# Likelihood values from Sigmoid [duplicate]

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?

• How is the sigmoid being used as a PDF? Where do you see people using sigmoid to infer probability?
– Dave
Jul 6, 2021 at 18:24
• @Dave: arxiv.org/pdf/1402.3722.pdf View page 3. Jul 6, 2021 at 18:31
• @Dave : dropbox.com/s/qiq2c85cle9ydb6/Chapter3.pdf?dl=0 View Page 7, last but 1th paragraph. Jul 6, 2021 at 18:39
• These seem to be three very different questions, thematically linked together by the very broad question "Where does logistic regression come from?" I think the duplicates address this question.
– Sycorax
Jul 6, 2021 at 19:13