12.1 Bayesian inference
3 min read•august 16, 2024
is a powerful approach to based on new evidence. It uses to combine prior knowledge with observed data, giving us a more complete picture of uncertainty in our conclusions.
This method stands out in probability theory for its ability to handle complex models and small sample sizes. It's especially useful in fields like machine learning and personalized medicine, where adaptive decision-making is key.
Bayesian Inference Principles
Fundamentals and Comparison to Frequentist Inference
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- Bayesian inference updates probabilities of hypotheses as more evidence becomes available, based on Bayes' theorem
- Interprets probability as a degree of belief or plausibility, not long-run frequency (frequentist approach)
- Incorporates prior knowledge through prior distributions, updated with observed data to form posterior distributions
- Focuses on distribution of parameters given observed data, unlike frequentist sampling distribution of estimators
- Allows direct probability statements about parameters and hypotheses, not relying on p-values or confidence intervals
- Handles small sample sizes and complex models effectively
- Facilitates sequential learning and decision-making under uncertainty (machine learning, adaptive experimental design)
Key Components and Advantages
- represents initial beliefs about parameters before observing data
- quantifies probability of observing data given particular parameter values
- combines prior knowledge with observed data, providing updated parameter beliefs
- Naturally accounts for parameter uncertainty in estimates and decision-making
- Provides more intuitive and interpretable results in many cases
- Useful in fields requiring adaptive learning (robotics, personalized medicine)
Bayes' Theorem Application
Formula and Components
- Bayes' theorem expresses of hypothesis given observed data:
- P(H): (initial beliefs about hypothesis)
- P(D|H): Likelihood (probability of data given hypothesis)
- P(D): (normalizing constant ensuring posterior integrates to 1)
- P(H|D): Posterior probability (updated beliefs about hypothesis)
Practical Considerations
- simplify calculations by resulting in posterior distributions of same family as prior
- Example: Beta prior with binomial likelihood yields beta posterior
- or complex models may require numerical methods (MCMC sampling)
- allows incorporation of new data without reprocessing all previous data
- Example: Updating product quality estimates in manufacturing as new batches are tested
Credible Intervals for Parameter Estimation
Types and Interpretation
- provide range of plausible parameter values given observed data and prior beliefs
- Direct probabilistic interpretation: parameter lies within interval with specified probability
- contains most probable parameter values
- Useful for asymmetric posterior distributions (skewed data)
- contain central (1-α)% of posterior distribution
- α: desired significance level (0.05 for 95% credible interval)
Properties and Applications
- Naturally account for parameter uncertainty in estimates
- Width reflects precision of parameter estimate (narrower intervals indicate more precise estimates)
- Allow for joint inference on multiple parameters, capturing dependencies
- Used for decision-making and hypothesis testing in Bayesian frameworks
- Example: Determining if a new drug is more effective than a placebo based on overlap of credible intervals
Bayesian Inference with MCMC Sampling
MCMC Methods and Algorithms
- samples from complex probability distributions, approximating posterior distributions
- : general MCMC method for sampling from any target distribution
- Proposes new parameter values and accepts/rejects based on posterior ratio
- : effective for multivariate distributions with easily sampled conditional distributions
- Updates one parameter at a time, conditioning on current values of other parameters
- : uses gradient information to improve sampling efficiency in high-dimensional spaces
- Particularly useful for complex hierarchical models
Implementation and Diagnostics
- Convergence diagnostics assess reliability of MCMC results:
- Trace plots: visual inspection of parameter values over iterations
- Autocorrelation plots: check for independence between samples
- Gelman-Rubin statistic: compares within-chain and between-chain variances
- Posterior predictive checks compare simulated data from posterior to observed data
- Helps assess model fit and identify potential issues
- Software packages (Stan, PyMC, JAGS) provide MCMC implementations and Bayesian inference tools
- Facilitate practical applications in various fields (ecology, finance, social sciences)
Key Terms to Review (24)
Bayes' Theorem: Bayes' Theorem is a mathematical formula used to update the probability of a hypothesis based on new evidence, allowing us to revise prior beliefs when presented with new data. This theorem connects various concepts in probability, such as conditional probability and independence, by demonstrating how to compute conditional probabilities when dealing with joint distributions or mass functions.
