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Negative Log-Likelihood Objective for Softmax Regression
To optimize a classification model using maximum likelihood estimation, we compare our predicted conditional probabilities with the actual labels. Assuming the dataset's labels are independent given the features , the probability of observing the correct labels is the product of individual probabilities:
Because maximizing a product of many small probabilities is numerically unstable and computationally awkward, we take the negative logarithm. This transforms the problem into minimizing the negative log-likelihood, turning the product into a manageable sum of individual losses:
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Negative Log-Likelihood Objective for Softmax Regression