5.1 Derivation of posterior distributions
7 min read•august 21, 2024
Deriving posterior distributions is a crucial skill in Bayesian statistics. It allows us to update our beliefs about parameters based on observed data, combining prior knowledge with new . This process forms the foundation for Bayesian inference, enabling us to quantify uncertainty and make informed decisions.
The derivation process involves identifying prior distributions, specifying likelihood functions, and calculating marginal likelihoods. Understanding , , and is essential for handling various scenarios. Proper interpretation of results, including and , is key to drawing valid conclusions.
Fundamentals of posterior distributions
- Posterior distributions form the cornerstone of Bayesian inference allowing updated beliefs about parameters based on observed data
- Combines prior knowledge with new evidence to yield a probability distribution over possible parameter values
- Enables quantification of uncertainty and facilitates decision-making in various fields (finance, medicine, engineering)
Definition of posterior distribution
Top images from around the web for Definition of posterior distribution
Bayes' theorem - Wikipedia View original
Is this image relevant?
Help me understand Bayesian prior and posterior distributions - Cross Validated View original
Is this image relevant?
Chapter 3 A Hands-on Example | Bayesian Basics View original
Is this image relevant?
Bayes' theorem - Wikipedia View original
Is this image relevant?
Help me understand Bayesian prior and posterior distributions - Cross Validated View original
Is this image relevant?
1 of 3
Top images from around the web for Definition of posterior distribution
Bayes' theorem - Wikipedia View original
Is this image relevant?
Help me understand Bayesian prior and posterior distributions - Cross Validated View original
Is this image relevant?
Chapter 3 A Hands-on Example | Bayesian Basics View original
Is this image relevant?
Bayes' theorem - Wikipedia View original
Is this image relevant?
Help me understand Bayesian prior and posterior distributions - Cross Validated View original
Is this image relevant?
1 of 3
- Probability distribution of parameters conditioned on observed data
- Represents updated beliefs after incorporating new information
- Expressed mathematically as
- Proportional to the product of likelihood and prior
Bayes' theorem review
- Fundamental principle for updating probabilities based on new evidence
- States
- Applied to parameter estimation becomes
- Allows inverse probability calculations crucial for inference
Components: prior, likelihood, evidence
- represents initial beliefs about parameters before observing data
- measures probability of observing data given parameter values
- Evidence normalizes ensuring it integrates to 1
- Relationship expressed as
Derivation process
- Deriving posterior distributions involves systematically combining prior knowledge with observed data
- Process requires careful specification of model components and mathematical manipulation
- Yields a probability distribution that can be used for inference and decision-making
Identifying prior distribution
- Select appropriate probability distribution to represent initial beliefs about parameters
- Consider domain knowledge, previous studies, or expert opinions
- Choose uninformative priors (uniform, Jeffreys) when little prior information exists
- Ensure prior distribution covers full range of plausible parameter values
Specifying likelihood function
- Define probability model for data generation process
- Express as function of parameters given observed data
- Common models include (Gaussian, Poisson, Binomial)
- Account for data collection methods and measurement uncertainties
Calculating marginal likelihood
- Compute evidence term
- Involves integrating product of likelihood and prior over all possible parameter values
- Often challenging to calculate analytically especially for complex models
- May require numerical approximation methods (Monte Carlo integration)
Conjugate priors
- Conjugate priors simplify posterior derivation by ensuring prior and posterior belong to same distribution family
- Play crucial role in Bayesian analysis by enabling closed-form solutions
- Facilitate sequential updating of beliefs as new data becomes available
Definition and importance
- Prior distribution yielding posterior of same functional form when combined with likelihood
- Simplifies calculations by avoiding complex integrals
- Allows for analytical solutions in many common scenarios
- Provides intuitive interpretation of prior as "pseudo-observations"
Common conjugate pairs
- with for proportion estimation
- with for rate parameter inference
- with Normal likelihood for mean estimation (known variance)
- Inverse-Gamma prior with Normal likelihood for variance estimation (known mean)
