Posterior probability
A posterior probability or distribution expresses the probability assigned to an event, hypothesis, or unknown quantity conditional on observed evidence and prior information.
Grokking Machine Learning
A posterior probability is the probability assigned to an event or hypothesis after incorporating observed evidence.
- How it is used
- Uses observed evidence to refine an initial probability for medical diagnosis, weather estimation, and spam detection.
- How deep it goes
- Defines prior, event, and posterior, then works through counts and a naive Bayes implementation without deriving Bayes’ theorem formally.
- Examples
- Uses rain in the Amazon, medical testing, lottery emails, and a real email dataset with Python predictions.
- What you can do after
- Apply it via naive Bayes to estimate whether emails are spam from their observed words.
Read how Grokking Machine Learning explains posterior probability →
Math and Architectures of Deep Learning
The updated distribution over model parameters after combining prior and likelihood.
- How it is used
- Combines prior beliefs and observed training data for Bayesian parameter estimation and latent-variable inference.
- How deep it goes
- Derives posterior families and MAP estimates for Gaussian models, then defers intractable VAE normalization to ELBO approaches.
- Examples
- Uses Gaussian mean and precision estimation, GMM component responsibilities, Neighborville height data, and VAE latent encodings.
- What you can do after
- Apply it via PyTorch distributions to compute posterior parameter distributions and extract MAP estimates for Gaussian models.
Read how Math and Architectures of Deep Learning explains posterior probability →
Fabulous Adventures in Data Structures and Algorithms
A posterior distribution is a probability distribution representing beliefs about an unknown quantity after incorporating observed evidence.
- How it is used
- Updates beliefs about possible causes after observing effects, supporting evidence-based diagnosis and uncertainty estimates.
- How deep it goes
- Derives the discrete update through Bayes’ theorem and distribution operations, then implements continuous posterior sampling with Metropolis.
- Examples
- Uses a cold-and-sneeze survey, a rare-disease test, repeated coin flips, and sampled coin fairness.
- What you can do after
- Build it from scratch with discrete distribution operations, or apply Metropolis to sample a continuous posterior from a prior and likelihood.
Read how Fabulous Adventures in Data Structures and Algorithms explains posterior probability →
Bayesian Optimization in Action
A posterior distribution is a probability distribution representing updated beliefs about unknown quantities after incorporating observed data.
- How it is used
- Uses updated uncertainty to learn from observations in Gaussian processes and compare informative search actions.
- How deep it goes
- Defines posterior updates and works through Gaussian conditioning effects, but defers the closed-form derivation.
- Examples
- Illustrates updates with house prices, correlated variables, repeated psychic guesses, and a remote-control search.
- What you can do after
- Evaluate/understand how relevant observations shift predictions and reduce uncertainty under a Bayesian model.
Read how Bayesian Optimization in Action explains posterior probability →
Build a Robo-Advisor with Python (From Scratch)
A posterior distribution is a probability distribution representing updated beliefs about an unknown quantity after incorporating data or observations.
- How it is used
- Updates equilibrium returns with uncertain investor views for Black–Litterman asset allocation.
- How deep it goes
- Defines Bayesian updating, gives multivariate normal posterior formulas and Python functions, and defers their mathematical derivation.
- Examples
- Uses an NBA-and-height conditional-probability scenario and uncertain views about future asset returns.
- What you can do after
- Apply it via NumPy functions to calculate posterior return means and covariances for allocation optimization.
Read how Build a Robo-Advisor with Python (From Scratch) explains posterior probability →
AI-Powered Search
A posterior distribution is a probability distribution representing an updated belief after incorporating observations into a prior distribution.
- How it is used
- Deploys posterior probability to stabilize document relevance estimates in SDBN click-model ranking.
- How deep it goes
- Explains priors, parameter updates, posterior means, and prior-strength tuning, then stops short of other click-model implementations.
- Examples
- Uses blue-ray query results, document click counts, beta-adjusted dryer grades, and Transformers soundtrack relevance.
- What you can do after
- Apply it via a beta-distribution calculation to update document relevance from clicks and examinations.
Machine Learning Algorithms in Depth
A posterior distribution is the probability distribution of unknown quantities after incorporating observed data into prior beliefs.
- How it is used
- Uses the posterior to represent beliefs about model parameters after observing data.
- How deep it goes
- Derives the prior–likelihood relationship and normalization, then motivates approximation when high-dimensional integration is intractable.
- Examples
- Uses a sequential Bayes-engine figure and a Dirichlet–multinomial posterior exercise.
- What you can do after
- Practice/perform a Dirichlet–multinomial posterior calculation and distinguish MCMC sampling from variational approximation.
Read how Machine Learning Algorithms in Depth explains posterior probability →
Causal AI
A posterior distribution is the conditional probability distribution of unknown quantities given observed data and prior assumptions.
- How it is used
- Uses observed data and prior assumptions to express uncertainty about unknown parameters for prediction and inference.
- How deep it goes
- Defines posterior distributions, priors, hyperparameters, and latent-variable targets, but does not detail computational inference algorithms.
- Examples
- Uses parameter values, posterior samples, observed variables, and joint parameter–latent-variable distributions as illustrative artifacts.
- What you can do after
- Evaluate parameter estimates by representing their uncertainty and incorporating it into predictions or other inferences.
Related concepts
- bayesian statisticsproduced by · 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.
- prior probabilityproduced by · A prior probability distribution represents beliefs or assumptions about an unknown quantity before relevant observations or evidence are incorporated.
- Bayes' theoremproduced by · Bayes' theorem is a probability rule that expresses a conditional probability in terms of a prior probability, a likelihood, and the evidence.
- markov chain monte carloused by · Markov chain Monte Carlo is a family of sampling algorithms that generate samples using a Markov chain whose stationary or limiting distribution is a target probability distribution.
- fully bayes parameter estimationproduced by · Fully Bayesian parameter estimation combines a prior distribution with observed-data likelihood to produce a posterior distribution over unknown parameters.
These guides are generated from the text of Manning books by Manning's knowledge graph and link to the sections they draw on. The full books are available on liveBook and at manning.com.