10.4 Sequential decision making

Sequential decision making is a key concept in Bayesian statistics, allowing for dynamic updating of beliefs and actions as new information becomes available. This approach aligns with the Bayesian philosophy of iterative learning, making it particularly relevant in complex, uncertain environments.

The topic covers various frameworks like Markov Decision Processes, Partially Observable MDPs, and Bayesian Reinforcement Learning. These methods provide powerful tools for modeling and solving sequential decision problems, balancing exploration and exploitation in uncertain scenarios.

Fundamentals of sequential decisions

• Sequential decision making forms a crucial component of Bayesian statistics, allowing for dynamic updating of beliefs and actions based on new information
• This approach aligns with the Bayesian philosophy of iterative learning and adaptation, making it particularly relevant in complex, uncertain environments

Bayesian decision theory basics

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• Foundations built on Bayes' theorem $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ for updating probabilities given new evidence
• Incorporates prior beliefs, likelihood functions, and posterior distributions to make informed decisions
• Utilizes expected utility theory to evaluate and compare different decision options
• Applies the principle of maximum expected utility to choose optimal actions

Sequential vs single-stage decisions

• Single-stage decisions involve making one choice based on available information
• Sequential decisions unfold over time, allowing for multiple decision points and information updates
• Temporal dependencies between decisions create complex decision trees or networks
• Incorporates the concept of value of information for future decision-making opportunities

Dynamic programming principles

• Breaks down complex problems into smaller, more manageable subproblems
• Utilizes the principle of optimality to solve problems recursively
• Employs backward induction to determine optimal decisions at each stage
• Introduces the concept of state value functions to represent expected future rewards
• Implements the Bellman equation to relate current and future values $V(s) = \max_a(R(s,a) + \gamma \sum_{s'} P(s'|s,a)V(s'))$

Markov decision processes

• Markov Decision Processes (MDPs) provide a mathematical framework for modeling sequential decision-making under uncertainty
• MDPs play a crucial role in Bayesian statistics by allowing for the incorporation of probabilistic transitions and rewards in decision-making scenarios

MDP components and structure

• States (S) represent the possible configurations of the system
• Actions (A) denote the choices available to the decision-maker
• Transition probabilities P(s'|s,a) describe the likelihood of moving from one state to another given an action
• Reward function R(s,a) quantifies the immediate benefit of taking an action in a given state
• Discount factor γ balances immediate and future rewards
• Policy π maps states to actions, guiding the decision-maker's behavior

Bellman equation

• Formulates the recursive relationship between value functions of different states
• Expresses the value of a state as the sum of immediate reward and discounted future value
• Optimal value function satisfies $V^*(s) = \max_a(R(s,a) + \gamma \sum_{s'} P(s'|s,a)V^*(s'))$
• Serves as the foundation for various MDP solution methods
• Enables the computation of optimal policies through iterative methods

Value iteration vs policy iteration

• Value iteration focuses on computing the optimal value function
  • Iteratively updates state values until convergence
  • Derives the optimal policy from the converged value function
• Policy iteration alternates between policy evaluation and improvement
  • Evaluates the current policy to obtain its value function
  • Improves the policy based on the computed value function
  • Continues until the policy stabilizes
• Both methods guarantee convergence to the optimal policy
• Trade-offs include computational efficiency and speed of convergence

Partially observable MDPs

• Partially Observable Markov Decision Processes (POMDPs) extend MDPs to scenarios with incomplete state information
• POMDPs align closely with Bayesian principles by explicitly modeling uncertainty in state observations

POMDP framework

• States (S), actions (A), and rewards (R) similar to MDPs
• Observations (O) represent incomplete or noisy information about the true state
• Observation function O(o|s',a) defines the probability of observing o after taking action a and transitioning to state s'
• Transition function T(s'|s,a) describes state dynamics
• Discount factor γ balances immediate and future rewards
• Policy π maps belief states to actions, accommodating partial observability

Belief state representation

• Belief state b(s) represents a probability distribution over possible states
• Captures the agent's uncertainty about the true state of the environment
• Updated using Bayes' rule after each action and observation
• Belief update equation: $b'(s') = \eta O(o|s',a) \sum_s T(s'|s,a)b(s)$
• Serves as a sufficient statistic for decision-making in POMDPs