Advantages in derivation
- Closed-form expressions for posterior parameters
- Efficient updating of beliefs with new data
- Reduced computational complexity compared to numerical methods
- Facilitates interpretation of prior strength in terms of sample size
Analytical derivation techniques
- Analytical methods provide exact solutions for posterior distributions
- Require mathematical manipulation of probability density functions
- Yield closed-form expressions for posterior parameters and moments
- Often limited to specific combinations of priors and likelihoods
Integration methods
- Use calculus techniques to solve integrals in
- Apply substitution, integration by parts, or partial fractions
- Utilize special functions (Beta, Gamma) to simplify expressions
- Handle multidimensional integrals through iterated integration
Transformation of variables
- Change variables to simplify integration or distribution form
- Apply Jacobian determinant to maintain proper probability scaling
- Utilize logarithmic transformations for products of distributions
- Implement polar or spherical coordinates for multivariate problems
Moment generating functions
- Employ MGFs to derive posterior moments directly
- Utilize properties of expectation to simplify calculations
- Apply differentiation to obtain higher-order moments
- Facilitate derivation of mean, variance, and other summary statistics
Numerical approximation methods
- Numerical methods approximate posterior distributions when analytical solutions unavailable
- Enable handling of complex models and non-conjugate prior-likelihood pairs
- Provide flexible approaches for high-dimensional parameter spaces
- Trade-off between computational cost and accuracy of approximation
Importance sampling
- Generates samples from proposal distribution to estimate posterior
- Assigns weights to samples based on importance ratios
- Approximates expectations and integrals using weighted samples
- Effective for low-dimensional problems with well-chosen proposal distributions
Markov Chain Monte Carlo
- Constructs Markov chain with stationary distribution equal to target posterior
- Generates correlated samples through iterative algorithms (Metropolis-Hastings, )
- Provides asymptotically exact representation of posterior distribution
- Handles high-dimensional and complex posterior landscapes
Variational inference
- Approximates posterior with simpler, tractable distribution
- Minimizes Kullback-Leibler divergence between approximate and true posterior
- Offers faster convergence compared to MCMC for large-scale problems
- Provides lower bound on for model comparison
Posterior distribution properties
- Properties of posterior distributions provide insights into parameter estimates and uncertainties
- Enable quantification of credible intervals and prediction of future observations
- Facilitate comparison between prior and posterior beliefs
- Guide decision-making and hypothesis testing in Bayesian framework
Mean and variance
- Posterior mean represents of parameters
- Calculated as expected value
- Posterior variance quantifies uncertainty in parameter estimates
- Computed as
Credible intervals
- Provide range of plausible parameter values given observed data
- Calculated as intervals containing specified probability mass of posterior distribution
- 95% contains parameter with 0.95 probability
- Differ from frequentist confidence intervals in interpretation and calculation
Posterior predictive distribution
- Represents distribution of future observations given current data and model
- Calculated by integrating over posterior distribution of parameters
- Expressed as
- Used for model checking, outlier detection, and forecasting
Challenges in derivation
- Deriving posterior distributions often involves overcoming various technical and computational hurdles
- Requires careful consideration of model complexity, prior choices, and available computational resources
- Necessitates development of advanced techniques to handle challenging scenarios
- Drives ongoing research in Bayesian methodology and computational statistics
Non-conjugate priors
- Lack closed-form solutions for posterior distributions
- Require numerical approximation methods (MCMC, )
- Increase computational complexity of inference process
- May lead to challenges in interpreting and summarizing results
High-dimensional parameter spaces
- Suffer from curse of dimensionality in sampling and integration
- Require specialized MCMC algorithms (Hamiltonian Monte Carlo, No-U-Turn Sampler)
- Increase computational cost and convergence time
- Necessitate careful diagnostics to ensure reliable posterior estimates
Computational complexity
- Involves trade-offs between accuracy and computational resources
- Requires efficient algorithms for large-scale data and complex models
- May necessitate parallel computing or GPU acceleration
