Negative log-likelihood
Negative log-likelihood is the negative logarithm of the likelihood a probabilistic model assigns to observed data, commonly used as an objective or loss to minimize.
Optimization Algorithms: AI techniques for design, planning, and control problems
Negative log likelihood is a loss equal to the negative logarithm of the probability assigned to the true label.
- How it is used
- Uses probability assigned to the correct class as a loss for training multiclass neural networks.
- How deep it goes
- Defines NLL and explains its behavior, then demonstrates batch computation but stops short of deriving optimization details.
- Examples
- Demonstrates NLL inside NumPy forward propagation for a neural network trained on the Penguins classification dataset.
- What you can do after
- Apply it via NumPy to compute and average classification loss from logits, probabilities, and true labels.
Reinforcement Learning from Human Feedback
Negative log-likelihood is the negation of a model's log-likelihood, used as a loss for probabilistic predictions.
- How it is used
- Optimizes preference probabilities for reward modeling and DPO, and regularizes preferred completions to retain language-modeling quality.
- How deep it goes
- Derives pairwise Bradley–Terry and DPO NLL objectives, their sigmoid forms, and the DPO gradient; also relates SFT NLL to forward KL.
- Examples
- Uses chosen and rejected completions, scalar reward scores, policy-to-reference likelihood ratios, and a length-normalized winning-completion penalty.
- What you can do after
- Apply it to train Bradley–Terry reward models, optimize DPO preference pairs, and add a winning-completion NLL penalty.
Read how Reinforcement Learning from Human Feedback explains negative log-likelihood →
Deep Learning with PyTorch, Second Edition
A loss function that penalizes low predicted probability assigned to the correct class by taking the negative logarithm of that probability and summing over samples.
- How it is used
- Uses predicted class likelihoods to train a supervised image classifier.
- How deep it goes
- Defines the formula, explains probability sensitivity, gives calculation steps, and demonstrates PyTorch's log-probability convention.
- Examples
- Demonstrates CIFAR bird-versus-airplane classification with nn.LogSoftmax and nn.NLLLoss.
- What you can do after
- Apply it via PyTorch by producing log probabilities and passing them with target class labels to nn.NLLLoss.
Read how Deep Learning with PyTorch, Second Edition explains negative log-likelihood →
Natural Language Processing in Action, Second Edition
Negative log likelihood is the negative of the logarithm of a model-assigned likelihood, commonly minimized as a probabilistic prediction loss.
- How it is used
- Uses NLLLoss to train an RNN and translate prediction scores into communicable confidence estimates.
- How deep it goes
- Explains logarithm inversion and likelihood interpretation, then stops short of a deeper derivation of the loss.
- Examples
- Uses a nationality-prediction model, the name Khalid, ranked log-loss values, and exponentiated likelihoods.
- What you can do after
- Evaluate/understand model confidence by converting log-loss outputs into likelihoods and inspecting ranked predictions.
Read how Natural Language Processing in Action, Second Edition explains negative log-likelihood →
Causal AI
The negative log-likelihood loss function is the negative logarithm of the likelihood and is minimized to estimate parameters.
- How it is used
- Uses likelihood-based estimation as a machine-learning loss optimized for model parameters.
- How deep it goes
- Defines the likelihood transformation and optimization role, then connects it to numerical optimization and regularization.
- Examples
- Uses linear regression, neural networks, and L2 regularization as estimation and optimization examples.
- What you can do after
- Apply it via numerical optimization to estimate parameters when likelihood maximization lacks a closed-form solution.
Machine Learning Algorithms in Depth
A negative log likelihood is a loss equal to the negative logarithm of the likelihood assigned to observed outcomes by a probabilistic model.
- How it is used
- Uses likelihood-based loss to train probabilistic classifiers and neural networks that predict continuous probability densities.
- How deep it goes
- Derives gradients and implements SGD for logistic regression, then constructs constrained Gaussian-mixture loss, stopping short of broader likelihood theory.
- Examples
- Uses binary logistic regression with synthetic two-class data and a Gaussian mixture density network trained on synthetic multivariate data.
- What you can do after
- Builds NLL objectives from scratch, optimizes logistic regression with SGD, and applies NLL in a Keras/TensorFlow mixture density network.
Read how Machine Learning Algorithms in Depth explains negative log-likelihood →
Ensemble Methods for Machine Learning
Negative log-likelihood is the negative logarithm of the likelihood assigned by a probabilistic model to observed data.
- How it is used
- Connects assumed residual distributions with regression losses and their optimization behavior.
- How deep it goes
- Defines the optimization role and compares distribution-linked losses, stopping short of detailed derivations.
- Examples
- Uses Gaussian, Laplacian, and Huber losses for continuous-valued targets, alongside comparisons across target types.
- What you can do after
- Evaluate how distribution assumptions change loss penalties, differentiability, outlier sensitivity, and boosting suitability.
Read how Ensemble Methods for Machine Learning explains negative log-likelihood →
Related concepts
- cross entropy lossequivalent to · Cross-entropy loss measures the expected negative log probability that a predicted distribution assigns to outcomes drawn from a target distribution.
- mean squared errorequivalent to · Mean squared error is the arithmetic mean of the squared differences between corresponding predicted and target numerical values.
- DPO lossused by · The DPO loss is a preference-training objective that optimizes a policy to assign higher relative likelihood to preferred responses than to rejected responses compared with a reference policy.
- NLLLossused by · NLLLoss is a PyTorch loss function that computes negative log-likelihood loss from log-probability outputs and target classes.
- mixture density networkused by · A mixture density network is a neural network that predicts the parameters of a conditional mixture probability distribution, optionally at each sequence step.
- variational lossy autoencodermeasures · A variational lossy autoencoder is an autoencoder variant that intentionally constrains its latent representation so reconstruction preserves selected information rather than every input detail.
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