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I understood this wiki example of likelihood (likelihood function).

given the observed data HH, the likelihood that the model parameter $p_\text{H}$ equals 0.5 is 0.25. Mathematically, this is written as

${\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5\mid {\text{HH}})=0.25.} \tag{eq 1}$

This CMU Machine Learning course with timestamp gives this notation

$L(Outcomes \mid Me)$

Later, that course gives this notation

$L(O_1, ..., O_n\mid M)$, where $O_i$ denotes ith outcome, $M$ denotes Model.

per my previous understanding Model is a structured set of parameters, corresponds to $p_{\text{H}}$ in eq1, which means this course denote likelihood in a inverse order.

what am I missing? or both order are correct?

JJJohn
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  • Nothing. You aren't missing anything. – spaceisdarkgreen Dec 12 '19 at 05:21
  • @spaceisdarkgreen So, both order are correct? – JJJohn Dec 12 '19 at 05:22
  • Well... it's notation.. there's no one "correct" way. I usually put the parameter first since it's usually best to view the likelihood as a function of the parameters rather than the data. But it's also equal to the probability of the data given the parameters, which in conditional probability notation is typically in the reverse order. – spaceisdarkgreen Dec 12 '19 at 05:29
  • @spaceisdarkgreen That comment has the makings of a correct answer. – David K Dec 12 '19 at 06:31
  • @spaceisdarkgreen It seems to me that the Wikipedia article and the CMU lecturer seem to use the word "given" in opposite ways in their statements about likelihood--"given the data" in one case and "given the model" in the other. I'm curious what you make of that. – David K Dec 12 '19 at 06:40
  • @DavidK Well, if it helps dispel some doubt, it's completely implausible that $L(p_h=0.5\mid\text{HH})$ means "the probability that the coin is fair given that heads came up twice". Probability that the coin is fair compared to what? Surely there's some continuous distribution of biases and the probability of $p_h=.5$ is zero (i.e. it's best thought of as a density). On the other hand, it's incontrovertable that the probability of HH given the coin is fair is $1/4.$ – spaceisdarkgreen Dec 12 '19 at 06:54
  • @DavidK The lecturer is using "likelihood" synonymously with "probability", which would be fine if the subject of statistics hadn't been badly maimed by a handful of pedants in the twentieth century. Instead of saying "we choose the model that gives the largest probability for the data we saw" we say "we choose the model with maximum likelihood given the data we saw" and (re)define the word "likelihood" so THESE MEAN THE EXACT SAME THING. It's truly horrifying. – spaceisdarkgreen Dec 12 '19 at 07:00
  • @spaceisdarkgreen Yes, following the link (once it was posted), it was clear what the notation did and didn't mean. (It did not occur to me when first reading this question--completely out of context--that HH meant "two heads". Oops!) – David K Dec 12 '19 at 07:05

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