- Drives development of approximate inference methods (variational Bayes, expectation propagation)
Applications in Bayesian inference
- Bayesian inference using derived posterior distributions finds applications across various domains
- Enables robust decision-making under uncertainty
- Facilitates integration of prior knowledge with observed data
- Provides framework for continuous updating of beliefs as new information becomes available
Parameter estimation
- Infer unknown quantities in statistical models
- Provide point estimates (posterior mean, median) and uncertainty measures
- Handle complex hierarchical models with multiple levels of parameters
- Allow incorporation of domain expertise through informative priors
Model selection
- Compare competing models using Bayes factors or posterior model probabilities
- Account for model complexity through automatic Occam's razor effect
- Perform model averaging to combine predictions from multiple models
- Handle nested and non-nested model comparisons
Decision making
- Utilize posterior distributions to inform optimal decisions
- Minimize expected loss or maximize expected utility
- Account for parameter uncertainty in risk assessment
- Facilitate sequential decision-making in dynamic environments
Interpretation of results
- Proper interpretation of derived posterior distributions crucial for drawing valid conclusions
- Requires understanding of both statistical and domain-specific aspects
- Involves assessing practical significance alongside statistical measures
- Necessitates clear communication of results to stakeholders and decision-makers
Posterior vs prior comparison
- Assess how much beliefs have changed after observing data
- Quantify information gain using Kullback-Leibler divergence
- Visualize shifts in distribution shape, location, and spread
- Identify parameters most affected by new information
Uncertainty quantification
- Characterize parameter uncertainty through posterior standard deviations or credible intervals
- Assess impact of uncertainty on predictions and decisions
- Identify areas requiring additional data collection or model refinement
- Communicate uncertainty to stakeholders for informed decision-making
Sensitivity analysis
- Evaluate robustness of conclusions to prior choices and model assumptions
- Vary prior distributions to assess impact on posterior inferences
- Investigate sensitivity to likelihood function specification
- Identify critical assumptions driving results and potential areas of model misspecification
Key Terms to Review (28)
A/B Testing: A/B testing is a statistical method used to compare two versions of a variable to determine which one performs better. This technique is commonly applied in marketing, product development, and web design, where different versions (A and B) are presented to users, and their responses are analyzed. The goal is to make data-driven decisions based on the performance of each version, ensuring that changes lead to improved outcomes.
Analytical techniques: Analytical techniques refer to a set of mathematical methods and procedures used to derive insights and extract information from data. In the context of Bayesian statistics, these techniques are crucial for calculating posterior distributions, which involve updating prior beliefs with new evidence. They include methods for deriving formulas, conducting simulations, and performing numerical integration, all of which are essential for accurately modeling uncertainty and making inferences based on observed data.
Bayes' Theorem: Bayes' theorem is a mathematical formula that describes how to update the probability of a hypothesis based on new evidence. It connects prior knowledge with new information, allowing for dynamic updates to beliefs. This theorem forms the foundation for Bayesian inference, which uses prior distributions and likelihoods to produce posterior distributions.
Bayesian Updating: Bayesian updating is a statistical technique used to revise existing beliefs or hypotheses in light of new evidence. This process hinges on Bayes' theorem, allowing one to update prior probabilities into posterior probabilities as new data becomes available. By integrating the likelihood of observed data with prior beliefs, Bayesian updating provides a coherent framework for decision-making and inference.
Beta prior: A beta prior is a specific type of prior distribution used in Bayesian statistics, characterized by its flexible shape, which can take on various forms depending on its parameters. This distribution is particularly useful for modeling probabilities because it is defined on the interval [0, 1], making it ideal for representing beliefs about the success probability of Bernoulli trials. The beta prior serves as a conjugate prior for the binomial likelihood, simplifying the process of deriving posterior distributions.
