by mathematicians: Risk Assessment of Novel Coronavirus COVID-19 Outbreaks Outside China.

6

Recently found this :
https://staff.math.su.se/tom.britton/
Maybe relevant.

5

Let $A={\rm diag}[a_1,\dots,a_n]$ and $B={\rm diag}[b_1,\dots,b_n]$. Let $\Delta(a)$ be the Vandermonde product in the $a_j$, and similarly $\Delta(b)$ be the Vandermonde product in the $b_k$. Suppose ${\rm min} \, a_j \ge {\rm max} \, b_k$. Let ${\rm d}U$ denote the normalised Haar measure on the unitary group $U(n)$. Then by Eq. (3.21) in Gross and ...

5

One example where this happens is when $P$ and $Q$ are "antipodal" distributions -- say, $p=(p_1,\ldots,p_n)$, $q=(q_1,\ldots,q_n)$,
$q_i=1-p_i$, and
$$P=\mathrm{Ber}(p_1)\times\mathrm{Ber}(p_2)\times\ldots
\mathrm{Ber}(p_n)$$
and $Q$ is defined analogously. Then $KL(P||Q)=KL(Q||P)$. These play a role in optimal decision theory, see
http://jmlr.org/beta/...

5

We're looking at the equation $$-\sum_{x\in\mathcal{X}} P(x) \log\left(\frac{Q(x)}{P(x)}\right)
=
-\sum_{x\in\mathcal{X}} Q(x) \log\left(\frac{P(x)}{Q(x)}\right).
$$
In the Bernoulli case,
$$-p \log\left(\frac{q}{p}\right)
-(1-p) \log\left(\frac{1-q}{1-p}\right)
=
-q \log\left(\frac{p}{q}\right)
-(1-q) \log\left(\frac{1-p}{1-q}\right)
$$
The only solutions ...

answered Nov 2 '19 at 19:54

Bjørn Kjos-Hanssen

22.6k33 gold badges4545 silver badges100100 bronze badges

5

Polya's criterion says that if $f:\mathbb{R}\to \mathbb{R}$ is even, convex on $[0,\infty)$, with $f(0)>0$ and zero limit at infinity, then $c(x,y) = f(\vert x-y\vert)$ is a positive definite kernel, hence the existence of the Gaussian random field. It would apply to your function for $p$ less than $1$ for example (to be checked).

5

There are already nice negative answers by Steve and Rémi Peyre. In the comments, user111 mentioned this post by David Reeb who gives a bound on the difference of entropies in terms of the KL-divergence when $p$ and $q$ are probability distributions on a finite set. I want to mention two other such bounds.
Suppose that $p$ and $q$ are distributions on a ...

5

The following paper is extremely important because it has informed the decisions of the UK government that realised (announced) on Monday 16/03/2020 that it can not afford "Herd immunity". The paper only shows the outcomes of the model and speaks about its parameters. It would of course be extremely interesting to know what exactly is the mathematics behind ...

5

### What is limiting behavior for the median of $2^X+2^Y$, where $X$ and $Y$ are iid binomial variables?

The limit is $\sqrt2$. Indeed, let $(h_n)$ be such that $h_n/\sqrt n\to\infty$ but $h_n/n\to0$ (as $n\to\infty$).
Then, by the central limit theorem, with probability $\to1$ we have $n/2-h_n\le X_n\le n/2+h_n$ and $n/2-h_n\le Y_n\le n/2+h_n$ and hence
\begin{equation}
2^{n/2-h_n}\le 2^{X_n}+2^{Y_n}\le2^{n/2+h_n+1}.
\end{equation}
So, eventually (for all ...

5

Even if $X_n\to c$ in probability for some real constant $c$, it is not necessary that $P(Y\le X_n)\to P(Y\le c)$ -- you also need to require that $P(Y=c)=0$.
More generally, if the limit of $X_n$ is not a constant, then you need to assume the convergence, not just of the distribution of $X_n$, but of the joint distribution of $X_n$ and $Y$. In particular, ...

4

Studeny says in the abstract of his 1992 paper that:
However, under the assumption that CIRs [conditional-independence relations] are grasped the existence of a countable characterization of CIRs is shown.
Since Studeny seems to be calling a "complete finite axiomatization" a "finite characterization", this seems to suggest that Studeny is ...

4

Just a quick follow-up: Very recently (well, actually in 2015) three teams came up independently and almost simultaneously with the same construction of a new "optimal-transport-like" distance on the space of Radon measures $\mathcal M^+$, which somehow interpolates continuously between Wasserstein and Fisher-Rao. This distance now goes by the name of ...

4

I think the problem is that "data science" means many different things to different people. To you it connotes applying statistics to marketing, but for others it covers large swaths of probability, statistics, machine learning, even things like geometry, etc.
But this can be an opportunity too. If I wade into the politics of hiring just a little and also ...

4

The analogous estimate in the range where your formula is valid (that is, $a_i=\frac nk+o(n^{1/2})$ - note Wikipedia claims the binomial case is valid for $a_i=\frac n2+o(n^{2/3})$, but I wasn't able to reproduce that estimate) is:
$$
\binom{n}{a_1\ a_2\ \ldots\ a_k}
\sim
\frac{k^n}{(2\pi n)^{(k-1)/2}}\exp\left(-\frac kn\sum_{i=1}^k b_i^2\right),
$$
where $...

4

The answer to your main question, about the Kullback--Leibler divergence, is no. Indeed, let the vectors $p=(1-s,s)$, $q=(1-t,t)$, and $r=p$ represent the corresponding probability distributions on (say) the set $\{1,2\}$, where $s\downarrow0$ and $t:=e^{-1/s^2}$. Then
$$KL(p||q)=(1-s)\ln\frac{1-s}{1-t}+s\ln\frac st\sim\frac1s\to\infty,
$$
$$KL(q||r)=KL(q||...

4

Just a partial answer, but the proposed inequality doesn't hold.
Take $p = [0.2, 0.8], q = [0.1, 0.9]$.
Then $H(p) = 0.2 \log(5) + 0.8 \log(1/0.8) \approx 0.5$,
$H(q) = 0.1 \log(10) + 0.9 \log(1/0.9) \approx 0.33$
and $KL(p, q) = 0.2 \log(2) + 0.8 \log(0.8/0.9) \approx 0.04$.

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