Glossary

9.2 Prior and posterior distributions

5 min read•august 13, 2024

Bayesian methods revolutionize statistical analysis by combining prior knowledge with observed data. This approach allows us to update our beliefs about parameters as new information comes in, providing a more nuanced understanding of uncertainty.

Prior and posterior distributions are the heart of . Priors represent our initial beliefs, while posteriors show our updated understanding after considering the data. This process lets us make more informed decisions based on all available information.

Prior Distributions in Bayesian Analysis

Concept and Purpose

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  • Prior distributions represent the initial beliefs or knowledge about the parameters of interest before observing the data
  • Priors can be based on subjective knowledge, previous studies, or expert opinions, allowing the incorporation of external information into the analysis
  • The choice of affects the posterior inference, especially when the sample size is small or the prior is informative (strong prior beliefs)
  • Priors can be used to regularize the model, prevent overfitting, and handle non-identifiability issues (multiple parameter values yielding the same likelihood)
  • The use of priors distinguishes Bayesian analysis from frequentist approaches, which rely solely on the observed data

Role in Bayesian Inference

  • Priors are combined with the , which represents the information from the observed data, to obtain the
  • The posterior distribution is the updated belief about the parameters after considering both the prior knowledge and the evidence from the data
  • Priors allow incorporating domain-specific knowledge or external information that is not captured by the data alone
  • The influence of the prior on the posterior inference depends on the relative strength of the prior and the likelihood (sample size and data informativeness)
  • Priors provide a formal way to quantify and update uncertainty about the parameters as more data becomes available

Choosing Prior Distributions

Types of Priors

  • are mathematically convenient as they result in posterior distributions from the same family as the prior, simplifying the computation (Beta prior for Bernoulli likelihood)
  • , such as uniform or Jeffreys priors, are used when there is little or no prior knowledge about the parameters, allowing the data to dominate the inference
  • Informative priors incorporate specific knowledge about the parameters and can be based on historical data, expert elicitation, or theoretical considerations (prior mean and variance for normal distribution)
  • introduce additional structure by placing priors on the of the prior distribution, allowing for more flexibility and borrowing of information across related parameters
  • combine multiple prior distributions to capture uncertainty or divergent beliefs about the parameters

Considerations for Prior Selection

  • The choice of prior should reflect the available information and the researcher's beliefs while being robust to misspecification and sensitive to the data
  • Priors should be chosen to avoid unintended consequences, such as inadvertently favoring certain parameter values or leading to improper posteriors
  • The prior's impact on the posterior inference should be carefully assessed, especially in small sample sizes or when the prior is highly informative
  • Sensitivity analysis can be performed to evaluate the robustness of the posterior inference to different prior choices and identify the prior's influence
  • In some cases, using multiple priors or model averaging techniques can help account for prior uncertainty and provide a more comprehensive analysis

Deriving Posterior Distributions

Bayes' Theorem

  • The posterior distribution is proportional to the product of the prior distribution and the likelihood function, as stated by : P(θ∣y)∝P(θ)×P(y∣θ)P(\theta|y) \propto P(\theta) \times P(y|\theta)
  • The likelihood function P(y∣θ)P(y|\theta) represents the probability of observing the data yy given the parameters θ\theta and is derived from the assumed statistical model
  • The prior distribution P(θ)P(\theta) encodes the initial beliefs or knowledge about the parameters before observing the data
  • The posterior distribution P(θ∣y)P(\theta|y) is obtained by normalizing the product of the prior and the likelihood to ensure it integrates to one: P(θ∣y)=P(θ)×P(y∣θ)∫P(θ)×P(y∣θ)dθP(\theta|y) = \frac{P(\theta) \times P(y|\theta)}{\int P(\theta) \times P(y|\theta) d\theta}

Computation Methods

  • In conjugate cases, where the prior and likelihood belong to the same family of distributions, the posterior distribution can be derived analytically using known mathematical properties (Beta-Bernoulli, Gamma-Poisson)
  • In non-conjugate cases, numerical methods like are used to approximate the posterior distribution by sampling from it iteratively
  • MCMC methods, such as the or the Gibbs sampler, construct a Markov chain that converges to the posterior distribution as its stationary distribution
  • is another approach that approximates the posterior distribution by minimizing the Kullback-Leibler divergence between a simpler variational distribution and the true posterior
  • can be used to approximate the posterior distribution with a Gaussian distribution centered at the mode of the posterior, which is useful for quick approximations

Interpreting Posterior Distributions

Summarizing Posterior Information

  • The posterior distribution represents the updated belief about the parameters after combining prior knowledge with the observed data
  • The , median, or mode can be used as point estimates for the parameters, depending on the shape of the distribution and the loss function (quadratic loss for mean, absolute loss for median)
  • Credible intervals can be constructed from the posterior distribution to quantify the uncertainty around the parameter estimates (95% highest posterior density interval)
  • The posterior distribution allows for probabilistic statements about the parameters, such as the probability of the parameter falling within a specific range or exceeding a threshold
  • Marginal posterior distributions can be obtained by integrating out other parameters to focus on the parameters of interest

