# Daily Archives: December 6, 2008

## Even more predictive models

In the last post, I presented a comparison of different ways of doing prediction. A natural follow-up question is whether or not there are even better functions?  I observed in the last post that a straight line performs better than the logistic function which has a downward hump.  A natural set of functions to explore then has an upwards hump.   First let’s plot the two probability functions from the previous post along with several new ones (note the color scheme is different):

 Function Label Predictive Likelihood $b - \exp(-c x + d)$ $\psi_b, b=1$ -11.4085 … $\psi_b, b=0.8$ -11.402 … $\psi_b, b=0.75$ -11.4003 $\pi_0 + (\pi_1 - \pi_0)\delta_{>r}(x)$ threshold -11.7929 $a x^2 -2 a x + c$ $\psi_q$ -11.3968 $u \exp(- \frac{(x - 1)^2}{v^2})$ $\psi_g$ -11.4704 $u + w * \exp(- \log(x)^2 /2)$ $\psi_l$ -11.3747
As it happens, we actually do better by making the hump protrude upwards (i.e., making the function concave.). $\psi_b, \psi_q$ and $\psi_l$ both perform better than the previously proposed functions; $\psi_q$ is easy since it has a simple quadratic form. In other words, it could be a second approximation of something, but of what? $\psi_l$ does a little bit better than that. Maybe we’re looking a posterior normality of log likelihoods?  If I knew the answer to this question, I think I’d be able to get a grip on a function which is actually justifiably better…