🎲Data, Inference, and Decisions Unit 10 – Bayesian Inference & Decision Making

Foundations of Probability

• Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain)
• Joint probability is the probability of two or more events occurring simultaneously, calculated by multiplying individual event probabilities if events are independent
• Conditional probability measures the probability of an event A given that event B has occurred, denoted as P(A|B) and calculated as P(A∩B) / P(B)
  • Helps update probabilities based on new information or evidence
  • Essential for understanding and applying Bayes' Theorem
• Marginal probability is the probability of an event occurring regardless of the outcome of another event, calculated by summing joint probabilities across all possible outcomes of the other event
• Independence of events occurs when the probability of one event does not affect the probability of another event
  • If events A and B are independent, then P(A|B) = P(A) and P(B|A) = P(B)
  • Allows for simplifying probability calculations in complex scenarios
• Random variables assign numerical values to outcomes of a random experiment, can be discrete (countable) or continuous (uncountable)
• Probability distributions describe the likelihood of different outcomes for a random variable
  • Examples include binomial (discrete) and normal (continuous) distributions

Bayes' Theorem Explained

• Bayes' Theorem is a fundamental rule in probability theory that describes how to update probabilities based on new evidence
• Mathematically, Bayes' Theorem is stated as: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
  • $P(A|B)$ is the posterior probability of event A given evidence B
  • $P(B|A)$ is the likelihood of observing evidence B given event A
  • $P(A)$ is the prior probability of event A before considering evidence B
  • $P(B)$ is the marginal probability of evidence B
• Bayes' Theorem allows for incorporating prior knowledge (prior probability) with new evidence (likelihood) to obtain an updated belief (posterior probability)
• The theorem is widely applied in various fields, including machine learning, medical diagnosis, and decision-making under uncertainty
• Example: In a medical context, Bayes' Theorem can be used to calculate the probability of a patient having a disease given a positive test result
  • Prior probability: Prevalence of the disease in the population
  • Likelihood: Probability of a positive test result given the patient has the disease
  • Evidence: Probability of a positive test result in the general population
• Bayes' Theorem provides a rational framework for updating beliefs based on evidence, making it a cornerstone of Bayesian inference and decision-making

Bayesian vs. Frequentist Approaches

• Bayesian and frequentist approaches are two main paradigms in statistical inference, differing in their interpretation of probability and treatment of parameters
• Bayesian approach:
  • Treats probability as a measure of belief or uncertainty about an event
  • Assumes parameters are random variables with prior distributions reflecting prior knowledge
  • Updates prior distributions using observed data to obtain posterior distributions
  • Focuses on quantifying uncertainty and making probabilistic statements about parameters
• Frequentist approach:
  • Treats probability as the long-run frequency of an event in repeated trials
  • Assumes parameters are fixed, unknown constants to be estimated from data
  • Relies on sampling distributions and point estimates (confidence intervals, p-values) to make inferences
  • Focuses on the properties of estimators and hypothesis testing
• Bayesian methods incorporate prior information and provide a natural way to update beliefs as new data becomes available
• Frequentist methods are often more computationally simple and have well-established procedures for hypothesis testing and confidence intervals
• Bayesian approach is more flexible in handling complex models and can provide more intuitive interpretations of results
• The choice between Bayesian and frequentist approaches depends on the problem context, available prior information, and computational resources

Prior and Posterior Distributions

• Prior distribution represents the initial belief or knowledge about a parameter before observing data
  • Reflects subjective or objective information available before the analysis
  • Can be informative (strong prior beliefs) or non-informative (vague or uniform priors)
• Posterior distribution is the updated belief about a parameter after incorporating observed data
  • Combines prior distribution with the likelihood of the data to obtain an updated distribution
  • Represents the revised knowledge about the parameter given the evidence
• The updating process from prior to posterior is the core of Bayesian inference
  • Bayes' Theorem is used to calculate the posterior distribution: $P(\theta|D) \propto P(D|\theta)P(\theta)$
    • $P(\theta|D)$ is the posterior distribution of parameter $\theta$ given data $D$
    • $P(D|\theta)$ is the likelihood of observing data $D$ given parameter $\theta$
    • $P(\theta)$ is the prior distribution of parameter $\theta$
• The posterior distribution summarizes the uncertainty about the parameter after considering the data
  • Can be used to make point estimates (posterior mean, median, mode) or interval estimates (credible intervals)
  • Provides a complete description of the parameter's probability distribution
• The choice of prior distribution can impact the posterior, especially when data is limited
  • Sensitivity analysis can be performed to assess the robustness of the posterior to different prior choices
• As more data is collected, the posterior distribution typically becomes more concentrated around the true parameter value, reflecting increased certainty