POMDP solution methods

• Value iteration over belief space (computationally intensive)
• Point-based value iteration focuses on reachable belief points
• Policy graphs represent policies as finite state controllers
• Monte Carlo tree search explores the belief tree efficiently
• Approximate methods (QMDP, AMDP) trade off optimality for computational efficiency
• Online planning algorithms compute actions in real-time

Bayesian reinforcement learning

• Bayesian Reinforcement Learning (BRL) integrates Bayesian inference with reinforcement learning principles
• BRL provides a principled approach to balancing exploration and exploitation in uncertain environments

Exploration vs exploitation

• Exploration involves gathering information about the environment
• Exploitation focuses on maximizing rewards based on current knowledge
• Trade-off arises from the need to learn optimal behavior while maximizing cumulative rewards
• ε-greedy strategy balances exploration and exploitation through random action selection
• Upper Confidence Bound (UCB) algorithms encourage exploration of uncertain options

Thompson sampling

• Probabilistic algorithm for balancing exploration and exploitation
• Maintains a posterior distribution over environment parameters
• Samples from the posterior to select actions
• Naturally adapts exploration based on uncertainty in the environment
• Provides a Bayesian approach to the exploration-exploitation dilemma

Posterior sampling for RL

• Extends Thompson sampling to sequential decision-making problems
• Maintains a posterior distribution over MDPs
• Samples an MDP from the posterior at each episode
• Acts optimally with respect to the sampled MDP
• Balances exploration and exploitation through posterior uncertainty
• Converges to optimal behavior as more information is gathered

Multi-armed bandit problems

• Multi-armed bandit problems serve as a simplified model for sequential decision-making under uncertainty
• These problems provide a foundation for understanding exploration-exploitation trade-offs in Bayesian statistics

Bandit algorithms

• ε-greedy selects the best-known arm with probability 1-ε and explores randomly with probability ε
• Upper Confidence Bound (UCB) algorithms balance exploitation and exploration using confidence intervals
• Thompson Sampling draws samples from posterior distributions to select arms
• Gittins index computes a dynamic allocation index for each arm
• Softmax exploration uses a temperature parameter to control exploration

Regret analysis

• Regret measures the difference between optimal and achieved rewards
• Cumulative regret $R_T = T\mu^* - \sum_{t=1}^T \mu_{a_t}$ quantifies long-term performance
• Logarithmic regret bounds achieved by efficient algorithms (UCB, Thompson Sampling)
• Lower bounds on regret (Lai-Robbins bound) establish theoretical limits
• High-probability regret bounds provide stronger guarantees on algorithm performance

Contextual bandits

• Extend standard bandits by incorporating contextual information
• Context vector x_t influences arm rewards at each time step
• Linear contextual bandits assume linear relationship between context and rewards
• LinUCB and Thompson Sampling adapted for contextual settings
• Applications include personalized recommendations and adaptive clinical trials

Sequential hypothesis testing

• Sequential hypothesis testing allows for dynamic decision-making in statistical inference
• This approach aligns with Bayesian principles by updating beliefs as new data becomes available

Sequential probability ratio test

• Wald's SPRT tests between two simple hypotheses
• Computes likelihood ratio $\Lambda_n = \frac{P(x_1, ..., x_n | H_1)}{P(x_1, ..., x_n | H_0)}$ after each observation
• Continues sampling if A < Λ_n < B, where A and B are predetermined thresholds
• Accepts H_1 if Λ_n ≥ B, accepts H_0 if Λ_n ≤ A
• Minimizes expected sample size for fixed error probabilities

Optional stopping

• Allows for data-dependent decisions to stop or continue sampling
• Challenges traditional fixed-sample size approaches
• Requires careful consideration of stopping rules to maintain statistical validity
• Bayesian approaches naturally accommodate optional stopping
• Frequentist methods may require adjustment for multiple testing

Bayesian sequential analysis

• Updates posterior probabilities after each observation
• Computes Bayes factors to compare hypotheses
• Defines stopping rules based on posterior probabilities or Bayes factors
• Incorporates prior information into the decision-making process
• Allows for continuous monitoring and adaptive designs in clinical trials

