Poissonian Fish

After catching a fishy phrase — "fraction of failing breams", which I corrected to "fraction of failing beams" — in my manuscript draft (though Juan G. Restrepo has suggested the dystopian "fraction of failing dreams" as a possible correction), I noticed in a Google Scholar e-mail that my work was cited recently by a paper called The Contact Structure of Great Britain's Salmon and Trout Aquaculture Industry (which has nothing to do with any baseball players on the Angels). This made me think of "failed salmon" and "failed trout" and gives possible inspiration for writing a paper with a title like Percolation on Fish(y) Networks. My mind continued to descend in a fishy direction, so I decided to plot Poisson distributions of fish (as you can see in my tweet).

Here are some Poisson distributions of fish. pic.twitter.com/wZ367E4GVg

— Mason Porter (@masonporter) May 11, 2019

Finding Identical Packs of Skittles

One blogger has examined the probability of finding identical packs of skittles and posted their data online.

In my new lower-division mathematical modeling course (Math 42; "Introduction to Data-Driven Mathematical Modeling: Life, The Universe, and Everything"), we started our unit on discrete stochastic modeling today, so this post makes me very happy! I have, of course, e-mailed my students about this.

(Tip of the cap to Jeremy Jay.)

The Perils of Generating Functions (and Submitting a Note from a Parent with an Exam)

When I taught Math 1d, my exam instructions included a joke line that went something along the lines of "You may also use a note from your mother, though you won't need one." One student (and fellow Caltech undergrad) actually got her mother to write this note, which is fantastic (despite the threats against both Lloyd House and the Los Angeles Dodgers).

I had this posted on my door at Georgia Tech when I was a postdoc, and I found the note a year ago today when I was going through an old spiral notebook.

As a @Caltech undergraduate, I taught a probability course (Math 1d) and introduced freshmen to the glories of generating functions.

My exam instructions included a line like: "You may also use a note from your mother, though you won't need one."

One student took me up on it. pic.twitter.com/I2Ocif9ttA

— Mason Porter (@masonporter) March 27, 2018

A Multiplex Network of Relations Between Probability Distributions

OK, quickly: Which distribution is the most central? :)

(There are quite a few comments on the tweet. I haven't looked at them, but I wonder if people are picking apart inaccuracies? I haven't spent the time to vet this diagram, but I really like the idea!)

how probability distributions are related pic.twitter.com/1i8VHEHdy5

— hardmaru (@hardmaru) August 2, 2017

Perceptions of the Probability of Ambiguous Statements

You can completely mess up this chart in the UK simply by using the word "quite". That will do quite an ambiguous number on the perceived probabilities.

(More seriously, I really like this visualization.)

Fantastic enumeration/visualization of qualitative statements re probability via @amandacox https://t.co/XtFnHD2Zgp | @rsimmon @johnjhorton pic.twitter.com/DeByHToazi

— Nicholas Christakis (@NAChristakis) April 25, 2017

A Random Walk Through Public Broadcasting

A few days ago, PBS posted a video introducing random walks to a public audience! Sweet!

Update: This is an episode of a mathematics show called "Infinite Series". The above episode refers to an episode about Markov chains. Take a look at their YouTube channel and Twitter feed.

Tales from the ArXiv: "Dreidel Fairness Study"

There's a new paper on the arXiv preprint server called Dreidel Fairness Study.

I am amused by the following line from the abstract: "Although an unfair dreidel does not necessarily make the game itself unfair, it is conjectured that hundreds of pounds of chocolate have been distributed during Chanukah under false pretenses."

Also, the abstract mentions a dreidel with the face of Santa Claus on it. I guess they're mixing metaphors?

P.S. Somewhere in my parents' home lie my massive collection of dreidels. I can do a spiffy job of spinning them both in the standard fashion and upside-down.

"Mathematics, Failure, and the World’s Most Famous Walking Event"

Here is my latest post for the Improbable Research blog.

