Bayesian statistics
Bayesian statistics is a framework for statistical inference that combines prior probability distributions with observed-data likelihoods to update probabilities over hypotheses or unknown quantities.
Grokking Machine Learning
A probability identity that expresses a conditional probability in terms of a reversed conditional probability and the relevant prior probabilities.
- How it is used
- Updates a classification probability when new evidence arrives, especially for medical tests and spam detection.
- How deep it goes
- Develops priors and posteriors through counted examples and probability trees, then implements a naive Bayes email classifier.
- Examples
- Uses a rare-illness test, an email containing “lottery,” and predictions from a real email dataset.
- What you can do after
- Build it from scratch as a Python naive Bayes spam predictor that returns probabilities for new emails.
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Math and Architectures of Deep Learning
A parameter-estimation approach that combines observed data with a prior distribution to produce a posterior distribution over unknown quantities.
- How it is used
- Estimates generative-model parameters and class probabilities when training data may be limited or unlabeled.
- How deep it goes
- Derives Gaussian, gamma, Wishart, and normal-gamma posterior updates, including multivariate estimation and PyTorch implementations.
- Examples
- Uses Gaussian height samples, Gaussian mixture classification, and univariate and multivariate Gaussian parameter estimation.
- What you can do after
- Apply it via PyTorch to update Gaussian-family posteriors and obtain MAP parameter estimates.
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Fabulous Adventures in Data Structures and Algorithms
Bayesian inference updates probability distributions over hypotheses or parameters using observed evidence and Bayes' rule.
- How it is used
- Updates uncertainty about causes from observed effects so software can make evidence-based decisions.
- How deep it goes
- Works through discrete priors, likelihoods, joint distributions, and posteriors, then implements continuous posterior sampling with Metropolis.
- Examples
- Tests a rare medical condition, updates coin hypotheses after ten heads, and samples coin-fairness uncertainty after nine heads in ten flips.
- What you can do after
- Build it from scratch using distribution types to update beliefs, and apply Metropolis sampling to a continuous posterior.
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Bayesian Optimization in Action
A Bayesian update is the revision of a prior belief into a posterior belief using observed data and a likelihood.
- How it is used
- Uses observed evidence to revise beliefs about quantities such as house prices or competing explanations.
- How deep it goes
- Defines priors, likelihoods, and posteriors with toy examples but treats Bayes’ theorem only at a high level.
- Examples
- Illustrates updates through house-price comparisons and repeated number guesses by a supposed psychic.
- What you can do after
- Apply Bayesian reasoning to combine an initial belief with evidence and interpret the resulting posterior.
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Causal AI
Bayesian inference is statistical inference that updates prior beliefs about unknown quantities using observed data and a likelihood model.
- How it is used
- Frames uncertainty for probabilistic modeling, causal inference, parameter estimation, and causal decision-making.
- How deep it goes
- Derives Bayes rule from probability axioms and connects it to factorization, marginalization, distributions, and subjective uncertainty.
- Examples
- Uses die rolls, coin flips, shuffled decks, speech production, and climate-driven ocean-current changes as uncertainty examples.
- What you can do after
- Evaluate uncertainty in probabilistic and causal models using Bayes rule, probability factorization, marginalization, and Bayesian estimation.
Sutskever's List
Bayes' rule updates the probability of a hypothesis from its prior probability and the likelihood of observed data under that hypothesis.
- How it is used
- Uses Bayesian updating to assign probabilities to candidate explanations during inductive inference for universal intelligence.
- How deep it goes
- Defines priors, likelihoods, and posteriors, notes prior controversy, and stops short of a formal derivation.
- Examples
- Uses a stream of coin flips to contrast fair, biased, and fixed-pattern hypotheses.
- What you can do after
- Evaluate competing hypotheses by reweighting prior plausibility according to fit with observed data.
Effective Data Analysis
Bayesian statistics is an approach to statistical inference that updates probability distributions for hypotheses or parameters using observed evidence.
- How it is used
- Frames hypotheses as numerical expectations about group differences for analytics and decision-oriented investigation.
- How deep it goes
- Defines probability as belief and demonstrates quantification, but defers broader probability and Bayesian-statistics treatment.
- Examples
- Uses simulated treatment and control distributions plotted as overlapping histograms.
- What you can do after
- Evaluate/understand hypotheses by estimating plausible group differences from research, organizational data, and stakeholder knowledge.
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Inside AI
Bayes' theorem is a probability formula for updating the probability of a hypothesis using new evidence and prior probabilities.
- How it is used
- Frames probabilistic reasoning as a way to understand how evidence changes beliefs and how events influence one another.
- How deep it goes
- Defines conditional-probability updating and illustrates network dependencies with a weather example.
- Examples
- Uses cloud cover and rain probabilities to illustrate conditional dependence.
- What you can do after
- Understand how evidence updates event probabilities and how variable dependencies can be represented.
Related concepts
- posterior probabilityproduces · A posterior probability or distribution expresses the probability assigned to an event, hypothesis, or unknown quantity conditional on observed evidence and prior information.
- conditional probabilityuses · Conditional probability is the probability of an event or variable value given that specified information, an event, or another variable’s value is known.
- causationcontrasts with · Causation is a relationship in which one event, condition, or factor produces or influences a change in another.
- prior probabilityuses · A prior probability distribution represents beliefs or assumptions about an unknown quantity before relevant observations or evidence are incorporated.
- naive bayes classifierused by · A Naive Bayes classifier is a probabilistic classification model that applies Bayes’ theorem while assuming features are conditionally independent given the class.
- probability theoryuses · Probability theory is the branch of mathematics that formalizes uncertainty and analyzes random events and outcomes using probability measures.
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