Probability Distributions

Probability distributions form the backbone of many machine learning algorithms, determining model behavior and enabling uncertainty quantification:

Gaussian (Normal) Distribution: The foundation for numerous machine learning techniques, including linear regression, many neural network architectures, and various regularization approaches. In natural language processing, word embeddings often approximate Gaussian distributions. In reinforcement learning, Gaussian policies provide a natural way to balance exploration and exploitation.

Bernoulli and Binomial Distributions: Essential for binary classification problems and click-through prediction in recommendation systems. These distributions underlie logistic regression and inform evaluation metrics like precision and recall. In A/B testing for model deployment, they help establish statistical significance of conversion improvements.

Multinomial Distribution: Powers multi-class classification through categorical cross-entropy and softmax outputs in neural networks. Topic models like Latent Dirichlet Allocation use multinomial distributions to represent document-topic relationships. Text generation models often output multinomial probabilities over vocabulary tokens.

Exponential Family: This broader class of distributions connects to Generalized Linear Models, enabling the modeling of different response types. Natural gradient methods in optimization leverage the geometry of exponential family distributions for more efficient training.

Dirichlet Distribution: Serves as a prior for concentration parameters in many Bayesian models. In collaborative filtering, Dirichlet distributions help model user preference patterns. They're also crucial for variational inference in deep generative models.

Understanding these distributions helps in selecting appropriate algorithms, designing custom loss functions, and interpreting probabilistic outputs. For example, recognizing that linear regression assumes normally distributed errors guides when to apply transformations to skewed target variables or when to consider alternative models for heavy-tailed data.