To evaluate the log-likelihood function for the given data where:

• $X = 45$ (feature)
• Label $Y = 0$ (actual outcome)
• Predicted probability $\hat{p} = 0.95$

The log-likelihood function for binary logistic regression is given by:

$LL(\beta) = Y \cdot \log(\hat{p}) + (1 - Y) \cdot \log(1 - \hat{p})$

Substituting the given values:

• $Y = 0$
• $\hat{p} = 0.95$

$LL(\beta) = 0 \cdot \log(0.95) + (1 - 0) \cdot \log(1 - 0.95)$ $LL(\beta) = \log(0.05)$

Now, we calculate $\log(0.05)$:

$LL(\beta) = \log(0.05) \approx -2.9957$

Thus, the log-likelihood value is approximately -2.996.

Let me know if you want further details or have any questions!

Here are five related questions to expand on this:

1. How is the log-likelihood function derived in logistic regression?
2. What is the interpretation of the log-likelihood value in logistic models?
3. How does the log-likelihood change with different predicted probabilities?
4. What role does the log-likelihood play in model evaluation?
5. How can you use log-likelihood in the context of maximum likelihood estimation (MLE)?

Tip: In logistic regression, maximizing the log-likelihood function helps in estimating the model's parameters that best fit the observed data.