Background
##########

Introduction
==============

The multi-armed bandit problem is a classic problem in probability theory and reinforcement learning that exemplifies the exploration-exploitation tradeoff dilemma. In this problem, a fixed limited set of resources must be allocated between competing choices in a way that maximizes their expected gain, when each choice's properties are only partially known at the time of allocation.

Problem Description
====================

In the multi-armed bandit problem:

1. There are multiple "arms" (choices) available
2. Each arm provides a random reward from a probability distribution specific to that arm
3. The objective is to maximize the sum of rewards earned through a sequence of selections
4. The crucial tradeoff is between:
   - "Exploitation" of the arm that has the highest expected payoff
   - "Exploration" to get more information about the expected payoffs of other arms

This trade-off between exploration and exploitation is fundamental to many real-world problems, including:
- Managing research projects in large organizations
- Clinical trials in pharmaceutical companies
- Online advertising and recommendation systems
- Resource allocation in cloud computing
- Portfolio optimization in finance

Algorithms
============

The thompson package implements three main algorithms for solving the multi-armed bandit problem:

1. **Thompson Sampling**
   - A Bayesian approach that maintains probability distributions over the expected rewards
   - Samples from these distributions to select the next arm to pull
   - Naturally balances exploration and exploitation
   - Particularly effective when the reward distributions are unknown

2. **Upper Confidence Bound (UCB)**
   - A deterministic algorithm that uses confidence bounds
   - Selects arms based on their estimated rewards and uncertainty
   - Provides theoretical guarantees on performance
   - Good for problems with known reward distributions

3. **Randomized Sampling**
   - A baseline method that randomly selects arms
   - Useful for comparison with more sophisticated methods
   - Demonstrates the importance of intelligent exploration

Applications
==============

The multi-armed bandit algorithms implemented in this package can be applied to various real-world problems:

1. **Online Advertising**
   - Optimizing ad placement and selection
   - Maximizing click-through rates
   - A/B testing of different ad designs

2. **Clinical Trials**
   - Optimizing treatment allocation
   - Minimizing patient exposure to inferior treatments
   - Accelerating drug discovery

3. **Recommendation Systems**
   - Personalizing content recommendations
   - Optimizing user engagement
   - Balancing exploration of new items with exploitation of known preferences

4. **Resource Allocation**
   - Optimizing cloud resource allocation
   - Load balancing in distributed systems
   - Network routing optimization

Aim
===

The ``thompson`` package provides a comprehensive implementation of multi-armed bandit algorithms in Python, making it easy to:
- Apply these algorithms to real-world problems
- Compare different approaches
- Visualize and analyze results
- Make data-driven decisions in uncertain environments

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