Eigil Fjeldgren Rischel
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Recent content on Eigil Fjeldgren RischelHugo -- gohugo.ioen-usFri, 12 Jun 2020 00:00:00 +0200Bivariate Causal Inference
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Mon, 13 Apr 2020 00:00:00 +0200/bivariate-causal-inference/TLDR I give a very short introduction to the idea of “causality” in statistics, then talk about two ways to infer causal structure for two variables - i.e, without using conditional independence statements. The ideas here are mostly taken from Peters, Janzing, and Schölkopf: Elements of causal inference: foundations and learning algorithms.
What is causality? A “causal statistical model” is something like this:
Or this
Where a classical statistical model tells you what the probability of various outcomes are, (which you can then use to derive conditional probabilities, etc), a causal model gives you more information - it tells you how the distribution changes when the system is intervened on.The zero-one laws of Kolmogorov and Hewitt–Savage in categorical probability
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Sun, 26 Apr 2020 00:00:00 +0200/zero-one-laws-paper/TLDR This is a post about my paper The zero-one laws of Kolmogorov and Hewitt-Savage in categorical probability, joint with Tobias Fritz. This is a “companion piece” where I try to explain those ideas in a more understandable language. There are essentially three ideas in this paper:
“Markov categories for synthetic probability theory” - this is only treated briefly, since this is just the background that we’re building on top of.Complexity theory, probability
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Sat, 02 May 2020 00:00:00 +0200/complexity-theory-probability/Computationally bounded probability theory Probability theory is about how to manage incomplete information. One way to interpret a statement like “the probability of event \(X\) is \(p\)” is in terms of betting odds - you think the probability of \(X\) is \(p\) if you value a lottery ticket that pays out $1 if \(X\) happens at \(p\) dollars. From this interpretation, all the laws of probability theory (except arguably those involving infinite conjunctions of events) follow, if we add the requirement that your valuation is “inexploitable” - in other words, if we require that no smart bookie can get you to make a series of bets that always loses money.Notes from "Persistent Homotopy Theory"
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Sun, 10 May 2020 00:00:00 +0200/jardine-persistent-htpy/My notes from Persistent Homotopy Theory by JF Jardine. The goal of the paper is to study “filtered spaces”. By this is meant in general something like an assignment \(s \mapsto X_s\) of a “space” or simplicial set to each nonnegative real \(s \in [0,\infty)\). A prototypical example is the Vietoris-Rips complex of a metric space, \(V_s(X)\).
The idea being pointed towards is some sort of modification of model category theory to make ideas from persistent homology work more nicely.The Ax-Grothendieck theorem
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Sun, 17 May 2020 00:00:00 +0200/ax-grothendieck-model-theory/The Ax-Grothendieck theorem says the following: Let \(f: \mathbb{C}^n \to \mathbb{C}^n\) be a polynomial function. If it’s injective, then it’s surjective as well.
Here’s how to prove it:
The statement can be formulated as a first-order statement in the language of fields If a statement like that fails for \(\mathbb{C}\), there’s a disproof in the first-order theory of algebraically closed fields of characteristic zero. Such a proof is finite, so it only uses finitely many of the assumptions \(p \neq 0\) - hence the theorem also fails in algebraically closed fields of sufficiently high characteristic.Stochastic Stalks
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Sun, 24 May 2020 00:00:00 +0200/stochastic-stalks/Stalks and points Recall that a point of a topos \(\mathcal{E}\) is a geometric morphism from the topos \(Set\). My preferred way to think about this is to consider the sheaf topos \(Sh(X)\) on some (sober) topological space \(X\). Then given \(x \in X\) and a sheaf \(S\), we can form the stalk at \(x\)
\(S_x := \operatorname{colim}_{x \in U \subseteq X \text{ open}} S(U)\)
This determines the point uniquely, i.Compositionality for Transfer Learning
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Tue, 02 Jun 2020 00:00:00 +0200/compositionality-for-transfer-learning/Transfer learning is the idea that, after a machine learning system (or a non-machine learning system, for that matter, like a human) has learned to solve some problem, it should be able to transfer this knowledge to solving similar problems. Humans are pretty good at this, at least compared to current ML systems, which tend to suck.
Why do we expect transfer learning to work? It seems that, in general, we expect that the solution to a task can be decomposed into several pieces, some of which will still be useful for the related task.Jensen-Shannon divergence is compositional
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Fri, 12 Jun 2020 00:00:00 +0200/jsd-as-enrichment/Let \(\mathsf{FinStoch}\) be the category of finite sets and stochastic matrices. Given two stochastic matrices, \(f_1,f_2: X \to Y\), we can define their Jensen-Shannon distance as \(d(f_1,f_2) := \sup_x \sqrt{\operatorname{JSD}(f_1(x),f_2(x))}\), where JSD is the Jensen-Shannon divergence. It’s a standard result that the root of JSD defines a metric on the space of probability measures - hence the above defines a metric on the set \(\mathsf{FinStoch}(X,Y)\). My aim here is to show that this gives an enrichment of \(\mathsf{FinStoch}\) in the category \(\mathsf{Met}\) of metric spaces and short, i.How to Make A Website
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Fri, 28 Feb 2020 00:00:00 +0000/2020-02-28-how-to-make-a-website/Here’s what goes into a website.
A server (hardware). The software which runs on the server, also called a server. A domain (optional) An SSL certificate (technically optional but highly recommended) I will explain what these terms mean, and how to get your own stuff set up. Please note that this is a guide for people who want to do everything from scratch. There are plenty of easier ways to set up a website.Frequentist Statistics and Compositionality
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Thu, 27 Feb 2020 00:00:00 +0000/2020-02-27-compositional-frequentism/Time-saving blurb: This essay thingy eventually ends without any useful conclusion (I don’t manage to figure out how to make something compose). Also, it’s not clear that what’s here is particularly deep even if it could be made to work, which it hasn’t.
P-values For convenience I’ll only work with finite sets - I’m not aware of any serious problems extending this to more general spaces, but it would add some technicalities and it’s not really germane to what I’m doing.