15.2 Prior and Posterior Distributions
3 min read•august 9, 2024
Bayes' Theorem is the backbone of Bayesian inference, connecting prior beliefs, , and posterior probabilities. It's used in various fields to update our understanding of events based on new evidence.
Prior and posterior distributions are key players in Bayesian analysis. Priors represent initial beliefs, while posteriors show updated beliefs after considering new data. This process helps refine our knowledge and make better decisions.
Bayes' Theorem and Probability Concepts
Fundamental Components of Bayesian Inference
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- Bayes' Theorem forms the foundation of Bayesian inference, expressing the relationship between conditional probabilities
- Mathematically represented as
- denotes the initial belief about an event before observing new evidence
- Likelihood measures the probability of observing the data given a specific hypothesis or parameter value
- represents the updated belief after incorporating new evidence, combining prior knowledge with observed data
Applications and Interpretations
- Bayes' Theorem applies to various fields (medical diagnoses, spam filtering, machine learning)
- Prior probability can be based on historical data, expert knowledge, or when no prior information exists
- quantifies how well different parameter values explain the observed data
- Posterior probability provides a probabilistic estimate of the true parameter value given the observed data and prior beliefs
Types of Priors
Conjugate Priors and Their Properties
- Conjugate priors belong to the same probability distribution family as the posterior distribution
- Simplify Bayesian calculations by allowing closed-form solutions for the posterior distribution
- Common pairs include (, , )
- serves as a conjugate prior for binomial and Bernoulli likelihoods
- Beta distribution defined on the interval [0, 1], making it suitable for modeling probabilities
Informative and Uninformative Priors
- Informative priors incorporate existing knowledge or beliefs about the parameter of interest
- Can be based on expert opinion, previous studies, or theoretical considerations
- Uninformative priors aim to have minimal impact on the posterior distribution
- Uniform distribution often used as an when no prior information exists
- represents another type of uninformative prior, invariant under reparameterization
Posterior Analysis
Interpreting Posterior Distributions
- Credible intervals provide a range of plausible values for the parameter of interest
- Typically reported as 95% credible intervals, containing 95% of the posterior probability mass
- represents the distribution of future observations given the observed data
- Enables predictions and uncertainty quantification for new, unobserved data points
- allows for sequential incorporation of new data to refine parameter estimates
Estimation and Decision Making
- estimation finds the parameter value that maximizes the posterior probability
- MAP estimation balances prior beliefs with observed data to produce point estimates
- Can be viewed as a regularized version of maximum likelihood estimation
- and median serve as alternative point estimates derived from the posterior distribution
- uses posterior distributions to make optimal decisions under uncertainty
Key Terms to Review (20)
Bayesian Decision Theory: Bayesian Decision Theory is a statistical framework that incorporates prior knowledge, evidence, and the consequences of decisions to guide the process of making optimal choices under uncertainty. This approach combines prior distributions, which represent beliefs before observing data, and posterior distributions, which are updated beliefs after considering new evidence. The decision-making process emphasizes minimizing expected loss or maximizing expected utility based on these distributions.
Bayesian Updating: Bayesian updating is a statistical method that involves revising the probability estimate for a hypothesis as additional evidence or information becomes available. This process connects prior beliefs with new data, leading to updated beliefs known as posterior probabilities. It is essential in understanding how to incorporate uncertainty and improve decision-making based on new information.
Beta distribution: The beta distribution is a continuous probability distribution defined on the interval [0, 1], characterized by two shape parameters, α (alpha) and β (beta), which determine the distribution's shape. It is widely used in statistics, particularly in Bayesian analysis, to model random variables that are constrained within a finite interval, making it highly relevant in various applications including estimating probabilities and defining prior distributions.
Beta-Binomial: The beta-binomial distribution is a discrete probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, where the success probability is not constant but follows a beta distribution. This allows for modeling situations where the underlying probability of success varies, incorporating prior beliefs about the parameter through its connection to prior and posterior distributions.
Conjugate Prior: A conjugate prior is a prior distribution that, when combined with a likelihood function through Bayes' theorem, results in a posterior distribution that belongs to the same family as the prior. This concept simplifies the process of updating beliefs with new data, making it easier to derive the posterior distribution analytically. Using conjugate priors streamlines calculations and helps maintain consistency across Bayesian inference.
Credible Interval: A credible interval is a range of values within which an unknown parameter lies with a specified probability, according to a Bayesian framework. This concept is closely linked to prior and posterior distributions, as it utilizes the information provided by the prior beliefs about the parameter and updates these beliefs with observed data to form the posterior distribution. In essence, credible intervals provide a way to summarize uncertainty about an estimate after taking into account prior knowledge and evidence from new data.
