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The Inverse Problem and Bayes’ Theorem

Posted on March 6, 2016 Written by The Cthaeh 7 Comments

The domino effect illustrated with gray domino tiles in the process of falling one by one.The inverse problem is fundamentally related to the subject of causality. Think about a scenario…

What will happen if you grab a solid rock and throw it at your neighbor’s window? The most common result is that the window will break. If your neighbor later asks if you know anything about the incident, you can confidently inform him that his window was broken because you threw a rock at it earlier. Cause and effect — sounds pretty straightforward.

But what if it wasn’t you who broke the window and, in fact, you have no idea what broke it? Did someone else throw a rock at it? Was there a large temperature difference between the center and the periphery of the glass which caused a spontaneous breakage? Or was it a spontaneous breakage caused by a fabric defect?

You see, unlike the previous question, this one is actually not straightforward to answer. It involves solving the sо-called inverse problem: inferring the causes of a particular effect.

[Read more…]

Filed Under: Applications, Bayes' Theorem Tagged With: Causality

Bayes’ Theorem: An Informal Derivation

Posted on February 28, 2016 Written by The Cthaeh 12 Comments

A man, a dog, a cat, and a hamster staring outside from a high building's window with Bayes' theorem formula in the foreground

If you’re reading this post, I’ll assume you are familiar with Bayes’ theorem. If not, take a look at my introductory post on the topic.

Here I’m going to explore the intuitive origins of the theorem. I’m sure that after reading this post you’ll have a good feeling for where the theorem comes from. I’m also sure you will find the simplicity of its mathematical derivation impressive. For that, some familiarity with sample spaces (which I discussed in this post) would come in handy.

So, what does Bayes’ theorem state again?

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Filed Under: Bayes' Theorem Tagged With: Coin flip, Conditional probability, Sample space

What Is Bayes’ Theorem? A Friendly Introduction

Posted on February 22, 2016 Written by The Cthaeh 9 Comments

Bayes' theorem hand-written on a notebook. Additionally, two dice are rolled on top of the notebook from a black plastic cup.If you ever came across Bayes’ theorem, chances are you know it’s a mathematical theorem. This theorem has a central role in probability theory. It’s most commonly associated with using evidence for updating rational beliefs in hypotheses.

While this post isn’t about listing its real-world applications, I’m going to give the general gist for why it has such potential in the first place.

Alright, let’s get to it.

[Read more…]

Filed Under: Bayes' Theorem, Fundamental Concepts Tagged With: Conditional probability

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