Bayesian Decision Theory: Bayesian Decision Theory is a statistical framework that combines Bayesian inference with decision-making under uncertainty. It helps in making optimal decisions based on prior beliefs and observed evidence, allowing for the incorporation of both subjective judgment and empirical data. This approach is particularly useful when faced with incomplete information, as it quantifies the uncertainty surrounding various choices and evaluates the expected outcomes to guide decision-making.
Bayesian Inference: Bayesian inference is a statistical method that updates the probability for a hypothesis as more evidence or information becomes available. This approach combines prior knowledge with new data to make probabilistic inferences, allowing for a flexible framework that incorporates both existing beliefs and observed evidence. The method hinges on Bayes' theorem, which relates the conditional and marginal probabilities of random events.
Conjugate Priors: Conjugate priors are a specific type of prior probability distribution that, when combined with a certain likelihood function, results in a posterior distribution that is in the same family as the prior distribution. This relationship simplifies the computation of the posterior and provides an elegant way to update beliefs based on new evidence. Using conjugate priors facilitates Bayesian inference by maintaining consistency in the form of distributions throughout the updating process.
Credible Intervals: Credible intervals are a Bayesian counterpart to frequentist confidence intervals, providing a range of values within which a parameter is believed to lie with a certain probability based on prior information and observed data. Unlike confidence intervals, which can be misunderstood as probabilities about the parameter itself, credible intervals allow for direct probability statements about the parameter, making them particularly useful in Bayesian inference. This concept emphasizes the subjective nature of probability in Bayesian statistics, reflecting beliefs updated with new evidence.
Equal-tailed credible intervals: Equal-tailed credible intervals are a type of Bayesian interval estimation that provides a range for a parameter where the probability of the true parameter value lying within the interval is equal to a specified level of confidence, typically 95%. This means that the tails of the distribution are symmetric, so there is an equal probability of the parameter being below or above the interval. They are commonly used in Bayesian inference to summarize uncertainty about parameter estimates.
Evidence incorporation: Evidence incorporation refers to the process of integrating new evidence or information into a pre-existing framework of beliefs or knowledge. In the context of Bayesian inference, this concept plays a critical role in updating prior beliefs based on observed data, allowing for more informed decision-making and predictions. The essence of evidence incorporation lies in its ability to quantitatively adjust probabilities as new data is acquired, leading to a dynamic understanding of uncertainty.
Expected Utility: Expected utility is a concept in decision theory and economics that represents the anticipated satisfaction or benefit derived from a particular choice or action, calculated by weighing each possible outcome by its probability and the utility associated with that outcome. This approach helps individuals make rational choices under uncertainty by considering not just the potential gains but also their likelihood, allowing for a more informed decision-making process. Expected utility plays a significant role in Bayesian inference, where probabilities are updated based on new evidence to guide optimal choices.
Gibbs Sampling: Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm used to generate samples from a probability distribution, especially when the joint distribution is known but difficult to sample from directly. This technique allows for the estimation of posterior distributions in Bayesian inference by iteratively sampling from the conditional distributions of each variable given the others, making it particularly useful in high-dimensional spaces.
Hamiltonian Monte Carlo (HMC): Hamiltonian Monte Carlo (HMC) is a sophisticated sampling method used in Bayesian inference that leverages concepts from physics, specifically Hamiltonian dynamics, to efficiently explore the posterior distribution of parameters. By simulating the movement of particles in a potential energy landscape, HMC can generate samples that are correlated and more representative of the target distribution. This method enhances the efficiency of Markov Chain Monte Carlo (MCMC) techniques by minimizing random walk behavior and improving convergence.
Highest posterior density (hpd) interval: The highest posterior density (hpd) interval is a credible interval used in Bayesian statistics that contains the most probable parameter values given the observed data. This interval represents the range within which the true parameter value is likely to lie with a specified probability, and it is constructed to include the highest density regions of the posterior distribution. The hpd interval is especially important because it provides a way to summarize uncertainty about parameter estimates derived from Bayesian inference.
Laplace: Laplace refers to Pierre-Simon Laplace, a prominent French mathematician and astronomer, known for his significant contributions to probability theory and statistics. His work laid the foundation for various concepts in statistical inference and moment-generating functions, helping to formalize how probabilities are calculated and interpreted in different contexts.
