## Fixed Points of Logistic Function

ModelingStatisticsposted by John Cook June 15, 2018

Here’s an interesting problem that came out of a logistic regression application. The input variable was between 0 and 1, and someone asked when and where the logistic transformation f(x) = 1/(1 + exp(a + bx)) has a fixed point, i.e. f(x) = x. So given logistic regression parameters a and b, when does... Read more

## Relative Error in the Central Limit Theorem

ModelingStatisticsposted by John Cook June 12, 2018

If you average a large number independent versions of the same random variable, the central limit theorem says the average will be approximately normal. That is the absolute error in approximating the density of the average by the density of a normal random variable will be small. (Terms... Read more

## Quantifying Uncertainty with Bayesian Statistics

ModelingStatisticsposted by Mat Leonard June 5, 2018

Whenever we’re working with data, there is necessarily uncertainty in our results. Firstly, we can’t collect all the possible data, so instead we randomly sample from a population. Accordingly, there is a natural variance and uncertainty in any data we collect. There is also uncertainty from... Read more

## Robustness and Tests for Equal Variance

ModelingStatisticsposted by John Cook May 30, 2018

The two-sample t-test is a way to test whether two data sets come from distributions with the same mean. I wrote a few days ago about how the test performs under ideal circumstances, as well as less than ideal circumstances. This is an analogous post for testing whether two... Read more

## “I hate math!” – Education and Artificial Intelligence to find a meaning

ModelingStatisticsposted by Pablo Casas May 21, 2018

Well, what you hate is the way that math was taught to you. That soup of equations, abstractions, and solutions to problems that we don’t know, It’s hard to enjoy the things we don’t feel part of. But how about relating some math techniques from the... Read more

## Least Squares Solutions to Over- or Underdetermined Systems

ModelingStatisticsposted by John Cook May 17, 2018

It often happens in applications that a linear system of equations Ax = b either does not have a solution or has infinitely many solutions. Applications often use least squares to create a problem that has a unique solution. Overdetermined systems Suppose the matrix A has dimensions m by n and the right hand side vector b has dimension m. Then... Read more

## Ask America Anything Results: Broad Support for Programs that Benefit Children

ModelingStatisticsposted by Civis Analytics Team May 14, 2018

Here at Civis Analytics, we recently discovered that an estimated eighty-one percent of Americans support increased federal spending on programs that benefit children — programs including strong support for enhancing children’s access to healthcare, affordable housing, quality K-12 education, and food — according to a survey we ran on behalf... Read more

## Datasets Are Books, Not Houses

Deep LearningModelingStatisticsposted by Brendan O'Brien May 2, 2018

What’s content addressing? What does it have to do with datasets? Why am I on this site in the first place? Read on, dear reader. Read on. The world of linked data is built on shaky foundations that prevent a true data commons from emerging. The... Read more

## Eight-bit Floating Point

ModelingStatisticsposted by John Cook April 25, 2018

Researchers have discovered that for some problems, deep neural networks (DNNs) can get by with low precision weights. Using fewer bits to represent weights means that more weights can fit in memory at once. This, as well as embedded systems, has renewed interest in low-precision floating... Read more

## Small p hacking

ModelingStatisticsposted by Thomas Lumley April 23, 2018

The proposal to change p-value thresholds from 0.05 to 0.005 won’t die. I think it’s targeting the wrong question:  many studies are too weak in various ways to provide the sort of reliable evidence they want to claim, and the choices available in analysis and publication... Read more