Is it not necessarily to put observed data at the right of the bar symbol $\mid$ when computing likelihood?

This Machine Learning course with timestamp is talking about PERFORMANCE MEASURE.

Assume there are 3 loan officers A, B and C (correspond to you, blank, your "Friend" on the blackboard) , assigned the probabilities (predictions) of default to 2 loans.

the column on the rightmost is the actual outcomes of loans, namely, not default and default.

the column on the leftmost is predictions made by A, 0.2 chance of default for both loans.

B's predictions are 0.1 and 0.5

C predicted 0.3 chance of default for both loans.

the bottom is the likelihood that A's estimates applied to the actual complete set of outcomes, according to A, which is 0.8*0.2, why is that?

wiki gives this formulas to calculate the likelihood for Discrete probability distribution

$${\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x),}$$

question

wiki's formula put observed data at the right of the bar symbol $$\mid$$, which means

over the outcome x, not over the parameter $$\theta$$

while that lecturer put "Me" at the right of the bar symbol $$\mid$$, which is implying

over the parameter $$\theta$$, not over the outcome x.

which is a opposite meaning.

What am I missing?

• This question is unclear. What do you mean by 'with a different meaning'? Dec 14, 2019 at 15:17
• @SextusEmpiricus I've updated my question, would you please take a loot at that? Dec 14, 2019 at 23:12
• So your question is about the notation and particularly the use of the bar symbol. It does indeed seem like a different use (but note a mistake is easily made since $L (\theta|x) = P (x|\theta)$ and from a certain perspective the bar symbol is not very meaningfull). Could you explain the terms on the blackboard. The 'actual complet set of outcome' and 'ME'. Dec 15, 2019 at 10:14
• This question inspired me to write: stats.stackexchange.com/questions/440910/… Dec 15, 2019 at 10:30

1 Answer

A gives an estimate of 0.2 for each of the two loans to default.
For the first loan (not default), that's an estimate of 0.8 for the correct result.
For the second loan (default), that's an estimate of 0.2 for the correct result.
Taken together, the likelihood of the correct result is 0.2 * 0.8