Gamma-Poisson: The gamma-Poisson model is a Bayesian statistical approach used to model count data, where the count variable follows a Poisson distribution and the rate parameter itself is treated as a random variable with a gamma distribution. This framework provides a flexible method for incorporating uncertainty about the rate of occurrence, allowing for more robust predictions and inferences in situations with limited data. It highlights the relationship between prior and posterior distributions, as the gamma distribution serves as a conjugate prior for the Poisson likelihood.
Informative Prior: An informative prior is a type of prior distribution in Bayesian statistics that incorporates specific, relevant information or beliefs about a parameter before observing data. This approach contrasts with non-informative priors, which aim to have minimal influence on the posterior distribution. Informative priors are often derived from previous studies, expert opinions, or established theories, making them crucial for situations where limited data is available but prior knowledge is valuable.
Jeffreys Prior: Jeffreys Prior is a non-informative prior distribution used in Bayesian statistics that is based on the concept of invariant measures under reparameterization. It provides a way to assign prior probabilities in a manner that reflects the underlying parameter space without imposing strong subjective beliefs. This prior is particularly useful when there is little or no prior information available about the parameters being estimated, making it widely applicable in various statistical models.
Likelihood: Likelihood is a statistical concept that measures how well a particular model or hypothesis explains the observed data. It plays a crucial role in Bayesian inference, allowing us to update our beliefs about the parameters of a model based on new evidence. The likelihood function quantifies the probability of the observed data given specific parameter values, bridging the gap between prior beliefs and posterior conclusions.
Likelihood Function: The likelihood function is a mathematical function that measures the plausibility of a statistical model given specific observed data. It provides a way to update beliefs about model parameters based on new data, making it a cornerstone in both frequentist and Bayesian statistics, especially in estimating parameters and making inferences about distributions.
Maximum a posteriori (MAP): Maximum a posteriori (MAP) estimation is a statistical technique used to estimate an unknown parameter by maximizing the posterior distribution. This approach combines prior beliefs about the parameter with observed data, allowing for a more informed estimate. In essence, MAP provides a way to incorporate prior knowledge into statistical inference, resulting in estimates that are not solely reliant on the observed data.
Normal-normal: Normal-normal refers to a scenario in Bayesian statistics where both the prior and the likelihood (the data model) are modeled using normal distributions. This creates a situation where the posterior distribution, after updating the prior with the data, also results in a normal distribution. This property makes normal-normal pairs particularly convenient in Bayesian analysis since they allow for easy interpretation and computation.
Posterior Mean: The posterior mean is the expected value of a random variable given new evidence, reflecting the average of all possible outcomes based on prior beliefs and updated data. This concept is central in Bayesian statistics as it combines prior distributions and likelihoods to create a posterior distribution, which helps in making informed decisions and estimates after observing data.
Posterior median: The posterior median is the value that separates the highest half of a posterior distribution from the lowest half, serving as a robust measure of central tendency in Bayesian statistics. It provides a point estimate for an unknown parameter after considering both the prior distribution and the likelihood of observed data. This concept is closely tied to prior and posterior distributions, emphasizing how prior beliefs are updated with new information, as well as to Bayesian estimation, where it is often used to create credible intervals that provide a range for where the true parameter likely lies.
Posterior predictive distribution: The posterior predictive distribution is the distribution of future observations given the data already observed and the parameters estimated from that data. It combines the information from both the prior distribution and the likelihood of the observed data to predict new data points, reflecting uncertainty about the model parameters. This concept plays a crucial role in Bayesian statistics, particularly in making predictions and constructing credible intervals around those predictions.
Posterior Probability: Posterior probability is the probability of a hypothesis being true after observing new evidence, calculated using Bayes' Theorem. This term highlights the importance of updating our beliefs in light of new data, making it essential for Bayesian inference and understanding how prior beliefs evolve when new information is introduced.
Prior Probability: Prior probability is the initial estimation of the likelihood of an event or hypothesis before considering new evidence. It forms the backbone of Bayesian inference, serving as the starting point for updating beliefs based on observed data. Understanding prior probability is crucial as it influences the posterior probability, which combines prior beliefs with new evidence to refine predictions or conclusions.
Uniform Distribution: Uniform distribution is a probability distribution where all outcomes are equally likely within a certain range. This means that every value in the defined interval has the same chance of occurring, leading to a flat, even graph when plotted. Understanding uniform distribution helps in grasping the basics of probability and serves as a foundation for comparing it to other distributions like normal distribution and for understanding prior and posterior distributions in Bayesian statistics.
Uninformative prior: An uninformative prior is a type of prior distribution in Bayesian statistics that is designed to have minimal influence on the posterior distribution. This prior is often used when there is little to no prior knowledge about the parameter being estimated, allowing the data to play a dominant role in shaping the results. By using an uninformative prior, analysts aim to avoid biasing their conclusions while still incorporating Bayesian principles into their analysis.