Binomial Likelihood: Binomial likelihood refers to the probability of observing a given number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. This concept is crucial in Bayesian statistics for estimating parameters, as it forms the basis for deriving posterior distributions when combined with prior beliefs about those parameters.
Conjugate Priors: Conjugate priors are a type of prior distribution that, when combined with a certain likelihood function, results in a posterior distribution that belongs to the same family as the prior. This property simplifies the process of updating beliefs with new evidence, making calculations more straightforward and efficient. The use of conjugate priors is particularly beneficial when dealing with Bayesian inference, as it leads to easier derivation of posterior distributions and facilitates model comparison methods.
Credible Interval: A credible interval is a range of values within which an unknown parameter is believed to lie with a certain probability, based on the posterior distribution obtained from Bayesian analysis. It serves as a Bayesian counterpart to the confidence interval, providing a direct probabilistic interpretation regarding the parameter's possible values. This concept connects closely to the derivation of posterior distributions, posterior predictive distributions, and plays a critical role in making inferences about parameters and testing hypotheses.
Evidence: In the context of Bayesian statistics, evidence refers to the information or data that informs the likelihood of a hypothesis being true. It plays a crucial role in updating beliefs and making decisions based on observed data, influencing how we incorporate new information into our existing knowledge. Understanding evidence helps in calculating posterior probabilities, applying Bayes' theorem, and interpreting results in machine learning models.
Gamma prior: A gamma prior is a type of probability distribution used in Bayesian statistics, specifically as a prior for modeling positive continuous variables. It is particularly popular for parameters that are rates or scales, like the rate of events in a Poisson process or the scale parameter in an exponential distribution. The gamma prior is notable for being a conjugate prior, which means that when combined with certain likelihood functions, it yields a posterior distribution of the same family, simplifying calculations.
Gibbs Sampling: Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm used to generate samples from a joint probability distribution by iteratively sampling from the conditional distributions of each variable. This technique is particularly useful when dealing with complex distributions where direct sampling is challenging, allowing for efficient approximation of posterior distributions in Bayesian analysis.
Importance Sampling: Importance sampling is a statistical technique used to estimate properties of a particular distribution while only having samples generated from a different distribution. It allows us to focus computational resources on the most important areas of the sample space, thus improving the efficiency of estimates, especially in high-dimensional problems or when dealing with rare events. This method connects deeply with concepts of random variables, posterior distributions, Monte Carlo integration, multiple hypothesis testing, and Bayes factors by providing a way to sample efficiently and update beliefs based on observed data.
Likelihood Function: The likelihood function measures the plausibility of a statistical model given observed data. It expresses how likely different parameter values would produce the observed outcomes, playing a crucial role in both Bayesian and frequentist statistics, particularly in the context of random variables, probabilities, and model inference.
Marginal likelihood: Marginal likelihood refers to the probability of the observed data under a specific model, integrating over all possible parameter values. It plays a crucial role in Bayesian analysis, as it helps in model comparison and selection, serving as a normalization constant in the Bayes theorem. Understanding marginal likelihood is essential for determining how well a model explains the data, influencing various aspects such as the likelihood principle, the derivation of posterior distributions, and the computation of posterior odds.
Markov Chain Monte Carlo: Markov Chain Monte Carlo (MCMC) refers to a class of algorithms that use Markov chains to sample from a probability distribution, particularly when direct sampling is challenging. These algorithms generate a sequence of samples that converge to the desired distribution, making them essential for Bayesian inference and allowing for the estimation of complex posterior distributions and credible intervals.
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 works by constructing a Markov chain that has the desired distribution as its equilibrium distribution, allowing us to obtain samples that approximate this distribution even in complex scenarios. This algorithm is particularly valuable in deriving posterior distributions, as it enables the exploration of multi-dimensional spaces and the handling of complex models.
