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I am reading a paper by Bolton and Chapman about predicting horse races and I am having trouble understanding a certain formula. To give you the context, the paragraph is called: "Estimating the Parameters of the Multinomial Logit Model". It says:

The likelihood function associated with a particular sample of races can be written in the following form for the multinomial logit model: $$exp(L) = \prod_{j=1}^J P_{jh*}$$ where the j subscript denotes a race (j = 1, 2, ..., J), h* in equation (10) is the horse that is observed to win eace j, and L refers to the log-likelihood function.

L is said to be the log-likelihood but it is not clear how it fits in here. My question is then: how should one interpret this exp(L), given that L is used as an argument to exp but is never used on the right side of the equation?

Note that it is not introduced previously in the text either.

Is it perhaps some shorthand notation or a convention one needs to be familiar with?

pswies
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  • I notice the paper is from the 80's. I don't know if this is a problem in the field this paper is from, but in my studies, I notice a plague of poorly formulated notation that began in about 1982 and slowly died off starting around 1991. – The Count Apr 24 '17 at 13:08
  • At a glance, I can't see the meaning either, and it is likely using notation from a cited paper in the text, as a guess. – The Count Apr 24 '17 at 13:09

2 Answers2

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I think they are just assuming you know that if you have a parametric model (either a parametric PMF or a parametric PDF) $p(x \mid \theta)$, and $n$ independent samples $\{ x_i \}_{i=1}^n$ from the distribution being modeled, then the likelihood for this sample is defined as $\ell(\theta)=\prod_{i=1}^n p(x_i \mid \theta)$. This product can be awkward to handle, so we often consider the log-likelihood $L(\theta):=\log(\ell(\theta))=\sum_{i=1}^n \log(p(x_i \mid \theta))$ instead.

Why they decided to refer to the exponential of the log-likelihood instead of just the likelihood itself, I couldn't tell you.

Ian
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  • Yes! This is what I have been missing. After your answer I also read https://en.wikipedia.org/wiki/Likelihood_function#Definition and the picture becomes clear.

    So as far as I understand, the L is what matters and they add the exp around it to simplify the right side.

     – pswies Apr 24 '17 at 15:09
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Yes. $\exp(L)$ is another way to rewrite $e^L$

amWhy
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Ziad Fakhoury
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