Repost of Mathemetics StackExchange question.
There are multiple doubts of mine associated around this theme:
In MLE, we try to find the PDF parameters ($\theta$) which maximise the likelihood of the observed data ($L(\theta | data)$). To get likelihood for a given data point for $\theta = \theta_1$ we simply evaluate the PDF for that data point. Now, we know that probability at any one particular point of a PDF is $0$. What is the correct reasoning behind evaluating the PDF at $x=x_1$ for its likelihood?
Clearly, the Sigmoid Function is not a PDF. But in the MLE estimates of Logistic Regression we see Sigmoid being used as if it is a PDF. Is my understanding correct ? If not, how to see it correctly? If yes, what is the reason behind it?
This is related to the previous question. I have seen at multiple places that people take the Sigmoid to infer probability. However there is not any constraint put to ensure that sum of all those probabilities must be $1$. What is the correct explanation behind it?