Re: Statistics Models
IIRC, Rutter uses a "heirarchical model"... the general concept is that say you are flipping coins... you know all coins are milled somewhat differently but behave similar to each other... so while they may take on different values we have some reason to believe that certain rates are more frequent than others. We then can flip several coins and use the knowledge from the other coins to provide sharper estimates of each of the coins by borrowing from a common structure (which we will often give a parameteric form... see beta distribution).
In this case, the concept is the same, but then its a question of linkage and calculation. I believe Rutter uses the linearized form as explained in the previous post but then entwines a hierarchy where he assumes each beta_i has a normal distribution with mean zero and variance sigma (but unknown). The idea is similar, you assume a common super-distribution (hyper-distribution, heirarchy, hyper-paramters, so on) and so you can use this to better couch the estimates... it serves in some case as a deflationary influence and as such is useful against extreme values. Rutter, as a Bayesian, also employs what is termed a "prior" on the unknown sigma... often this can be refered to as "prior belief", "subjective probability", etc. though there are forms that try to be objective. Even the heirarchy imposed here can be seen as a Bayesian application. The general idea is you start with a vague initial state and you use your learned knowledge (data) to refine that belief. Anyhoo... there are a lot of other parts that one can argue about or disagree with... but the important notion here is that he ties all the teams through a common distribution under the notion that hockey teams should exhibit some amount of spread and that spread can be calculated. He's also making a rough notional call to the "Stein Estimation" problem. Its an interesting phenomena in science (things that are true in 2-D in science are often not true in 3-D and beyond... I had a professor who said he had a professor who speculated that this is why our universe works)... so while one may argue that hockey teams don't necessarily come from a common pool there is some utility in doing so because it mitigates the overall degree of error.
As such its a bit technical of an approach but certainly something that I personally see as a reasonable model. Hopefully I've made somewhat of an understandable pitch without being too technical.
Originally posted by Numbers
View Post
In this case, the concept is the same, but then its a question of linkage and calculation. I believe Rutter uses the linearized form as explained in the previous post but then entwines a hierarchy where he assumes each beta_i has a normal distribution with mean zero and variance sigma (but unknown). The idea is similar, you assume a common super-distribution (hyper-distribution, heirarchy, hyper-paramters, so on) and so you can use this to better couch the estimates... it serves in some case as a deflationary influence and as such is useful against extreme values. Rutter, as a Bayesian, also employs what is termed a "prior" on the unknown sigma... often this can be refered to as "prior belief", "subjective probability", etc. though there are forms that try to be objective. Even the heirarchy imposed here can be seen as a Bayesian application. The general idea is you start with a vague initial state and you use your learned knowledge (data) to refine that belief. Anyhoo... there are a lot of other parts that one can argue about or disagree with... but the important notion here is that he ties all the teams through a common distribution under the notion that hockey teams should exhibit some amount of spread and that spread can be calculated. He's also making a rough notional call to the "Stein Estimation" problem. Its an interesting phenomena in science (things that are true in 2-D in science are often not true in 3-D and beyond... I had a professor who said he had a professor who speculated that this is why our universe works)... so while one may argue that hockey teams don't necessarily come from a common pool there is some utility in doing so because it mitigates the overall degree of error.
As such its a bit technical of an approach but certainly something that I personally see as a reasonable model. Hopefully I've made somewhat of an understandable pitch without being too technical.
Comment