京都大学 情報学研究科 知能情報学専攻 2025年8月実施 専門科目 S-1

标签：

• Kyoto-University
• Beta-Bernoulli-Model
• Bayesian-Inference
• Correlation-Coefficient
• Selection-Bias
• Effect-Size

Author

itsuitsuki

Description

Q.1

Let $\theta$ be the probability of getting heads in a coin toss. Assuming that $\theta$ is a random variable, we want to estimate the probability distribution of $\theta$ from observed data $D$ of independent coin tosses (Bernoulli trials). Here, we introduce a prior distribution of $\theta$ ($p(\theta)$) that follows a beta distribution $\text{Beta}(\theta; \alpha, \beta)$, where $\alpha$ and $\beta$ are the parameters. The probability density function of the beta distribution is given by

$$f(\theta; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} \theta^{\alpha-1} (1 - \theta)^{\beta-1}, \quad (\alpha > 0, \ \beta > 0, \ 0 \leq \theta \leq 1)$$

where $B(\alpha, \beta) = \int_{0}^{1} \theta^{\alpha-1}(1 - \theta)^{\beta-1} d\theta$ is the beta function.

(1) Describe briefly what kind of distribution $\text{Beta}(\theta; 1, 1)$ is.

(2) Given that heads appeared $k$ times in $n$ trials, determine the likelihood function $p(D \mid \theta)$ for this observation $D$.

(3) According to Bayes' theorem, the posterior distribution of $\theta$ after observing $D$ can be calculated as $p(\theta \mid D) = p(\theta) \, p(D \mid \theta) / p(D)$. Here, $p(D)$ is a normalization constant that adjusts the integral of the posterior distribution to be 1. Given a prior distribution $\text{Beta}(\theta; \alpha, \beta)$ and observed data consisting of $k$ heads in $n$ trials, derive the posterior distribution of $\theta$.

Q.2

Let $M$ and $W$ be random variables representing scores in math and English, respectively. Assume $M$ and $W$ are independent and both normally distributed with mean 50 and standard deviation 10.

(1) Let $S$ be the total score $(M + W)$ and $D$ be the difference in scores $(M - W)$. Calculate the variances of $S$ and $D$: $\text{Var}(S)$ and $\text{Var}(D)$.

(2) Show that $S$ and $D$ are uncorrelated.

(3) If two random variables follow a bivariate normal distribution and are uncorrelated, they are independent. Using this property, calculate the covariance of $M$ and $W$ given that $S$ is fixed at $s$ (where $s$ is a constant) : $\text{Cov}(M, W \mid S = s)$.

(4) Calculate the correlation coefficient of $M$ and $W$ given $S = s$:

$$\rho = \frac{\text{Cov}(M, W \mid S = s)}{\sqrt{\text{Var}(M \mid S = s) \ \text{Var}(W \mid S = s)}}.$$

(5) Even when math and English scores are independent, a negative correlation emerges when selecting only examinees with high total scores. Briefly explain why, based on the results above.

Q.3

When reporting experimental results, it is recommended to include effect sizes alongside $p$-values and statistical significance. Cohen's $d$ (the difference between group means divided by within-group standard deviation) is a common effect size.

(1) Both the $t$-statistic and Cohen's $d$ are standardized measures of mean differences between two groups, yet they provide different types of information. Explain the difference using the term 'sample size.'

(2) When is it particularly important to report effect sizes? Describe the circumstances and explain why.

题目描述

1. 设 $\theta$ 为一次抛硬币出现正面的概率，并把 $\theta$ 本身视为随机变量。希望根据相互独立的抛硬币观测数据 $D$（Bernoulli 试验）估计 $\theta$ 的概率分布。令 $\theta$ 的先验分布为参数为 $\alpha,\beta$ 的 Beta 分布，其密度为

   $$f(\theta;\alpha,\beta) =\frac{1}{B(\alpha,\beta)} \theta^{\alpha-1}(1-\theta)^{\beta-1}, \qquad \alpha>0,\ \beta>0,\ 0\leq\theta\leq1,$$

   其中

   $$B(\alpha,\beta) =\int_0^1\theta^{\alpha-1}(1-\theta)^{\beta-1}\,d\theta$$

   为 Beta 函数。

   （1）简要说明 $\operatorname{Beta}(\theta;1,1)$ 是怎样的分布。

   （2）若 $n$ 次试验中出现 $k$ 次正面，求该观测数据 $D$ 的似然函数 $p(D\mid\theta)$。

   （3）根据 Bayes 定理，

   $$p(\theta\mid D) =\frac{p(\theta)p(D\mid\theta)}{p(D)},$$

   其中 $p(D)$ 是使后验密度积分为 1 的归一化常数。给定先验 $\operatorname{Beta}(\theta;\alpha,\beta)$ 及 $n$ 次试验中 $k$ 次正面的数据，推导 $\theta$ 的后验分布。

2. 令随机变量 $M,W$ 分别表示数学与英语成绩。假设二者相互独立，且都服从均值 50、标准差 10 的正态分布。

   （1）令总分 $S=M+W$，分差 $D=M-W$，计算 $\operatorname{Var}(S)$ 与 $\operatorname{Var}(D)$。

   （2）证明 $S$ 与 $D$ 不相关。

   （3）可使用性质：服从二元正态分布的两个随机变量若不相关，则相互独立。给定总分固定为常数 $s$，计算

   $$\operatorname{Cov}(M,W\mid S=s).$$

   （4）计算给定 $S=s$ 时 $M,W$ 的相关系数

   $$\rho= \frac{\operatorname{Cov}(M,W\mid S=s)} {\sqrt{\operatorname{Var}(M\mid S=s)\operatorname{Var}(W\mid S=s)}}.$$

   （5）即使数学和英语成绩原本独立，只选择总分高的考生后也会出现负相关。根据上述结果简要解释原因。

3. 报告实验结果时，建议在 $p$ 值和统计显著性之外同时报告效应量。Cohen's $d$（两组均值之差除以组内标准差）是一种常用效应量。

   （1）$t$ 统计量与 Cohen's $d$ 都是两组均值差的标准化度量，但二者提供的信息不同。使用“样本量”这一概念解释区别。

   （2）在哪些情况下报告效应量尤其重要？说明具体情形及其原因。