Can we extrapolate? Can we say, because yesterday it was raining, today it will rain?
Can we just say, because Germany won the last championship, it will this one also?
Clearly our past data say so, but we know its not. I claim, if we fitting a model it does
overfitting.

Thanks Bob!
I already knew both the Andrew World Cupâ€™s model and the Milad model for the Premier League as well. In fact I have been largely inspired by these models and I enjoyed reading.

Actually, instead of the softmax parametrization, I used the alternative multinomial logistic parametrization here, (https://en.wikipedia.org/wiki/Multinomial_logistic_regression) modeling K-1 =2 probabilities and the K-th (the draw in this case ) as:

1/{1+\sum_{k=1}^{K-1}exp { beta_k x}

However, I realized now there is a typo since i did not exponentiate the etas in the denominators, and the sum is from 1 to K: thus, thanks!

As I motivated in the Andrewâ€™s blog in the comments section (http://andrewgelman.com/2018/06/15/stan-goes-world-cup/#comments), this table only represents the estimated probabilities obtained after simulating the World Cup 10000 times before each game is playedâ€¦ Thus, the reason why Germany is favored is mainly due to a high FIFA ranking, rather than past historical results

The nomenclature around all this is very inconsistent and confusing. What youâ€™re calling â€śmultinomial logisticâ€ť is just softmax with one of the inputs pinned to 0. The 0 in the version youâ€™re using (1 after exp(0)) identifies the model, but comes with the disadvantage that priors become asymmetric. Thereâ€™s a discussion in the manual around K vs. K - 1 parameter parameterizations of multinomial logistic regression.

\eta_{n.} not have an intercept resp. home advantage parameter. Is there any reason for that?
At the same time you have u_{att} in att_t, same for defense. This is a constant for all t, so
both \eta_{n.} gets added these. Is this a case for an identifiability problem?

Mmh, I still have to think about it. Anyway, at the time being, mu_att and mu_def do not appear in the model anymore.
See my website for model and predictions updates about the quarter of finals starting today!

I had no idea what sensor fusion was, thanks for the suggestion!