Proofs from the notebook Olivier's open lab book.

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Global bounds on the Jensen functional

Given a convex function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ and $X$ a random variable on $\mathbb{R}$, the Jensen functional of $\varphi$ and $X$ is defined as

The well-known Jensen inequality states that $\mathcal{J}(\varphi, X) \geq 0$. For instance, if $\varphi(x) = x^2$, then $\mathcal{J}(\varphi, X) = \text{Var}(X) \geq 0$. If $\mu$ and $\nu$ are two probability measures, $X \sim \nu$ and $\varphi$ is convex with $\varphi(1) = 0$, then $\mathcal{J}(\varphi, \tfrac{d\mu}{d\nu}(X)) =: D_\varphi(\mu, \nu)$ is a so-called $f$-divergence between probability measures such as the total variation distance, the Kullback-Leibler divergence, the $\chi^2$ divergence, etc.

If $X$ is bounded, then a converse to the Jensen inquality can be easily obtained as follows: let $m$ and $M$ be the infimum and maximum of $X$, and write $X = \alpha m + (1-\alpha)M$ for some random variable $\alpha$ taking values in $[0,1]$. Then $\mathbb{E}[\alpha] = (M - \mu)/(M-m)$ and consequently with $\mu:= \mathbb{E}[X]$,

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The motivation for the *Guide des bourses*

I’ve been TAing for first year analysis and statistics classes past semester. My job was to animate weekly problem solving sessions, i.e. every week I had lists of problems that I showed them how to think about and solve.

At the start of the weekly sessions, I tried to give students some helpful hints about studying at UQAM - the type of things that I wished people told me when I was just beginning. These are things like:

  • Info about their student insurance, how they can use it to get free glasses, free dentist appointments, etc.
  • Info about mental health support at UQAM.
  • What’s going on in the mathematics community: conferences that might interest them, events and activities.
  • How they should look out for scholarships and research internship opportunities, and some tips on how to improve their CVs.

Unfortunately, even though I tried my best, I don’t think I did nearly a good enough job at informing them of what’s going on and at helping them for post-graduate professional life:

  • I’m unfortunately not good at talking about mental health.
  • I have a very limited knowledge of what’s going on in the math undergrad world.
  • I only know about the FRQNT and CRSNG scholarships to which I applied (because somebody once told me about them).

(Even though I was bad at it, I was still one of the very few who talked about these things.)

And if it’s still hard for me to get a clear picture and thorough information on this subject, even after going through a bachelor and a master at UQAM, imagine what it’s like for first or second year students! I also had no clue when I was beginning. Getting some professional development advice early on would have greatly helped me.

So that’s the story behind the Guide des bourses (Guide to Scholarship and Awards). Now I want to put together all of the ressources that would have helped me when going through my studies at UQAM, so that all students have easy access to this information.

For now, the guide is still very much focused on my own personal experiences and the similar ones of my friends in the statistics department. However, anyone can contribute to the Guide on Github and I’m working on extending it. Hopefully we’ll gather more ressources in the weeks to come and present a broader persective on the opportunities that are available to students.

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J’ai eu la chance au cours de la dernière année d’être admis à Stanford, à Duke et à l’Université de Toronto pour mon doctorat en statistique. J’ai aussi reçu la bourse Alexander-Graham-Bell du Canada (105 000$ sur trois ans) en étant classé premier au Canada parmi toutes les demandes en sciences mathématiques, et j’ai obtenu la bourse doctorale du FRQNT.

C’est une belle récompense pour je crois beaucoup d’efforts, et j’ai bien hâte de poursuivre à Duke l’automne prochain. :^)

How to make a blog like this one

Here’s my new math blogging workflow:

  • I write my posts in Typora, which is a wysiwyg Markdown editor with Latex math support. (It’s really nice and you should check it out!)
  • I then push my project folder to Github and I get published in a few seconds.

It’s very easy to write math in this way. No need to struggle with formatting and rendering errors on Wordpress. You have an editor with instant preview and complete control over the appearence of the site.

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