📊Probability and Statistics Unit 11 – Bayesian Inference & Decision Theory

Key Concepts and Foundations

• Bayesian inference is a statistical approach that updates the probability of a hypothesis as more evidence or information becomes available
• Relies on Bayes' theorem to compute and update probabilities
• Incorporates prior knowledge or beliefs about a parameter or hypothesis before observing data
• Combines prior knowledge with observed data to obtain an updated posterior probability distribution
• Differs from frequentist inference which relies solely on the likelihood of the observed data
• Allows for the incorporation of subjective beliefs and provides a framework for updating those beliefs based on evidence
• Useful in various fields such as machine learning, data science, and decision-making under uncertainty

Bayes' Theorem Explained

• Bayes' theorem is a fundamental concept in Bayesian inference that describes the probability of an event based on prior knowledge and new evidence
• Mathematically, Bayes' theorem is expressed as: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
  • $P(A|B)$ represents the posterior probability of event A given event B has occurred
  • $P(B|A)$ represents the likelihood of observing event B given event A is true
  • $P(A)$ represents the prior probability of event A
  • $P(B)$ represents the marginal probability of event B
• Allows for the updating of probabilities based on new information or evidence
• Provides a way to incorporate prior beliefs or knowledge into the inference process
• Can be used to compute the probability of a hypothesis given observed data
• Helps in making decisions under uncertainty by combining prior information with observed evidence

Probability Distributions in Bayesian Analysis

• Probability distributions play a crucial role in Bayesian inference as they represent the uncertainty about parameters or hypotheses
• Prior distributions express the initial beliefs or knowledge about a parameter before observing data
  • Can be based on domain expertise, previous studies, or subjective opinions
  • Common prior distributions include uniform, beta, gamma, and normal distributions
• Likelihood functions describe the probability of observing the data given a specific value of the parameter
  • Quantifies how well the observed data supports different parameter values
  • Depends on the assumed statistical model and the nature of the data
• Posterior distributions represent the updated beliefs about the parameter after combining the prior distribution with the observed data through Bayes' theorem
  • Provides a complete description of the uncertainty about the parameter
  • Can be used to make inferences, predictions, and decisions
• Conjugate priors are prior distributions that result in posterior distributions belonging to the same family as the prior
  • Simplifies the computation of the posterior distribution
  • Examples include beta-binomial, gamma-Poisson, and normal-normal conjugate pairs

Prior and Posterior Distributions

• Prior distributions represent the initial beliefs or knowledge about a parameter before observing any data
  • Reflect the subjective opinions or previous information available about the parameter
  • Can be informative (specific knowledge) or non-informative (vague or objective)
• The choice of prior distribution can have a significant impact on the posterior inference, especially when the sample size is small
• Posterior distributions are obtained by updating the prior distribution with the observed data using Bayes' theorem
  • Combine the prior information with the likelihood of the data
  • Provide an updated representation of the uncertainty about the parameter after considering the evidence
• The posterior distribution is proportional to the product of the prior distribution and the likelihood function
  • $P(\theta|D) \propto P(D|\theta)P(\theta)$, where $\theta$ is the parameter and $D$ is the observed data
• As more data becomes available, the posterior distribution becomes less influenced by the prior and more dominated by the likelihood of the data
• Posterior distributions can be used to make inferences, estimate parameters, and quantify uncertainty

Bayesian Inference Methods

• Bayesian inference involves updating prior beliefs about parameters or hypotheses based on observed data to obtain posterior distributions
• Maximum a Posteriori (MAP) estimation finds the parameter value that maximizes the posterior probability
  • Provides a point estimate of the parameter
  • Can be seen as a regularized version of maximum likelihood estimation
• Markov Chain Monte Carlo (MCMC) methods are used to sample from the posterior distribution when it is analytically intractable
  • Includes algorithms such as Metropolis-Hastings and Gibbs sampling
  • Generates a Markov chain that converges to the posterior distribution
  • Allows for the estimation of posterior quantities and uncertainty intervals
• Variational Inference (VI) is an alternative to MCMC that approximates the posterior distribution with a simpler distribution
  • Minimizes the Kullback-Leibler (KL) divergence between the approximate and true posterior
  • Faster and more scalable than MCMC but may provide less accurate approximations
• Bayesian model selection compares different models based on their posterior probabilities
  • Uses Bayes factors or posterior odds ratios to quantify the relative evidence for each model
  • Allows for the selection of the most plausible model given the observed data

