# Tag Archives: probability

## Ancient Probability

I had a need to look in the first-ever probability textbook this weekend. It isn’t really ancient, but it is definitely old. And it’s all on the Internet Archive as a pdf. Good times.

Abraham de Moivre, The doctrine of chances: or, A method of calculating the probabilities of events in play (1756)

Filed under education

## Random Graphs in NetworkX: My Spatial-Temporal Preferred Attachment Diversion

To take my mind off my meetings, I spent a little time modifying the Spatial Preferred Attachment model from Aiello, Bonato, Cooper, Janssen, and Prałat’s paper A Spatial Web Graph Model with Local Influence Regions so that it changes over time. Continue reading

Filed under combinatorics, probability

## MCMC in Python: PyMC for Bayesian Model Selection

(Updated 9/2/2009, but still unfinished; see other’s work on this that I’ve collected)

I never took a statistics class, so I only know the kind of statistics you learn on the street. But now that I’m in global health research, I’ve been doing a lot of on-the-job learning. This post is about something I’ve been reading about recently, how to decide if a simple statistical model is sufficient or if the data demands a more complicated one. To keep the matter concrete (and controversial) I’ll focus on a claim from a recent paper in Nature that my colleague, Haidong Wang, choose for our IHME journal club last week: Advances in development reverse fertility declines. The title of this short letter boldly claims a causal link between total fertility rate (an instantaneous measure of how many babies a population is making) and the human development index (a composite measure of how “developed” a country is, on a scale of 0 to 1). Exhibit A in their case is the following figure:

An astute observer of this chart might ask, “what’s up with the scales on those axes?” But this post is not about the visual display of quantitative information. It is about deciding if the data has a piecewise linear relationship that Myrskyla et al claim, and doing it in a Bayesian framework with Python and PyMC. But let’s start with a figure where the axes have a familiar linear scale! Continue reading

Filed under MCMC, statistics

## MCMC in Python: PyMC for Bayesian Probability

I’ve got an urge to write another introductory tutorial for the Python MCMC package PyMC.  This time, I say enough to the comfortable realm of Markov Chains for their own sake.  In this tutorial, I’ll test the waters of Bayesian probability.

Tom Bayes

Now, what better problem to stick my toe in than the one that inspired Reverend Thomas in the first place? Let’s talk about sex ratio. This is also convenient, because I can crib from Bayesian Data Analysis, that book Rif recommended me a month ago.

Bayes started this enterprise off with a question that has inspired many an evolutionary biologist: are girl children as likely as boy children? Or are they more likely or less likely? Laplace wondered this also, and in his time and place (from 1745 to 1770 in Paris) there were birth records of 241,945 girls and 251,527 boys. In the USA in 2005, the vital registration system recorded 2,118,982 male and 2,019,367 female live births [1]. I’ll set up a Bayesian model of this, and ask PyMC if the sex ratio could really be 1.0.

Filed under MCMC, probability

## MCMC in Python: PyMC to sample uniformly from a convex body

This post is a little tutorial on how to use PyMC to sample points uniformly at random from a convex body.  This computational challenge says: if you have a magic box which will tell you yes/no when you ask, “Is this point (in n-dimensions) in the convex set S”, can you come up with a random point which is nearly uniformly distributed over S?

MCMC has been the main approach to solving this problem, and it has been a great success for the polynomial-time dogma, starting with the work of Dyer, Frieze, and Kannan which established the running-time upper bound of $\mathcal{O}\left(n^{23}(\log n)^5\right)$.  The idea is this: you start with some point in S and try to move to a new, nearby point randomly.  “Randomly how?”, you wonder.  That is the art. Continue reading