📊Bayesian Statistics Unit 10 – Bayesian decision theory

What's Bayesian Decision Theory?

• Framework for making optimal decisions under uncertainty by combining prior knowledge with observed data
• Utilizes Bayes' theorem to update beliefs (prior probabilities) based on new evidence (likelihood) to obtain posterior probabilities
• Incorporates decision-maker's preferences and values through the use of utility functions and loss functions
• Aims to minimize expected loss or maximize expected utility when choosing among different actions or decisions
• Provides a principled approach to balance exploration and exploitation in sequential decision-making problems (multi-armed bandits)
• Applicable to a wide range of fields, including statistics, machine learning, economics, and psychology
• Differs from classical (frequentist) decision theory by explicitly incorporating prior knowledge and updating beliefs based on data

Key Concepts and Terminology

• Prior probability: Initial belief or knowledge about a parameter or hypothesis before observing data
• Likelihood: Probability of observing the data given a specific parameter value or hypothesis
• Posterior probability: Updated belief about a parameter or hypothesis after incorporating the observed data
• Bayes' theorem: Mathematical rule for updating prior probabilities based on new evidence to obtain posterior probabilities
• Utility function: Quantifies the decision-maker's preferences and assigns a numerical value to each possible outcome
  • Represents the relative desirability or satisfaction associated with different outcomes
• Loss function: Measures the cost or penalty incurred for making a specific decision when the true state of nature is known
  • Common loss functions include squared error loss, absolute error loss, and 0-1 loss
• Expected utility: Average utility of an action, weighted by the probabilities of different outcomes
• Expected loss: Average loss incurred by an action, weighted by the probabilities of different states of nature
• Bayes risk: Minimum expected loss achievable by any decision rule for a given prior distribution and loss function

Probability Basics Refresher

• Probability: Measure of the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain)
• Joint probability: Probability of two or more events occurring simultaneously, denoted as $P(A, B)$
• Conditional probability: Probability of an event A occurring given that another event B has already occurred, denoted as $P(A|B)$
• Marginal probability: Probability of an event A occurring, regardless of the outcome of other events, obtained by summing or integrating joint probabilities
• Independence: Two events A and B are independent if the occurrence of one does not affect the probability of the other, i.e., $P(A|B) = P(A)$
• Random variable: Variable whose value is determined by the outcome of a random experiment
  • Discrete random variables take on a countable number of distinct values (integers)
  • Continuous random variables can take on any value within a specified range (real numbers)
• Probability distribution: Function that assigns probabilities to the possible values of a random variable
  • Examples include binomial, Poisson, normal, and exponential distributions

Bayes' Theorem in Decision Making

• Bayes' theorem allows updating prior beliefs (probabilities) about a parameter or hypothesis based on observed data to obtain posterior beliefs
• Mathematical formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$, where A is the parameter or hypothesis and B is the observed data
• Prior probability $P(A)$ represents the initial belief about A before observing data
• Likelihood $P(B|A)$ represents the probability of observing data B given that A is true
• Posterior probability $P(A|B)$ represents the updated belief about A after incorporating the observed data B
• Bayesian decision-making involves choosing the action that minimizes expected loss or maximizes expected utility based on the posterior distribution
• Enables incorporating domain knowledge and expert opinions through the specification of informative prior distributions
• Provides a framework for sequential decision-making and learning from data as it becomes available (Bayesian updating)