Normal prior: A normal prior is a type of probability distribution that expresses beliefs about a parameter before observing any data, characterized by its bell-shaped curve. This prior is particularly popular in Bayesian statistics due to its mathematical properties, making it easy to work with when deriving posterior distributions. Using a normal prior can help in situations where we assume the parameter being estimated follows a normal distribution, which can lead to convenient calculations and interpretations.
Numerical methods: Numerical methods are mathematical techniques used to approximate solutions for complex problems that cannot be solved analytically. They play a crucial role in computing, especially when deriving posterior distributions in Bayesian statistics, as they allow researchers to obtain practical estimates and understand uncertainties in their models.
Pierre-Simon Laplace: Pierre-Simon Laplace was a French mathematician and astronomer who made significant contributions to statistics, astronomy, and physics during the late 18th and early 19th centuries. He is renowned for his work in probability theory, especially for developing concepts that laid the groundwork for Bayesian statistics and formalizing the idea of conditional probability.
Point Estimate: A point estimate is a single value or statistic that is used to estimate an unknown parameter of a population. It represents the best guess of that parameter based on observed data and is often derived from a sample. In Bayesian statistics, the point estimate can be obtained from the posterior distribution, reflecting both prior beliefs and the evidence provided by the data.
Poisson likelihood: Poisson likelihood refers to the statistical model used for count data that describes the probability of a given number of events happening in a fixed interval of time or space, given a constant mean rate of occurrence. It is based on the Poisson distribution, which is characterized by its parameter $ extlambda$ that represents the average rate of events. In Bayesian analysis, the Poisson likelihood plays a crucial role in deriving posterior distributions when combined with prior information about the parameter.
Posterior Distribution: The posterior distribution is the probability distribution that represents the updated beliefs about a parameter after observing data, combining prior knowledge and the likelihood of the observed data. It plays a crucial role in Bayesian statistics by allowing for inference about parameters and models after incorporating evidence from new observations.
Posterior predictive distribution: The posterior predictive distribution is a probability distribution that provides insights into future observations based on the data observed and the inferred parameters from a Bayesian model. This distribution is derived from the posterior distribution of the parameters, allowing for predictions about new data while taking into account the uncertainty associated with parameter estimates. It connects directly to how we derive posterior distributions, as well as how we utilize them for making predictions about future outcomes.
Prior Distribution: A prior distribution is a probability distribution that represents the uncertainty about a parameter before any data is observed. It is a foundational concept in Bayesian statistics, allowing researchers to incorporate their beliefs or previous knowledge into the analysis, which is then updated with new evidence from data.
Sensitivity Analysis: Sensitivity analysis is a method used to determine how the variation in the output of a model can be attributed to different variations in its inputs. This technique is particularly useful in Bayesian statistics as it helps assess how changes in prior beliefs or model parameters affect posterior distributions, thereby informing decisions and interpretations based on those distributions.
Thomas Bayes: Thomas Bayes was an 18th-century statistician and theologian known for his contributions to probability theory, particularly in developing what is now known as Bayes' theorem. His work laid the foundation for Bayesian statistics, which focuses on updating probabilities as more evidence becomes available and is applied across various fields such as social sciences, medical research, and machine learning.
Uncertainty quantification: Uncertainty quantification is the process of quantifying the uncertainty in model predictions or estimations, taking into account variability and lack of knowledge in parameters, data, and models. This concept is crucial in Bayesian statistics, where it aids in making informed decisions based on probabilistic models, and helps interpret the degree of confidence we have in our predictions and conclusions across various statistical processes.
Variational Inference: Variational inference is a technique in Bayesian statistics that approximates complex posterior distributions through optimization. By turning the problem of posterior computation into an optimization task, it allows for faster and scalable inference in high-dimensional spaces, making it particularly useful in machine learning and other areas where traditional methods like Markov Chain Monte Carlo can be too slow or computationally expensive.