概率 Expected Value Of Information

问题分析

You are considering betting on an event. The probability, $p$, of the event occurring is unknown but is modeled as a uniform random variable between 0 and 1 ( $p \sim Uniform(0, 1)$ ).
You have the option to bet on the event. If you bet, it costs you 0.50 dollars. If the event occurs, the bet pays y

解法

根据题目要求实现相应功能。核心逻辑需要：

// 核心数据结构和方法——根据题目 API 约定实现
// 1. 确定状态表示——选择支持所需操作的数据结构
// 2. 实现核心算法——确保 O(·) 时间复杂度和正确性
// 3. 处理边界条件——空输入、极值参数、并发访问

验证

用具体输入验证：构造已知输入的测试用例，确认输出匹配预期结果。

复杂度与边界

• 时间复杂度：取决于选用的算法
• 空间复杂度：取决于数据规模
• 关键边界条件：空输入、极值参数、并发场景下的正确性保证

英文解析

Analysis

You are considering betting on an event. The probability, p, of the event occurring is unknown but is modeled as a uniform random variable between 0 and 1 (p ~ Uniform(0, 1)). You have the option to bet on the event. If you bet, it costs you 0.50 dollars. If the event occurs, the bet pays you 1 dollar (net profit of 0.50). If it does not occur, you lose your 0.50 bet. The expected value of betting without additional information is E[profit] = E[p * 0.50 + (1-p) * (-0.50)] = E[p - 0.50] = 0, since E[p] = 0.5 for Uniform(0,1). The Expected Value of Perfect Information (EVPI) measures how much you would pay to know p exactly before deciding.

Solution

double evpi_bet() {
    // Without info: E[bet profit] = 0, E[skip profit] = 0
    // With perfect info: bet only when p > 0.5
    // E[profit with info] = E[max(p - 0.5, 0)]
    double ev_with_info = 0;
    int N = 100000;
    for (int i = 0; i < N; ++i) {
        double p = (i + 0.5) / N;  // Uniform sampling
        double profit_if_bet = p * 0.5 + (1 - p) * (-0.5);
        ev_with_info += std::max(profit_if_bet, 0.0) / N;
    }
    // EVPI = E[optimal with info] - E[optimal without info]
    // Without info, optimal is indifferent (EV=0)
    // With info, only bet when p>0.5: E[p*0.5-(1-p)*0.5 | p>0.5] * P(p>0.5)
    // Analytical: EVPI = integral from 0.5 to 1 of (p-0.5) dp = 0.125
    return ev_with_info;  // Should be ~0.125
}

Complexity & Edge Cases

• Time complexity: O(N) for numerical integration, O(1) for analytical solution
• Space complexity: O(1)
• Edge cases: (1) When bet has zero expected value, EVPI equals the expected gain from optimal decision (2) EVPI decreases as prior becomes more informative (3) For non-uniform priors, integration changes accordingly

Verification

Verify analytical result: EVPI = integral from 0.5 to 1 of (2p-1)/2 dp = 0.125. Numerical approximation should converge to this value. Test with different prior distributions.

Key Considerations

EVPI quantifies the maximum value of information before making a decision. In quantitative trading, this concept applies to research investment - how much should you spend on data analysis before committing capital? EVPI provides a theoretical upper bound on research spending: if the cost of obtaining information exceeds EVPI, the research is not worthwhile.