Likelihood and Evidence

• Likelihood measures the probability of observing the data given a specific value of the parameter
  • Denoted as $P(D|\theta)$, where $D$ is the observed data and $\theta$ is the parameter
  • Quantifies how well the parameter value explains the observed data
  • Likelihood is a function of the parameter, not a probability distribution
• Maximum likelihood estimation (MLE) is a frequentist method that finds the parameter value that maximizes the likelihood of the observed data
  • Provides a point estimate of the parameter without considering prior information
  • Often used as a starting point for Bayesian inference or when prior information is unavailable
• In Bayesian inference, the likelihood is combined with the prior distribution to obtain the posterior distribution
  • The likelihood acts as an updating factor, adjusting the prior beliefs based on the observed data
  • The shape of the likelihood function influences the shape of the posterior distribution
• Evidence, also known as marginal likelihood, is the probability of observing the data marginalized over all possible parameter values
  • Calculated as $P(D) = \int P(D|\theta)P(\theta)d\theta$, integrating the likelihood times the prior over the parameter space
  • Measures the overall fit of the model to the data, considering both the likelihood and the prior
  • Used for model comparison and selection in Bayesian inference, as it automatically penalizes complex models (Occam's razor)
• The likelihood principle states that all information about the parameter from the data is contained in the likelihood function
  • Implies that inferences should be based on the likelihood, not on the sampling distribution or other frequentist concepts
  • Supports the use of Bayesian methods, which naturally incorporate the likelihood in the updating process

Bayesian Inference in Practice

• Bayesian inference involves specifying a prior distribution, defining a likelihood function, and computing the posterior distribution
• Prior elicitation is the process of translating expert knowledge or previous studies into a prior distribution
  • Can be done through discussions with domain experts, literature review, or using non-informative priors
  • The choice of prior should be carefully considered and justified based on the available information
• Likelihood specification involves defining a probabilistic model for the data generation process
  • Requires selecting an appropriate probability distribution (binomial, normal, Poisson, etc.) that captures the data characteristics
  • The likelihood function is then constructed based on the chosen probability distribution and the observed data
• Computing the posterior distribution often requires numerical methods, especially for complex models or high-dimensional parameter spaces
  • Markov Chain Monte Carlo (MCMC) methods, such as Metropolis-Hastings or Gibbs sampling, are commonly used to sample from the posterior distribution
  • Variational inference is another approach that approximates the posterior with a simpler distribution, trading off accuracy for computational efficiency
• Model checking and validation are essential to assess the fit and adequacy of the Bayesian model
  • Posterior predictive checks compare the observed data with data simulated from the posterior predictive distribution to identify model misspecification
  • Sensitivity analysis investigates the robustness of the posterior inferences to changes in the prior distribution or likelihood assumptions
• Bayesian decision-making involves using the posterior distribution to make optimal decisions under uncertainty
  • Requires specifying a loss function that quantifies the consequences of different actions
  • The optimal decision minimizes the expected loss over the posterior distribution of the parameters
• Bayesian inference provides a coherent framework for combining prior knowledge with observed data, quantifying uncertainty, and making probabilistic statements about parameters and future observations

Decision Theory Basics

• Decision theory is a framework for making optimal decisions under uncertainty
• A decision problem consists of:
  • A set of possible actions or decisions
  • A set of possible states of nature or outcomes
  • A loss function that quantifies the consequences of each action-state combination
• The goal is to choose the action that minimizes the expected loss, considering the probability distribution over the states
• In a Bayesian decision problem, the probability distribution over the states is given by the posterior distribution of the parameters
  • The posterior distribution summarizes the uncertainty about the parameters after observing the data
  • The expected loss for each action is calculated by integrating the loss function over the posterior distribution
• The Bayes action is the action that minimizes the expected loss under the posterior distribution
  • It represents the optimal decision given the available information and the specified loss function
• Common loss functions include:
  • Quadratic loss: Penalizes the squared difference between the true state and the decision
  • 0-1 loss: Assigns a loss of 1 for incorrect decisions and 0 for correct decisions
  • Absolute loss: Penalizes the absolute difference between the true state and the decision
• The choice of loss function should reflect the decision-maker's preferences and the problem context
• Bayesian decision theory provides a principled way to incorporate prior knowledge, observed data, and the consequences of decisions into a unified framework

Applying Bayesian Decision Making

• Bayesian decision making has numerous applications across various domains, including business, healthcare, and engineering
• In clinical trials, Bayesian methods can be used to:
  • Incorporate prior information from previous studies or expert opinion
  • Adapt the trial design based on interim results, allowing for early stopping or sample size adjustments
  • Make decisions about treatment effectiveness or safety based on the posterior probabilities
• In predictive maintenance, Bayesian decision making can help:
  • Estimate the probability of equipment failure based on sensor data and historical records
  • Determine the optimal maintenance schedule that balances the costs of preventive maintenance and unexpected failures
  • Update the maintenance strategy as new data becomes available
• In marketing and customer analytics, Bayesian methods can be applied to:
  • Segment customers based on their purchase behavior and demographic information
  • Predict the likelihood of a customer responding to a marketing campaign or making a purchase
  • Optimize marketing strategies and resource allocation based on the expected returns
• In finance and portfolio management, Bayesian decision making can assist in:
  • Estimating the expected returns and risks of different assets or investment strategies
  • Incorporating market trends, economic indicators, and expert opinions into the investment decisions
  • Rebalancing the portfolio based on the updated beliefs about the asset performance
• When applying Bayesian decision making, it is important to:
  • Clearly define the decision problem, including the available actions, possible outcomes, and the loss function
  • Specify a suitable prior distribution and likelihood function based on the available information and domain knowledge
  • Use appropriate computational methods to obtain the posterior distribution and calculate the expected losses
  • Perform sensitivity analysis to assess the robustness of the decisions to changes in the prior or loss function
  • Communicate the results and the underlying assumptions to stakeholders in a clear and transparent manner
• Bayesian decision making provides a formal and coherent framework for making optimal decisions under uncertainty, leveraging prior knowledge, observed data, and the consequences of actions