📊Bayesian Statistics Unit 3 – Prior distributions

What Are Prior Distributions?

• Prior distributions represent the initial beliefs or knowledge about a parameter before observing data
• Encapsulate subjective or objective information available before conducting an experiment or analysis
• Mathematically, a prior distribution is a probability distribution that expresses the uncertainty about the parameter of interest
• Denoted as $P(\theta)$, where $\theta$ represents the parameter
• Play a crucial role in Bayesian inference by combining with the likelihood function to obtain the posterior distribution
• Allow incorporating domain expertise, historical data, or theoretical considerations into the statistical analysis
• Enable a more comprehensive and informative approach to parameter estimation compared to frequentist methods

Types of Prior Distributions

• Conjugate priors result in a posterior distribution belonging to the same family as the prior distribution
  • Simplify the computation of the posterior distribution
  • Examples include Beta prior for Bernoulli likelihood, Gamma prior for Poisson likelihood
• Noninformative priors aim to minimize the influence of prior knowledge on the posterior distribution
  • Represent a state of ignorance or lack of strong prior beliefs
  • Commonly used noninformative priors include uniform distribution and Jeffreys prior
• Informative priors incorporate specific knowledge or beliefs about the parameter
  • Derived from domain expertise, previous studies, or theoretical considerations
  • Assign higher probabilities to parameter values considered more likely based on prior information
• Improper priors are not valid probability distributions but can still lead to proper posterior distributions
  • Integrate to infinity over the parameter space
  • Require careful handling to ensure the resulting posterior is proper

Choosing the Right Prior

• Consider the available prior information and its reliability
  • Incorporate strong prior beliefs when supported by solid evidence or expertise
  • Use noninformative priors when prior knowledge is limited or to let the data speak for itself
• Assess the sensitivity of the posterior distribution to the choice of prior
  • Conduct sensitivity analysis by comparing results obtained from different priors
  • Ensure that the posterior is robust to reasonable variations in the prior distribution
• Balance the influence of the prior with the strength of the observed data
  • With large sample sizes, the likelihood dominates, and the impact of the prior diminishes
  • With small sample sizes, the prior has a more substantial effect on the posterior
• Consider the computational tractability and convenience of the chosen prior
  • Conjugate priors offer analytical solutions and faster computations
  • More complex priors may require advanced sampling techniques like Markov Chain Monte Carlo (MCMC)

Conjugate Priors

• Conjugate priors combine with the likelihood function to yield a posterior distribution of the same family
• Provide analytical tractability and computational convenience in Bayesian inference
• Examples of conjugate priors include:
  • Beta prior for Bernoulli or binomial likelihood
  • Gamma prior for Poisson likelihood
  • Normal prior for normal likelihood with known variance
• Enable efficient updating of beliefs as new data becomes available
• Facilitate the derivation of closed-form expressions for the posterior distribution and posterior predictive distribution

Noninformative Priors

• Noninformative priors aim to minimize the influence of prior knowledge on the posterior distribution
• Represent a state of ignorance or lack of strong prior beliefs about the parameter
• Commonly used noninformative priors:
  • Uniform prior assigns equal probability to all possible parameter values within a specified range
  • Jeffreys prior is proportional to the square root of the Fisher information matrix
• Noninformative priors allow the data to dominate the posterior distribution
• Useful when there is little or no reliable prior information available
• Can lead to improper posteriors in some cases, requiring careful handling

Informative Priors

• Informative priors incorporate specific knowledge or beliefs about the parameter into the analysis
• Derived from various sources such as domain expertise, previous studies, or theoretical considerations
• Assign higher probabilities to parameter values considered more likely based on prior information
• Can be expressed using various probability distributions (normal, beta, gamma, etc.) depending on the nature of the parameter and prior knowledge
• Strengthen the inference by combining prior information with the observed data
• Particularly useful when dealing with small sample sizes or rare events
• Require careful elicitation and justification to ensure the prior accurately reflects the available knowledge

Impact on Posterior Distributions

• The choice of prior distribution directly influences the resulting posterior distribution
• Informative priors can shift the posterior distribution towards the prior beliefs
  • Stronger priors have a greater impact on the posterior, especially with limited data
  • Weaker priors allow the data to have more influence on the posterior
• Noninformative priors minimize the prior's impact, letting the data drive the posterior distribution
• Conjugate priors lead to analytically tractable posterior distributions, simplifying computations
• The posterior distribution combines the information from the prior and the likelihood, weighted by their relative strengths
• As more data is observed, the influence of the prior diminishes, and the posterior converges towards the true parameter value

Real-World Applications

• Bayesian clinical trials utilize informative priors to incorporate historical data or expert opinions, leading to more efficient and ethical trials
• In machine learning, priors are used for regularization and parameter shrinkage (Lasso, Ridge regression)
• Bayesian A/B testing employs priors to make informed decisions based on prior knowledge and observed data
• Bayesian networks use prior distributions to model the probabilistic relationships among variables in complex systems (medical diagnosis, risk assessment)
• Bayesian hierarchical models leverage priors to capture dependencies and borrowing of information across different levels of data (meta-analysis, spatial modeling)
• Bayesian forecasting incorporates prior knowledge to improve the accuracy and uncertainty quantification of predictions (sales forecasting, stock market analysis)