Considerations for Interpretation

  • The interpretation of the posterior distribution should consider the prior assumptions, the model's limitations, and the data's quality and representativeness
  • Posterior inferences are conditional on the assumed model and the chosen prior, so model checking and sensitivity analysis are crucial for assessing the robustness of the conclusions
  • The posterior distribution provides a complete characterization of the uncertainty about the parameters, but it does not guarantee that the true parameter values are within the credible intervals
  • Bayesian hypothesis testing and model comparison can be performed using Bayes factors or posterior probabilities to assess the relative evidence for different hypotheses or models
  • The communication of posterior results should include the assumptions, limitations, and potential sources of uncertainty to facilitate accurate interpretation and decision-making

Key Terms to Review (27)

Bayes Factor: A Bayes Factor is a numerical value that quantifies the strength of evidence for one hypothesis over another, specifically in Bayesian statistical analysis. It compares the likelihood of the observed data under two competing hypotheses, often referred to as the null and alternative hypotheses. This concept is crucial when updating beliefs based on new evidence, as it helps in determining which hypothesis is more plausible given the data.
Bayes' Theorem: Bayes' Theorem is a mathematical formula used to update the probability of a hypothesis based on new evidence. It establishes a relationship between joint, marginal, and conditional probabilities, allowing us to make informed decisions by revising our beliefs when presented with new data. This theorem plays a crucial role in understanding how prior beliefs and new information interact, especially in Bayesian inference, where it is used to derive posterior distributions from prior distributions and observed data.
Bayesian inference: Bayesian inference is a statistical method that applies Bayes' theorem to update the probability of a hypothesis as more evidence or information becomes available. This approach emphasizes the importance of prior beliefs and knowledge, allowing for a systematic way to incorporate new data and refine predictions. The process involves calculating the posterior distribution, which combines prior distributions and likelihoods, enabling a coherent interpretation of uncertainty in the presence of incomplete information.
Bayesian Model Averaging: Bayesian Model Averaging (BMA) is a statistical method that accounts for model uncertainty by averaging predictions from multiple models, weighted by their posterior probabilities. This approach recognizes that there are often several plausible models for a given data set, and instead of selecting a single 'best' model, BMA incorporates the uncertainty associated with different models to produce more robust and accurate predictions.
Conjugate Priors: Conjugate priors are a specific type of prior distribution used in Bayesian statistics that, when combined with a likelihood function, results in a posterior distribution that is of the same family as the prior distribution. This property greatly simplifies the process of updating beliefs with new data because the form of the prior remains consistent throughout the analysis, allowing for easier mathematical manipulation and interpretation.
Credibility intervals: Credibility intervals are a Bayesian equivalent of confidence intervals, providing a range of values within which an unknown parameter is likely to fall, given the observed data and prior information. They reflect the uncertainty about the parameter's value based on both the data collected and the prior beliefs about the parameter's distribution. This concept is fundamental in Bayesian inference, where it helps to summarize the posterior distribution of parameters after considering prior information.
Evidence incorporation: Evidence incorporation refers to the process of integrating empirical data and prior beliefs into a formal statistical framework, particularly through the use of Bayesian methods. This approach emphasizes how prior distributions are updated with new evidence to produce posterior distributions, effectively combining what is already known with new information. This method is crucial for making informed decisions and predictions based on both historical data and recent observations.
Gibbs Sampling: Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for obtaining a sequence of observations approximating the joint probability distribution of multiple variables. It works by iteratively sampling from the conditional distributions of each variable given the current values of the other variables, thereby allowing for the estimation of complex posterior distributions, particularly when direct sampling is challenging.
Hierarchical Priors: Hierarchical priors are a statistical modeling approach where prior distributions are structured in a way that reflects relationships among parameters at different levels. This method allows for the sharing of information across groups or levels, improving estimation in situations with limited data and providing a more coherent framework for incorporating uncertainty. Hierarchical priors play a significant role in Bayesian statistics, particularly when dealing with multi-level data.
Hyperparameters: Hyperparameters are the parameters whose values are set before the learning process begins and control the training process of machine learning models. They are crucial as they determine the structure of the model, such as the learning rate, batch size, and the number of hidden layers. Adjusting these settings can greatly influence model performance and can lead to different outcomes in terms of accuracy and generalization to new data.
Informative Prior: An informative prior is a type of prior distribution in Bayesian statistics that incorporates existing knowledge or beliefs about a parameter before observing any data. This contrasts with a non-informative prior, which assumes no prior knowledge. Informative priors are crucial for shaping posterior distributions and play a significant role in Bayesian estimation and hypothesis testing, providing a framework that reflects previously established information.
Laplace Approximation: The Laplace approximation is a method used to approximate complex integrals, particularly in the context of Bayesian statistics. This technique is based on the idea that the posterior distribution can be approximated by a Gaussian distribution centered around the maximum a posteriori (MAP) estimate, simplifying the computation of posterior probabilities and integrals.
Likelihood Function: The likelihood function is a fundamental concept in statistics that measures the probability of observing the given data under various parameter values of a statistical model. It provides a way to estimate parameters by maximizing this function, indicating how likely the observed data is for different parameter settings. The likelihood function plays a crucial role in both maximum likelihood estimation and Bayesian inference, connecting it to the concepts of prior and posterior distributions.
