Thread by @MoritzSchauer

Let's do a beautiful #science #randomwalk It starts at a lake and in a moment raindrops will fall making little ripples

Where the ripples meet, cell boundaries form. A Voronoi tessellation of a random point process.

Let's connect rain drops which share a boundary to get a random graph

This is gives a random triangulation called Poisson-Delaunay triangulation. A model for local self-organisation.

Slightly random Voronoi tessellations show for example up in insects wings.

Random Delaunay triangulations are also good for Nerd snipes (you recall xkcd.com/356/)

Now there was a surprise! Turns out to solve questions about electric grids you can use a random walker exploring the Delaunay triangulation (Doyle/Snell: Random Walks and Electric Networks, arxiv.org/abs/math/00010…)

So I started simulating random walks on random triangulations

For example to answer @xkcd 's "Snipe" question you want to know: What is the probability that a random walker starting in A ever comes back to A on its journey before finding B? (math.dartmouth.edu/~doyle/docs/wa…, 1.3.4)

Although, random walks are not particular skilled at this

Opens the question: how do I make those

Good moment to revisit @rlmcelreath's Statistical Rethinking, second chapter: Bayesian statistics is just about counting ways to go from A to B. Counting for discrete spaces, or somehow easier: Integration for continuous spaces. xcelab.net/rm/statistical…

In this regard, the random walkers help exploration of space and the counting of paths:

High probability for a random walker to go from A to B in T steps ≃ many path which go from A to B in T steps ≃ small resistance

Let's help the walkers by guiding them from A to B. That makes some events more likely. We just need to count those events less often then, and nobody will notice the cheat!

How to guide? Make it a landscape, let the random walker gravitate to B.

(Shoot-out to @MakiePlots and github.com/JuliaGeometry/… !)

With less end less time left to reach the destination, this gravitational potential field becomes stronger and stronger making sure the walker reaches B at just the right time

A bit funny: We often control the process such that the event in which we are interested always happens. So how can you estimate the probability of something by making it always happen? Turns out you can, by measuring how much control was necessary.

Finding the perfect potential field is as hard as the original problem: You'd have to solve Kolmogorov's backward equations.

This means you'd have again to count all the paths to the destination with the right length - not so easy for random walks in random environments.

Good thing, in sampling methods you can approximate without making errors. For example the Poisson-Delaunay random walk looks familiar if you zoom out. Brownian motion!

Brownian motion has Gaussian transition densities. So that was why the landscape looked so quadratic... it is up to local fluctuations the log of a Gaussian density

So! Check out our (with @MeulenFrank) paper

https://twitter.com/MoritzSchauer/status/1314129899561586688?s=20

which has your Poisson-Delaunay bridges (Example 10.3.), guided agent based models for likelihood based inference, rules for automatic guiding in probabilistic programs etc.

Julia #julialang package: github.com/mschauer/Mitos…. Poisson-Delaunay bridge example: gist.github.com/mschauer/71716…. Image credit insect wing: commons.wikimedia.org/wiki/File:IC_G…

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