The Equivalence of Minimizing Cross-Entropy Loss and Maximizing Likelihood in Neural Networks

Ask a statistician how to train a classification model, and they will tell you to maximize the probability of the observed data. The goal is to find the exact parameters that make the training set as likely as possible, a framework known as Maximum Likelihood Estimation.

Ask a computer scientist the same question, and they will tell you to minimize the network's error. The goal is to penalize the model for incorrect predictions using a metric called cross-entropy loss, pushing the error rate as close to zero as the architecture allows.[2][5]

Both disciplines defend their approach as the fundamental basis of machine learning. The statistical view treats the neural network as a probabilistic engine, outputting a distribution over possible classes. The computer science view treats it as an information-processing system, measuring the "surprise" or inefficiency in the network's outputs.[4][6]

Yet, when an engineer writes a loss function in PyTorch or TensorFlow in 2026, they are satisfying both camps simultaneously. The two frameworks are not just philosophically aligned; they are algebraically identical.[7]

To understand how maximizing a probability is the exact same operation as minimizing a loss, we have to look at the mechanics of how neural networks evaluate their own performance. A network does not inherently know what a specific category is; it only outputs a vector of numbers between 0.0 and 1.0.[1]

Under the statistical framework, we want the network to assign a probability of 1.0 to the correct label. If we have 10,000 images in a training batch, we want the joint probability of all 10,000 correct predictions to be as high as possible.[2][4]

Because the images are assumed to be independent, calculating this joint probability requires multiplying 10,000 individual probabilities together. Multiplying thousands of numbers smaller than 1.0 quickly results in a value so microscopic that a computer's floating-point architecture rounds it to absolute zero—a hardware limitation known as numerical underflow.[2]

To bypass this, statisticians apply a logarithm. As the authors of the 2016 Deep Learning textbook note, "The argmax does not change when we scale the objective function." Taking the natural logarithm transforms the massive multiplication problem into a simple addition problem, creating the log-likelihood.[1]

Meanwhile, information theorists approach the problem through the lens of entropy, a concept introduced by Claude Shannon in 1948. Entropy measures the minimum number of bits required to encode a piece of information.[5][6]

Cross-entropy extends this to measure the difference between two probability distributions: the true distribution of the data (where the correct label has a 100 percent probability) and the predicted distribution from the model. If a model predicts a 0.01 probability for the correct class, the cross-entropy is extremely high, representing a massive penalty.[5]

If the model predicts a 0.99 probability, the cross-entropy drops near zero. The objective in information theory is to minimize this cross-entropy, forcing the model's predictions to perfectly match the true labels.[5][6]

The mathematical bridge between these two worlds relies on a simple property of optimization: maximizing a number is exactly the same as minimizing its negative.[2][7]

The log-likelihood is a negative number, because the logarithm of any probability between 0.0 and 1.0 is negative. To maximize this negative number, optimization algorithms instead minimize the Negative Log-Likelihood (NLL).[3][4]

When the true labels are one-hot encoded—meaning the correct class is represented as a 1 and all other classes as 0—the cross-entropy formula collapses. All the terms for the incorrect classes multiply by zero and vanish.[5][6]

The only term that remains in the cross-entropy calculation is the negative logarithm of the probability assigned to the correct class. This is exactly the formula for the Negative Log-Likelihood.[1][5]

As the Dive into Deep Learning authors explain, "Minimizing the cross-entropy is equivalent to maximizing the likelihood." The two equations reduce to the exact same string of variables.[2]

This equivalence is why modern deep learning frameworks use cross-entropy as their default classification loss. When combined with a softmax activation function, the derivative of cross-entropy simplifies to a remarkably clean equation: the predicted probability minus the true label.[1][6]

If a model predicts 0.8 for a true label of 1.0, the gradient is simply -0.2. This clean derivative allows the backpropagation algorithm to update millions of weights with minimal computational overhead, a mathematical elegance that makes training massive language models feasible.[4][7]

The equivalence does have boundaries. It assumes the training examples are independent and identically distributed. If the data points influence each other—such as in certain time-series or autoregressive models—the strict equivalence between cross-entropy and maximum likelihood requires complex adjustments to account for conditional probabilities.[1][3]