Markov Chain Monte Carlo (MCMC): Markov Chain Monte Carlo (MCMC) is a class of algorithms used to sample from probability distributions when direct sampling is difficult. MCMC relies on constructing a Markov chain that has the desired distribution as its equilibrium distribution, allowing researchers to generate samples that approximate the target distribution. This technique is particularly useful in Bayesian analysis, where prior and posterior distributions play a crucial role in estimating parameters and testing hypotheses.
Metropolis-Hastings Algorithm: The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo (MCMC) method used for sampling from probability distributions when direct sampling is difficult. It allows for the approximation of posterior distributions by generating a sequence of samples that converge to the target distribution. This algorithm is particularly useful in Bayesian statistics, where prior distributions are updated to form posterior distributions based on observed data.
Mixture priors: Mixture priors are a type of prior distribution that combine multiple probability distributions, allowing for greater flexibility in modeling uncertainty about parameters. They can capture complex relationships by blending different distributions, which helps reflect diverse sources of information or beliefs. This approach is particularly useful when dealing with heterogeneous data or when the underlying parameter distributions are unknown.
Non-informative priors: Non-informative priors are prior probability distributions that provide minimal or no information about a parameter before observing any data. They are designed to let the data speak for themselves in Bayesian analysis, aiming to avoid influencing the posterior distribution. By using non-informative priors, analysts can express a lack of prior knowledge or belief about the parameter being estimated, allowing the evidence from the data to dominate the inference process.
Posterior Distribution: The posterior distribution represents the updated beliefs about a parameter after observing data, incorporating both the prior distribution and the likelihood of the observed data. It combines prior knowledge and new evidence to provide a complete picture of uncertainty around the parameter of interest. This concept is fundamental in Bayesian statistics, where it is used for estimation and hypothesis testing.
Posterior mean: The posterior mean is the expected value of a parameter given the observed data and prior information, representing a central tendency in Bayesian statistics. It is calculated by taking the average of the parameter estimates after incorporating the likelihood of the observed data with the prior distribution. This concept highlights the importance of updating beliefs based on new evidence, providing a powerful tool for inference in various applications.
Posterior median: The posterior median is a measure of central tendency that represents the middle value of the posterior distribution, effectively summarizing the updated beliefs about a parameter after observing data. It serves as a point estimate that is particularly useful in Bayesian statistics, where the posterior distribution combines prior information with new evidence. This statistic provides insights into the likely values of a parameter while also accounting for uncertainty.
Posterior mode: The posterior mode is a statistical estimate that identifies the mode of the posterior distribution, which represents the updated beliefs about a parameter after considering new evidence. This concept is essential in Bayesian statistics, as it allows for the determination of the most likely value of a parameter given prior beliefs and observed data. Understanding posterior mode helps in making informed decisions based on data while incorporating prior knowledge.
Posterior Predictive Checks: Posterior predictive checks are a diagnostic tool used in Bayesian statistics to assess the fit of a statistical model by comparing observed data with data simulated from the posterior predictive distribution. This technique allows researchers to evaluate how well a model predicts new data, providing insight into the model's validity and potential areas for improvement. By utilizing prior and posterior distributions, posterior predictive checks help ensure that the chosen model adequately captures the underlying data structure.
Posterior variance: Posterior variance is a measure of the uncertainty of an unknown parameter after observing data, derived from the posterior distribution in Bayesian statistics. It quantifies how much the estimates of this parameter vary once the prior beliefs are updated with new evidence, reflecting the influence of both the prior distribution and the likelihood function. The smaller the posterior variance, the more precise the estimation of the parameter becomes as additional data is incorporated.
Prior Distribution: A prior distribution is a probability distribution that represents our beliefs or knowledge about a parameter before observing any data. It plays a crucial role in Bayesian statistics, as it is combined with the likelihood of observed data to produce a posterior distribution, which updates our beliefs based on evidence. This concept highlights how our prior beliefs can influence statistical inference and decision-making.
Uninformative prior: An uninformative prior is a type of prior distribution used in Bayesian statistics that conveys little to no information about the parameter being estimated. This approach allows the data to play a more significant role in determining the posterior distribution, ensuring that the analysis remains unbiased by previous beliefs or assumptions.
Updating beliefs: Updating beliefs is the process of revising one’s prior knowledge or assumptions in light of new evidence, leading to a refined understanding or perspective. This concept is central to Bayesian inference, where prior distributions represent initial beliefs and posterior distributions reflect updated beliefs after considering observed data. This iterative process allows for more accurate predictions and decisions based on evolving information.
Variational Inference: Variational inference is a technique in Bayesian statistics used to approximate complex posterior distributions through optimization. It transforms the problem of sampling from a posterior distribution into an optimization problem, where the goal is to find the closest simpler distribution that can serve as an approximation. This method allows for efficient inference in large datasets and complex models, where traditional sampling methods like Markov Chain Monte Carlo (MCMC) may be computationally expensive.
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