Likelihood Function: The likelihood function is a mathematical representation that measures the probability of observing the given data under various parameter values in a statistical model. It plays a critical role in estimation methods, allowing statisticians to determine which parameters make the observed data most probable. This function is particularly significant in methods such as maximum likelihood estimation and Bayesian inference, where it helps update beliefs about model parameters based on observed data.
Marginal Likelihood: Marginal likelihood refers to the probability of observing the data under a specific model, integrating out the parameters of that model. It plays a crucial role in Bayesian inference as it helps to compare different models based on how well they explain the observed data, allowing for model selection and evaluation. By focusing on the likelihood of the data while accounting for uncertainty in the parameters, marginal likelihood serves as a fundamental tool in assessing the plausibility of various hypotheses.
Markov Chain Monte Carlo (MCMC): Markov Chain Monte Carlo (MCMC) is a class of algorithms used for sampling from a probability distribution when direct sampling is difficult. It relies on constructing a Markov chain that has the desired distribution as its equilibrium distribution, allowing for efficient exploration of high-dimensional spaces, making it a crucial tool in Bayesian inference for estimating posterior distributions.
Metropolis-Hastings Algorithm: The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo (MCMC) method used to generate samples from a probability distribution when direct sampling is challenging. It is particularly useful in Bayesian inference for approximating posterior distributions, allowing for the estimation of complex models where analytical solutions are not feasible.
Non-conjugate priors: Non-conjugate priors are prior probability distributions that do not form a conjugate pair with the likelihood function in Bayesian inference. This means that when you combine them with the likelihood to get the posterior distribution, the resulting posterior does not belong to the same family of distributions as the prior. Non-conjugate priors are important as they allow for greater flexibility in modeling, accommodating complex situations where conjugate priors may be too restrictive.
Posterior distribution: The posterior distribution represents the updated beliefs about a parameter after observing data, calculated using Bayes' theorem. It combines prior information about the parameter with evidence from observed data, resulting in a new probability distribution that reflects both prior knowledge and the likelihood of the observed data. This concept is central to Bayesian inference, as it allows for continuous updating of beliefs as more data becomes available.
Posterior probability: Posterior probability is the probability of an event occurring after taking into account new evidence or information. It is a fundamental concept in Bayesian inference, where prior beliefs are updated in light of new data to form a revised belief about a hypothesis. This updated probability reflects the incorporation of evidence into the decision-making process, showcasing how new information can change our understanding of uncertainty.
Prior distribution: A prior distribution represents the initial beliefs or assumptions about a parameter before any evidence or data is taken into account. In Bayesian inference, it serves as the foundation for updating beliefs when new data is observed, allowing for a systematic approach to incorporating prior knowledge into statistical analysis.
Prior probability: Prior probability is the probability assigned to a hypothesis before any evidence is considered. It serves as the initial degree of belief about the truth of that hypothesis, forming the foundation for updating beliefs in light of new evidence through Bayesian inference. This initial probability can be based on previous knowledge, expert opinion, or historical data.
Sequential updating: Sequential updating is a method in Bayesian inference where probabilities are adjusted as new evidence becomes available, allowing for the continuous refinement of beliefs. This process involves revising prior beliefs using the likelihood of new data, leading to updated posterior probabilities. The approach reflects the dynamic nature of information, emphasizing how earlier assumptions can evolve with incoming data, which is central to effective decision-making in uncertain environments.
Thomas Bayes: Thomas Bayes was an 18th-century statistician and theologian known for his work in probability theory, particularly in developing what is now known as Bayesian inference. His most significant contribution is Bayes' theorem, which provides a mathematical framework for updating beliefs based on new evidence. This theorem has become foundational in statistical reasoning, influencing how data is analyzed and interpreted in various fields, including science, finance, and artificial intelligence.
Updating beliefs: Updating beliefs refers to the process of adjusting one's prior beliefs or hypotheses in light of new evidence or information. This concept is central to Bayesian inference, where individuals revise their probability estimates based on observed data, allowing for a dynamic approach to understanding uncertainty and making informed decisions.