Dataloopr

Here's a commonly asked question on logistic regression that appears in various forms across tech company interviews. It is designed to assess a candidate's understanding of key concepts such as probability, model interpretation, and evaluation metrics.

Can I say the following about logistic regression? Explain why or why not?

1) Minimizes cross-entropy loss

2) Models the log-odds as a linear function

3) Has a simple, closed form analytical solution

4) Is a classification method to estimate class posterior probabilities

Let's first refresh our concepts around logistic regression. 

Logistic regression is a classification algorithm that’s often used when you want to predict whether something belongs to one of two classes (for example, spam or not spam in email).

Let’s break down the four statements from before, one by one, in simple terms.

1. Minimizes Cross-Entropy Loss

• When logistic regression makes predictions, it estimates the probability that something belongs to a certain class (like the probability that an email is spam).
• The algorithm tries to make these predictions as accurate as possible by minimizing an error measure called cross-entropy loss (also called log loss). This measures how "off" the predicted probabilities are compared to the actual outcomes.
• In short: Logistic regression works by adjusting its predictions so that the errors (based on predicted probabilities vs. actual outcomes) become as small as possible.

2. Models the Log-Odds as a Linear Function

• Logistic regression predicts the probability of an outcome (like whether an email is spam or not), but instead of directly predicting the probability, it first works with the log-odds.
• The log-odds are a way to convert probabilities into numbers that can range from negative to positive infinity. This makes it easier for logistic regression to use a simple linear equation to make predictions. The equation looks like this:

• In simple terms: Logistic regression fits a straight line to the log-odds of the data, which it then uses to predict probabilities. It transforms probabilities into a linear relationship using log-odds.

3. Has a Simple, Closed-Form Analytical Solution

• This statement is false because logistic regression doesn’t have a simple formula we can solve directly. Instead, logistic regression needs to use optimization algorithms like gradient descent to find the best values for its parameters (like the coefficients in the equation).
• In short: Logistic regression doesn’t have a quick formula to solve the problem. Instead, it uses an iterative process to get the best solution.

4. Is a Classification Method to Estimate Class Posterior Probabilities

• Logistic regression is mainly used for classification. It estimates the probability of an input belonging to a class (like predicting the probability that an email is spam).
• This probability is called the posterior probability, which means "what is the probability that this is the correct class, given the data?"
• In short: Logistic regression helps us classify data by giving us the probability that something belongs to a certain class.

Summary:

• Logistic regression tries to predict the probability of an event happening (like predicting if an email is spam).
• It uses a linear function of the log-odds to make these predictions.
• It tries to minimize prediction error using something called cross-entropy loss.
• It’s a classification algorithm, but it doesn't have a simple formula for finding the answer, so it uses optimization techniques instead.

Here’s a summary of each option for logistic regression:

1. Minimizes cross-entropy loss:

   • True. Logistic regression minimizes the cross-entropy loss (also called log loss) during training. This loss function measures the difference between the predicted class probabilities and the actual labels.

2. Models the log-odds as a linear function:

   • True. Logistic regression models the log-odds (logarithm of the odds of the positive class) as a linear function of the input features. The formula is 

3. Has a simple, closed-form analytical solution:

   • False. Unlike linear regression, logistic regression does not have a closed-form solution because the likelihood function for logistic regression is not a simple linear equation. Instead, optimization techniques like gradient descent or Newton-Raphson are used to estimate the parameters.

4. Is a classification method to estimate class posterior probabilities:

   • True. Logistic regression is a classification method that estimates the posterior probabilities of classes. It predicts the probability of the output being in a particular class (often binary classification).