"A Unified Theory of Randomness"

This article is very cool. It talks about some exciting work that follows up that of people like Wendelin Werner, Oded Schramm, Martin Hairer, and others. The younger author of the pair who are doing many of these great things seems like somebody who is going to be considered strongly for a Fields Medal. (The article doesn't mention this, but this does appear to be at that level.)

In fact, several recent Fields Medals have been awarded for work with deep relationships to statistical mechanics (and related topics like kinetic theory and probability theory), and it took quite some time for these to be considered "acceptable" mathematics topics.

"Estimating Interevent Time Distributions from Finite Observation Periods in Communication Networks"

Here is the latest paper in my "please do this stuff more carefully" series.

People have been getting things wrong when it comes to examining inter-event time (IET) distributions, and there are methods from other fields (e.g., renewal processes) that allow one to correct for biases.

Here are the details of the paper.

Title: Estimating Interevent Time Distributions from Finite Observation Periods in Communication Networks

Authors: Mikko Kivelä and Mason A. Porter

Abstract: A diverse variety of processes––including recurrent disease episodes, neuron firing, and communication patterns among humans––can be described using interevent time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected without assuming any particular shape for the IET distribution. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. We analyze several well-known data sets from the literature, and we find that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

What Does Probability Mean in Your Profession?

Here are some drawings about the meaning of probability in various professions.

The one for philosophy is fantastic! Several of the others are also funny.

The so-called "actual meaning" among the drawings is not correct: "definitely" requires a probability of exactly 1, and "definitely not" requires a probability of exactly 0.

I would have loved to have seen a mathematical one, as then we would need to use "almost always", epsilons, and measures.

(Tip of the cap to Karen Kustedjo​.)

Using Statistical Mechanics to Capture a Skittish Lamb

I wrote a blurb on the Improbable Research blog on a recent paper on the arXiv for how to use ideas from statistical mechanics to capture a skittish lamb.

"Infinite Moments of Incredible Delight"

I just got a spam e-mail with the title of "Infinite Moments of Incredible Delight" (or something like that), and naturally the first thing I wondered about was which was the first moment that was no longer finite. Of course, it then struck me that that would make a great title for a paper (or section thereof) involving probability theory or power laws.

Teachers Get the Last Laugh

Damn straight.

I have pulled my share of things too, of course. For example, I need to go dig old my exam instructions that allow the use of 'a note from your mother' and the question about mystery meat with Poisson-distributed things in it. (One of the students actually submitted a note from her mother. It include threats against both Lloydies and the Los Angeles Dodgers. There was also a comment about the potential health problems caused by generating functions.)

(Tip of the cap to Sammy Kline --- and, of course, to Katie Noyes Rogstad and her mother.)

An Excellent Statistics Question

Now here's a truly excellent statistics question.

Those of you at Caltech who also took Alan Hájek's "Philosophy of Probability" course will know exactly what this question reminds me of. :)

Hajek's test was the best test ever when it comes to exam-taking paranoia: It was multiple choice. For each question, instead of picking a correct answer, we assigned a probability of correctness to each possibility, and our score on each question was determined by a formula that included a logarithm. Woe to anybody who assigned probability 0 of correctness to a choice that turned out to be correct, because then one got a grade of -infinity in the course. The result of this was to ask oneself nervously on every question: Am I truly sure that I am 100% confident that this answer isn't right? (Not only that, but exactly how confident am I on every possible answer for every question?)

[The scoring methodology was announced before the exam when I took the course, so I computed beforehand exactly what I should place instead of '0' on the questions in which I was maximally confident that some choice wasn't correct.]

(Tip of the cap to Tammy Porter and Danny Suiza.)

Sin-trees

Apparently, there is something called a sin-tree in probability theory. Obviously, they're a religious bunch. (Do a 'find' on the website to which I link if you want to see what a sin-tree is.)

Probability and O.J.

Here is Steve Strogatz's latest column in the New York Times. This one is on probability.