Consider a two-class classification problem, where the densities of the two competing classes are given by $f_1(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$ and $f_2(x) = \begin{cases} 2x & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$. Let $\pi_1$ and $\pi_2$ be the prior probabilities of these two classes. Now consider a classifier $\delta$, which classifies an observation $x$ to class $1$ if $x < 1/2$ and to class $2$ if $x \geq 1/2$.
To determine when the classifier \( \delta \) is the Bayes classifier and to find the average probability of misclassification, let's first understand the problem:
We have two classes with probability densities:
- Class 1 density: \(f_1(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}\)
- Class 2 density: \(f_2(x) = \begin{cases} 2x & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}\)
Priors are given as \( \pi_1 \) and \( \pi_2 \), representing the prior probabilities of the classes. A Bayes classifier \(\delta\) minimizes the probability of misclassification.
The given classifier \(\delta\) classifies:
- To class 1 if \( x < 1/2 \)
- To class 2 if \( x \geq 1/2 \)
According to the Bayes rule, \(\delta\) is optimal when it classifies by maximizing posterior probabilities. For \(\pi_1 = \pi_2\):
- When \(0 \leq x < 1/2\): \(f_1(x) > f_2(x) = 2x\) (since \(f_1(x) = 1\)), correctly classified to class 1.
- When \(1/2 \leq x \leq 1\): \(f_2(x) > f_1(x) = 1\) doesn't hold, but because densities are equal at \(1/2\) and for simplicity equal prior probabilities align decision boundary at \(1/2\).
Hence, when \(\pi_1 = \pi_2\), the classifier \(\delta\) is indeed a Bayes classifier.
Next, for the average probability of misclassification calculation:
- Misclassification in the region \(0 \leq x < 1/2\) happens when a class 2 point is chosen: \(\int_0^{1/2} 2x \, dx = [x^2]_0^{1/2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
- Misclassification in the region \(1/2 \leq x \leq 1\) happens when a class 1 point is chosen: \(\int_{1/2}^1 1 \, dx = [x]_{1/2}^1 = \frac{1}{2}\).
- The total probability of misclassification: \(\pi_1 \frac{1}{4} + \pi_2 \frac{1}{2}\\).
- If \(\pi_1 = \pi_2 = \frac{1}{2}\): \(\frac{1}{2} \left(\frac{1}{4}\right) + \frac{1}{2} \left(\frac{1}{2}\right) = \frac{1}{8} + \frac{2}{8} = \frac{3}{8}\).
Conclusively, if \(\pi_1 = \pi_2\), then \(\delta\) is the Bayes classifier, and the average probability of misclassification for \(\delta\) is \(\frac{3}{8}\).
Important Questions from Elementary Statistics (Notes)
Let $X_1, X_2, X_3$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For $i=1,2,3$, let $R_i$ denote the rank of $|X_i|$ among $|X_1|, |X_2|$ and $|X_3|$.
If $T^+ = \sum_{i=1, X_i>0}^3 R_i$
is the Willcoxon signed-rank statistic, then which of the following statements are true?,
- What is the geometric mean of 2, 4 and 8?
- In correlation analysis, the two variables
1. Are treated with distinction.
2. Are treated differently based on individual characteristics.
3. Are treated symmetrically.
4. Are regressed. - In statistics, standard error measures the
1. Specification error of the model.
2. Autocorrelation in the regression model.
3. Correlation between dependent and independent variables.
4. Precision of an estimate. - Linear regression model is
1. linear in explanatory variables but may not be linear in parameters
2. non-linear in parameters and must be linear in variables
3. linear in parameters and must be linear in variables
4. linear in parameters and may be linear in variables