Loss Functions and Utility

• Loss functions quantify the cost or penalty incurred for making a specific decision when the true state of nature is known
  • Squared error loss: $L(a, \theta) = (a - \theta)^2$, where a is the action and $\theta$ is the true parameter value
  • Absolute error loss: $L(a, \theta) = |a - \theta|$
  • 0-1 loss: $L(a, \theta) = \begin{cases} 0, & \text{if } a = \theta \ 1, & \text{if } a \neq \theta \end{cases}$
• Utility functions assign a numerical value to each possible outcome, representing the decision-maker's preferences
  • Higher utility values indicate more desirable outcomes
  • Utility functions can be ordinal (ranking outcomes) or cardinal (quantifying differences in desirability)
• Expected loss is the average loss incurred by an action, weighted by the probabilities of different states of nature
  • $E[L(a)] = \sum_{\theta} L(a, \theta) P(\theta)$ for discrete $\theta$
  • $E[L(a)] = \int L(a, \theta) p(\theta) d\theta$ for continuous $\theta$
• Expected utility is the average utility of an action, weighted by the probabilities of different outcomes
  • $E[U(a)] = \sum_{x} U(x) P(x|a)$ for discrete outcomes $x$
  • $E[U(a)] = \int U(x) p(x|a) dx$ for continuous outcomes $x$
• Bayes risk is the minimum expected loss achievable by any decision rule for a given prior distribution and loss function
  • Represents the optimal performance attainable under uncertainty

Bayesian vs. Frequentist Approaches

• Bayesian approach treats parameters as random variables with associated probability distributions (prior and posterior)
  • Incorporates prior knowledge and updates beliefs based on observed data
  • Focuses on the probability of parameters given the data, $P(\theta|x)$
  • Provides a natural framework for decision-making under uncertainty
• Frequentist approach treats parameters as fixed, unknown constants
  • Relies on the sampling distribution of estimators and the likelihood of the data given the parameters
  • Focuses on the probability of the data given the parameters, $P(x|\theta)$
  • Uses point estimates, confidence intervals, and hypothesis tests to make inferences about parameters
• Bayesian methods can incorporate prior information and provide a more intuitive interpretation of results
  • Particularly useful when prior knowledge is available or when dealing with small sample sizes
• Frequentist methods are often simpler to implement and have well-established theoretical properties
  • Suitable when prior information is unavailable or when objectivity is a concern
• Both approaches have their strengths and weaknesses, and the choice depends on the specific problem and the researcher's goals and assumptions

Real-World Applications

• Medical diagnosis: Updating the probability of a disease based on test results and patient characteristics
  • Prior probability from population prevalence, likelihood from test sensitivity and specificity
• Spam email classification: Determining the probability that an email is spam based on its content and metadata
  • Prior probability from overall spam prevalence, likelihood from the presence of specific words or features
• Recommender systems: Predicting user preferences based on their past behavior and similarities with other users
  • Prior probability from overall item popularity, likelihood from user-item interactions
• A/B testing: Comparing the effectiveness of different versions of a website or application
  • Prior probability from domain knowledge or previous experiments, likelihood from observed user behavior
• Portfolio optimization: Selecting investments to maximize expected returns while minimizing risk
  • Prior probability from market trends and expert opinions, likelihood from historical performance data
• Robotics and autonomous systems: Making decisions based on sensor data and prior knowledge about the environment
  • Prior probability from maps or previous experiences, likelihood from sensor measurements

Common Pitfalls and Misconceptions

• Ignoring or misspecifying the prior distribution can lead to biased or overconfident conclusions
  • Sensitivity analysis can help assess the robustness of results to different prior choices
• Overreliance on point estimates (posterior mean or mode) without considering the full posterior distribution
  • Credible intervals and posterior predictive checks can provide a more complete picture of uncertainty
• Confusing the likelihood $P(x|\theta)$ with the posterior $P(\theta|x)$, or the prior $P(\theta)$ with the marginal $P(x)$
  • Bayes' theorem helps clarify the relationship between these probabilities
• Assuming that Bayesian methods always require conjugate priors or analytical solutions
  • Markov Chain Monte Carlo (MCMC) and variational inference enable Bayesian analysis for complex models
• Neglecting the impact of the loss function or utility function on the optimal decision
  • Different loss functions can lead to different optimal actions for the same posterior distribution
• Interpreting Bayesian probabilities as long-run frequencies or objective probabilities
  • Bayesian probabilities represent degrees of belief and are conditioned on the available information
• Overlooking the computational complexity of Bayesian methods, especially for high-dimensional or large-scale problems
  • Efficient algorithms and approximations may be necessary for practical implementation