Friday, July 11, 2008

Importance of being normal

In my previous post I discussed how variance in abilities and needs in a local neighborhood is necessarily a result of efficient resource management by Nature for sustenance of life. Then how change is essential to keep the variance (multiple hypotheses) over time to avoid getting stuck in local optima. I also mentioned that being around normal value is important.

In statistics filtering is a problem of estimating posterior of a random variable given observations correlated with the variable over time. Life can almost be treated as a random variable with some moments. In a more global perspective it's hard to characterize these moments and hence the posterior is usually represented using just random samples. I had a related post about six months ago. But our lives are mostly dominated by local perspectives. Actually if we had true global perspectives all the time we would be super natural!

In statistics a very popular technique of estimating a complex non-linear probability distribution of a random variable is non-parametric kernel density estimation. Intuitively it says that any complex distribution can be approximated using sum of Gaussians (or normal distributions). Let's say if we can track these individual Gaussians then we automatically track the overlaying complex distribution. Kalman filters are useful when the Gaussians undergo linear changes that is the mean and variance of the Gaussian only undergo linear transformations. For reasonable non-linear changes there are linear approximations resulting in extended Kalman filters. But for highly non-linear transformations the approximations made in extended Kalman filters are not good enough. Hence people developed unscented Kalman filter which is a combination of sampling based and closed form trackers. The key elements in unscented Kalman filters are the a set of sample around the mean of the distribution. These "sigma points" are the ones that undergo non-linear transformations which can then lead to recovering the necessary moments! See it's quite important to be around the normal distribution especially in the era of highly non-linear changes to actually "participate/contribute" in successful propogation of moments.