Most people (at least those with college education) are well aware of how exponential growth works. The typical (correct) intuition is that when things are growing exponentially, they may initially look like nothing, in fact things may go very slow for quite a while, but eventually there is an explosion and exponential growth eventually outpaces everything sub-exponential. What is less commonly appreciated is that exponential decay works similarly - things exist, get smaller and effectively at some point become nonexistent. It is almost as if there was a discrete transition. Let us keep that in mind while we discuss some probability theory below.

Gauss and Cauchy

Gauss and Cauchy were two very famous mathematicians, both having countless contributions in various areas of mathematics. Coincidentally, two seemingly similarly looking probability distributions are named after these two individuals. And although many people working in data science and engineering have relatively good understanding of Gaussian distribution (otherwise known as "normal" distribution), Cauchy distribution is less known. It is also a very interesting beast, as it is an example of a much less "normal" distribution than Gaussian and most intuitions from typical statistics fail in the context of Cauchy. Although Cauchy like distributions are … Read more...

Introduction

One of the hallmarks of science is the reproducibility of results. It lies at the very foundation of our epistemology that objectivity of a result could only be assured if others are able to independently reproduce the experiment.

One could argue that science today actually has various issues with reproducibility, e.g. results obtained in a unique instrument (such as the LHC - Large Hydron Collider) cannot be reproduced anywhere, simply because nobody has another such instrument. At least in this case the results are in principle reproducible, and aside from the lack of another instrument, the basic scientific methodology can remain intact. Things get a bit more hairy with AI.

Determinism, reproducibility and randomness

The one hidden assumption with reproducibility is that the reality is roughly deterministic, and the results of the experiment depend deterministically on the experimental setup. After carefully tuning the initial conditions we expect the same experimental result. But things start to be more complex when our experiment itself is statistical in nature and relies on a random sample.

For example the experiment called elections: once the experiment is performed it cannot be reproduced, since the outcome of the first experiment affects substantially the system studied … Read more...