A probability distribution $P$ over $X \times \{0, 1\}$. $P$ can be defined in term of its marginal distribution over $X$ , which we will denote by $P_X$ and the conditional labeling distribution, which is defined by the regression function $$ \mu(x) = P_{ (x,y) \sim P} [y = 1 \mid x] $$ Consider a 2-dimensional Euclidean domain, that is $X = \mathbb R^2$, and the following process of data generation: The marginal distribution over $X$ is uniform over two square areas $[1, 2] \times [1, 2] \cup [3, 4] \times [1.5, 2.5]$. Points in the first square $Q_1 = [1, 2] \times [1, 2]$ are labeled 0 (blue) and points in the second square $Q_2 = [3, 4] \times [1.5, 2.5]$ are labeled 1 (red).

Describe the density function of $P_X$, and the regression function, Bayes predictor and Bayes risk of $P$.

In the image, I have defined the Probability density function. Is that correct? I want to find it over both the axis and in general in two dimensions.

A probability distribution $P$ over $X \times \{0, 1\}$. $P$ can be defined in term of its marginal distribution over $X$ , which we will denote by $P_X$ and the conditional labeling distribution, which is defined by the regression function $$ \mu(x) = P_{ (x,y) \sim P} [y = 1 \mid x] $$ Consider a 2-dimensional Euclidean domain, that is $X = \mathbb R^2$, and the following process of data generation: The marginal distribution over $X$ is uniform over two square areas $[1, 2] \times [1, 2] \cup [3, 4] \times [1.5, 2.5]$. Points in the first square $Q_1 = [1, 2] \times [1, 2]$ are labeled 0 (blue) and points in the second square $Q_2 = [3, 4] \times [1.5, 2.5]$ are labeled 1 (red).

Describe the density function of $P_X$, and the regression function, Bayes predictor and Bayes risk of $P$.

In the image, I have defined the Probability density function. I am having trouble in figuring out the pdf of this function in 2 dimensional space.

A probability distribution P over X × {0, 1}. P can be defined in term of its marginal distribution over X , which we will denote by $P_X$ and the conditional labeling distribution, which is defined by the regression function $$ µ(x) = P_{ (x,y)∼P} [y = 1 | x] $$ consider a 2-dimensional Euclidean domain, that is $X = R^2$, and the following process of data generation: The marginal distribution over X is uniform over two square areas [1, 2] × [1, 2] ∪ [3, 4] × [1.5, 2.5]. Points in the first square Q1 = [1, 2] × [1, 2] are labeled 0 (blue) and points in the second square Q2 = [3, 4] × [1.5, 2.5] are labeled 1 (red)

Describe the density function of $P_X$, and the regression function, Bayes predictor and Bayes risk of P.

In the image, I have defined the Probability density function. Is that correct? I want to find it over both the axis and in general in two dimensions.

A probability distribution $P$ over $X \times \{0, 1\}$. $P$ can be defined in term of its marginal distribution over $X$ , which we will denote by $P_X$ and the conditional labeling distribution, which is defined by the regression function $$ \mu(x) = P_{ (x,y) \sim P} [y = 1 \mid x] $$ Consider a 2-dimensional Euclidean domain, that is $X = \mathbb R^2$, and the following process of data generation: The marginal distribution over $X$ is uniform over two square areas $[1, 2] \times [1, 2] \cup [3, 4] \times [1.5, 2.5]$. Points in the first square $Q_1 = [1, 2] \times [1, 2]$ are labeled 0 (blue) and points in the second square $Q_2 = [3, 4] \times [1.5, 2.5]$ are labeled 1 (red).

Describe the density function of $P_X$, and the regression function, Bayes predictor and Bayes risk of $P$.

In the image, I have defined the Probability density function. Is that correct? I want to find it over both the axis and in general in two dimensions.