Bayesian optimization

• Bayesian optimization provides a framework for efficient global optimization of expensive black-box functions
• This approach leverages Bayesian principles to guide sequential sampling and decision-making

Gaussian process regression

• Models unknown function as a Gaussian process
• Defines prior distribution over functions
• Updates posterior distribution based on observed data
• Provides mean and variance estimates for unobserved points
• Enables uncertainty quantification in function predictions

Acquisition functions

• Guide the selection of next evaluation points
• Expected Improvement (EI) balances exploitation and exploration
• Probability of Improvement (PI) focuses on finding better solutions
• Upper Confidence Bound (UCB) controls exploration-exploitation trade-off
• Entropy Search maximizes information gain about the optimum
• Knowledge Gradient considers value of information for future decisions

Sequential model-based optimization

• Iteratively updates surrogate model and selects new evaluation points
• Balances exploration of uncertain regions and exploitation of promising areas
• Incorporates noise and constraints in optimization problems
• Adapts to changing environments and non-stationary objectives
• Applications include hyperparameter tuning and experimental design

Decision trees for sequential choices

• Decision trees provide a visual and analytical tool for modeling sequential decision-making processes
• This approach aligns with Bayesian decision theory by incorporating probabilities and expected utilities

Tree structure and notation

• Nodes represent decision points or chance events
• Branches denote possible actions or outcomes
• Leaf nodes contain final outcomes or payoffs
• Probabilities assigned to chance event branches
• Expected values calculated for each node

Backward induction

• Solves decision trees from leaves to root
• Calculates expected values at chance nodes
• Selects optimal decisions at decision nodes
• Propagates values upward through the tree
• Determines optimal strategy and overall expected value

Pruning techniques

• Reduce tree complexity while maintaining decision quality
• Cost-complexity pruning balances tree size and accuracy
• Reduced error pruning uses validation set to evaluate subtrees
• Minimum description length (MDL) principle for model selection
• Bayesian approaches incorporate prior beliefs in pruning decisions

Applications in Bayesian statistics

• Sequential decision-making principles find numerous applications in Bayesian statistical methods
• These applications leverage the iterative nature of Bayesian inference to adapt and optimize processes

Adaptive clinical trials

• Modify trial design based on accumulating data
• Implement response-adaptive randomization to allocate patients
• Use group sequential methods for interim analyses
• Employ Bayesian decision rules for early stopping or continuation
• Balance ethical considerations with statistical efficiency

Sequential Monte Carlo methods

• Particle filtering for online state estimation in dynamic systems
• Sequential importance sampling with resampling (SIR) algorithm
• Adaptive particle filters adjust proposal distributions
• Applications in target tracking and financial modeling
• Combine with MCMC for high-dimensional problems

Bayesian experimental design

• Optimize experimental conditions to maximize information gain
• Utilize expected Kullback-Leibler divergence as a design criterion
• Implement sequential designs that adapt based on interim results
• Apply decision-theoretic approaches to balance costs and benefits
• Incorporate prior knowledge to guide experimental choices

Challenges and limitations

• Sequential decision-making in Bayesian statistics faces several challenges that impact its practical implementation
• Understanding these limitations is crucial for developing robust and efficient methods

Curse of dimensionality

• Exponential growth of state space with increasing dimensions
• Computational intractability for high-dimensional problems
• Sparse data issues in high-dimensional spaces
• Difficulty in accurately representing and updating belief states
• Approximation methods (function approximation, dimensionality reduction) to mitigate the curse

Computational complexity

• Exact solutions often intractable for large-scale problems
• Value iteration and policy iteration scale poorly with state space size
• POMDP solving computationally expensive, limiting real-world applications
• Approximate methods trade off optimality for computational efficiency
• Parallel and distributed computing approaches to handle large-scale problems

Model uncertainty

• Misspecified models lead to suboptimal decisions
• Difficulty in accurately specifying prior distributions
• Robustness concerns when true environment differs from assumed model
• Model selection and averaging techniques to address uncertainty
• Bayesian nonparametric approaches for flexible model specification