Decision Theory Basics

• Decision theory provides a framework for making optimal decisions under uncertainty
• Involves specifying a set of possible actions, states of nature, and consequences or utilities associated with each action-state pair
• The goal is to choose the action that maximizes the expected utility or minimizes the expected loss
• Utility functions quantify the preferences or desirability of different outcomes
  • Assign numerical values to the consequences of actions
  • Higher utility values indicate more preferred outcomes
• Loss functions measure the cost or penalty incurred for making a particular decision
  • Quantify the discrepancy between the true state and the chosen action
  • Common loss functions include squared error loss and 0-1 loss
• Bayesian decision theory incorporates prior probabilities and posterior distributions into the decision-making process
  • Uses the posterior distribution to compute the expected utility or loss for each action
  • Selects the action that optimizes the expected utility or minimizes the expected loss
• The Bayes risk is the expected loss associated with a decision rule
  • Provides a measure of the overall performance of a decision-making strategy
  • Optimal Bayesian decision rules minimize the Bayes risk

Applying Bayesian Decision Making

• Bayesian decision making involves combining prior knowledge, observed data, and utility or loss functions to make optimal decisions
• Starts with specifying the prior distribution over the possible states of nature or hypotheses
• Observes data and updates the prior distribution to obtain the posterior distribution using Bayes' theorem
• Defines a utility function or loss function that quantifies the consequences of different actions under each state
• Computes the expected utility or expected loss for each action using the posterior distribution
  • Expected utility: $\mathbb{E}[U(a)] = \sum_{s} U(a, s) P(s|D)$, where $a$ is an action, $s$ is a state, and $D$ is the observed data
  • Expected loss: $\mathbb{E}[L(a)] = \sum_{s} L(a, s) P(s|D)$
• Selects the action that maximizes the expected utility or minimizes the expected loss
• The optimal decision rule is known as the Bayes rule or the Bayes optimal classifier
• Bayesian decision making allows for the incorporation of prior knowledge and the quantification of uncertainty in the decision-making process
• Can be applied in various domains such as medical diagnosis, spam email classification, and investment portfolio optimization

Real-World Applications and Examples

• Bayesian inference and decision theory have numerous real-world applications across different fields
• In medical diagnosis, Bayesian methods can be used to estimate the probability of a disease given observed symptoms and test results
  • Prior knowledge about disease prevalence and test accuracy can be incorporated
  • Helps in making informed decisions about treatment options
• Spam email classification utilizes Bayesian techniques to distinguish between spam and legitimate emails
  • Learns from labeled training data to estimate the probability of an email being spam based on its features
  • Continuously updates the probabilities as new emails are observed
• Bayesian methods are used in recommender systems to personalize product or content recommendations
  • Incorporates user preferences and past behavior as prior information
  • Updates recommendations based on user feedback and interactions
• In finance, Bayesian approaches are employed for portfolio optimization and risk management
  • Combines prior market knowledge with observed financial data to make investment decisions
  • Helps in estimating the probability of different market scenarios and optimizing asset allocation
• Bayesian techniques are applied in natural language processing for tasks such as sentiment analysis and topic modeling
  • Utilizes prior knowledge about language structure and word frequencies
  • Updates the models based on observed text data to improve performance
• In robotics and autonomous systems, Bayesian methods enable decision-making under uncertainty
  • Incorporates sensor data and prior knowledge about the environment
  • Allows for the estimation of robot localization, obstacle avoidance, and decision-making in real-time