Updating mean and variance estimates an improved method
So far, a beta distribution looks like a pretty appropriate choice based on the above histogram. Well, suppose the histogram had two peaks, or three, instead of one.Then we might need a mixture of Betas, or an even more complicated model). Now when we look at any individual to estimate their batting average, we’ll start with our overall prior, and update based on the individual evidence.The method is compared to several related methods, as well as a full Bayesian update, on a simple scalar example.
The second player, on the other hand, has a lot of evidence that he’s an above-average batter.
This post isn’t really about baseball, I’m just using it as an illustrative example. If you want a more technical version of this post, check out this great paper).
We give an error analysis of an algorithm for computing the sample variance due to Chan, Golub, and Le Veque [The American Statistician 7 (1983), pp. It is shown that this algorithm is numerically stable.
The algorithm computes the sample variance (and the sample mean) using just one pass through the sample data.
Now I’ll demonstrate the related method of empirical Bayes estimation, where the beta distribution is used to improve a large set of estimates. (For the sake of estimating the prior distribution, I’ve filtered out all players that have fewer than 500 at-bats, since we’ll get a better estimate from the less noisy cases.
Updating mean and variance estimates an improved method Ai webcam sex chat
What’s great about this method is that as long as you have a lot of examples, . I show a more principled approach in the Appendix).
This is why this process is sometimes called : we’ve moved all our estimates towards the average.
How much it moves these estimates depends on how much evidence we have: if we have very little evidence (4 hits out of 10) we move it a lot, if we have a lot of evidence (300 hits out of 1000) we move it only a little.
We would estimate his batting average as: How about the batter who went up only 10 times, and got 4 hits.
We would estimate his batting average as: Thus, even though , we would guess that the batter is better than the batter!