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REPOST from Data Science:

I've been toying with this idea for a while. I think there is probably some method in the text mining literature, but I haven't come across anything just right...

What is/are some methods for tackling a problem where the number of variables is its self a variable. This is not a missing data problem, but one where the nature of the problem fundamentally changes. Consider the following example:

Suppose I want to predict who will win a race, a simple multinomial classification problem. I have lots of past data on races, plenty to train on. Lets further suppose I have observed each contestant run multiple races. The problem however is that the number or racers is variable. Sometimes there are only 2 racers, sometimes there are as many as 100 racers.

One solution might be to train a separate model for each number or racers, resulting in 99 models in this case, using any method I choose. E.g. I could have 100 random forests.

Another solution might be to include an additional variable called 'number_of_contestants' and have input field for 100 racers and simply leave them blank when no racer is present. Intuitively, it seems that this method would have difficulties predicting the outcome of a 100 contestant race if the number of racers follows a Poisson distribution (which I didn't originally specify in the problem, but I am saying it here).

Thoughts?

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A popular method for modelling race outcomes (or more generally the winner of a multi-player game) is an order statistic model, popularized by Plackett in "The analysis of permutations" (1975). This type of model is used in the TrueSkill rating system to predict multi-player game outcomes. This approach has a natural interpretation for races: each racer samples their finishing time randomly from a distribution, and the one with the lowest time wins.

In TrueSkill, this concept is abstracted into a "performance" distribution for each player, and the player who draws the best performance wins. The performance of player $i$ is assumed to be Gaussian distributed with mean $s_i$ and standard deviation $\beta$. The skill of different players is captured by $s_i$, and can be learned from data by following the equations in the paper. Many different variations of this approach are possible by making different assumptions about how the performances are distributed. For example, Plackett's model assumes they are Gumbel distributed.

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