The Cthaeh, Author at Probabilistic World

What Are Bayesian Belief Networks? (Part 1)

Posted on November 3, 2016 Written by The Cthaeh 19 Comments

In my introductory Bayes’ theorem post, I used a “rainy day” example to show how information about one event can change the probability of another. In particular, how seeing rainy weather patterns (like dark clouds) increases the probability that it will rain later the same day.

Bayesian belief networks, or just Bayesian networks, are a natural generalization of these kinds of inferences to multiple events or random processes that depend on each other.

This is going to be the first of 2 posts specifically dedicated to this topic. Here I’m going to give the general intuition for what Bayesian networks are and how they are used as causal models of the real world. I’m also going to give the general intuition of how information propagates within a Bayesian network.

[Read more…]

2016 US Presidential Election Predictions (October 10 Update)

Posted on October 10, 2016 Written by The Cthaeh Leave a Comment

This is an update to the main post from September 20. Please click on the link for a detailed description of the election process and how we model it to make the predictions you see below. Click here to go back to the daily predictions page.

You can click on the date above the map to check the predictions for other dates. You can also click on different states on the map to see our state-specific predictions.

[Read more…]

2016 US Presidential Election Predictions (October 7 Update)

Posted on October 7, 2016 Written by The Cthaeh Leave a Comment

This is an update to the main post from two weeks ago. Please click on the link for a detailed description of the election process and how we model it to make the predictions you see below.

You can click on the date above the map to check the predictions for other dates. You can also click on different states on the map to see our state-specific predictions. Click here to go back to the daily predictions page.

[Read more…]

Predicting The 2016 US Presidential Election

Posted on September 20, 2016 Written by The Cthaeh 4 Comments

November 7 UPDATE: click here to view our final update post where we give our latest analyses just a day before the election!
Click here to go to our daily predictions page.

In the first 10 posts I mostly concentrated on theoretical topics. But the general focus of this blog is much broader. For the first time I’m going to show an actual application of probability theory for estimating real life events.

An ongoing event many people are closely following right now is the US presidential election. The primary season officially concluded at the end of July and now the general election battle is in full swing. The main clash is between former Secretary of State Hillary Clinton (D) and businessman Donald Trump (R). Clinton and Trump are challenged by 3^rd party candidates Gary Johnson from the Libertarian Party (a two-time former governor of New Mexico) and Jill Stein from the Green Party (a physician and a political activist).

[Read more…]

Frequentist And Bayesian Approaches In Statistics

Posted on June 16, 2016 Written by The Cthaeh 37 Comments

What is statistics about?

Well, imagine you obtained some data from a particular collection of things. It could be the heights of individuals within a group of people, the weights of cats in a clowder, the number of petals in a bouquet of flowers, and so on.

Such collections are called samples and you can use the obtained data in two ways. The most straightforward thing you can do is give a detailed description of the sample. For example, you can calculate some of its useful properties:

The average of the sample
The spread of the sample (how much individual data points differ from each other), also known as its variance
The number or percentage of individuals who score above or below some constant (for example, the number of people whose height is above 180 cm)
Etc.

You only use these quantities to summarize the sample. And the discipline that deals with such calculations is descriptive statistics.

But what if you wanted to learn something more general than just the properties of the sample? What if you wanted to find a pattern that doesn’t just hold for this particular sample, but also for the population from which you took the sample? The branch of statistics that deals with such generalizations is inferential statistics and is the main focus of this post.

The two general “philosophies” in inferential statistics are frequentist inference and Bayesian inference. I’m going to highlight the main differences between them — in the types of questions they formulate, as well as in the way they go about answering them.

But first, let’s start with a brief introduction to inferential statistics.

[Read more…]

Combinatorics: An Intuitive Introduction

Posted on May 3, 2016 Written by The Cthaeh 4 Comments

Combinatorics is a branch of mathematics with applications in fields like physics, economics, computer programming, and many others. In particular, probability theory is one of the fields that makes heavy use of combinatorics in a wide variety of contexts.

For example, when calculating probabilities, you often need to know the number of possible orderings or groupings of events, outcomes of experiments, or generally any kind of objects.

Here’s what I’m talking about.

[Read more…]