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I wonder how frequentist and bayesian calculate the winning probability for Tottenham versus Arsnel, Saturday.

For frequentists, the following is how I understand:

  • Imagine a population made of all the historical games between Tottenham versus Liverpool[Do not consider the specific day of the week because of the assumption that all games are played with the same condition.]. Here I don't exactly know what the population could be made of. I don't know if I can assume the population is made of all the historical data but, to me, the historical data seem to be just a sample to measure for the frequency.
  • Repeatedly draw samples.
  • Record frequencies of Tottenham that won the game versus Liverpool.

I don't know if the way I think about the frequentists' way is correct.


For Bayesians, I don't know yet. However, for the sake of simplicity, I can think of two factors of players to win the game over the opponent:

  • the number of goals each player scores
  • the number of attack assists

How can I think of the prior distribution for the two factors of players?

  • This is the the way I think about the prior distribution. I do not need two different prior distributions because the prior distribution is all about the probability distribution of winning the games. So I just need one prior distribution. If I am not correct, please feel free to correct me.

What could be the data to apply to the likelihood?

I also hope to know what issues that bayesians would say frequentists may run into for this probability calculation.

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Bayesians would begin by specifying a prior probability distribution for the probability that one team wins over the other. Then can then combine their priors with available data to update their model and obtain posterior distributions for the probability. This would be the simplest approach, but one may imagine adjusting that probability estimate based on any number of things (which stadium the two teams play in, if any players are injured, accounting for correlation between other teams who have also won/lost against their opponent, etc etc).

Bayesians might say that Frequentists are making very strong assumptions about their data model. If all Frequentists (or Bayesians for that matter) do is take the total number of wins Liverpool has against Tottenham and divide by all their matches, then the most straightforward objection is relevance of some data. Are matches from the 70s informative 50 years later? Maybe, but certainly not so informative so as to be given equal weight with matches played last year, or earlier in the season.

All in all, your question is very broad, and this answer provides just a taste of what could be done. If you have specific questions I'd be happy to expand.

EDIT:

This is going to be overkill, but I will write down a Bayesian model for the probability a player scores on a scoring chance given their position and possibly other covariates.

Let $p_i$ be the probability a given player scores on a scoring chance. Conditioned on the player's covariates, assume

$$ p_i \vert X \sim \operatorname{Beta}(\mu(X), \kappa) $$

Here, I have parameterized the beta distribution in terms of its mean and a seperate parameter which controls precision. See here for more. I've let $\kappa$ be constant, but there is no reason to assume it doesn't change with the covariates as well.

The mean could be modelled by

$$ \operatorname{logit}(\mu(x)) = X \beta $$

Where $X$ are covariates and beta are regression coefficients. This model allows for partial pooling of information between players from separate teams (though were I to actually fit this I think random effects for teams and positions might be a better model).

Finally, we require priors on the $\beta$ and $\kappa$. Prior selection is more of an art than an exact science, and so I will refrain from selecting priors and simply wave that detail away.

The expected number of goals is then the sum of the product of the probability of scoring multiplied by the expected number of scoring chances. It might be even more interesting to model scoring chances independently (or jointly) and then propagate uncertainty through the final computation.

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  • $\begingroup$ Although the frequentist way seems to have a strong assumption, I am just wondering if frequentists cannot even model the soccer match outcome because of the strong assumption or not. Please tell me If you see that I am not understand some points in this context. $\endgroup$ – StoryMay Nov 20 '20 at 4:06

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