计算机代写|机器学习代写machine learning代考|Transformation of Random Variables

Assume we have a set of $n$ continuous random variables, denoted as $\left{X_1, X_2, \cdots, X_n\right}$. If we arrange their values as a vector $\mathbf{x} \in \mathbb{R}^n$, we can represent their joint distribution (p.d.f.) as $p(\mathbf{x})$. We can apply some transformations to convert them into another set of $n$ continuous random variables as follows:
\begin{aligned} Y_1 &=f_1\left(X_1, X_2, \cdots, X_n\right) \ Y_2 &=f_2\left(X_1, X_2, \cdots, X_n\right) \ & \vdots \ Y_n &=f_n\left(X_1, X_2, \cdots, X_n\right) \end{aligned}
We similarly arrange the values of the new random variables $\left{Y_1, Y_2, \cdots, Y_n\right}$ as another vector $\mathbf{y} \in \mathbb{R}^n$, and we further represent the transformations as a single vector-valued and multivariate function:
$$\mathbf{y}=f(\mathbf{x}) \quad\left(\mathbf{x} \in \mathbb{R}^n, \mathbf{y} \in \mathbb{R}^n\right) .$$
If this function is continuously differentiable and invertible, we can represent the inverse function as $\mathbf{x}=f^{-1}(\mathbf{y})$. Under these conditions, we are able to conveniently derive the joint distribution for these new random variables, that is, $p(\mathbf{y})$.

We first need to define the so-called Jacobian matrix for these inverse transformations $\mathbf{x}=f^{-1}(\mathbf{y})$, as follows:
$$\mathbf{y})=\left[\frac{\partial x_i}{\partial y_j}\right]_{n \times n}=\left[\begin{array}{cccc} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2} & \cdots & \frac{\partial x_1}{\partial y_n} \ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} & \cdots & \frac{\partial x_2}{\partial y_n} \ \vdots & \vdots & \ddots & \vdots \ \frac{\partial x_n}{\partial y_1} & \frac{\partial x_n}{\partial y_2} & \cdots & \frac{\partial x_n}{\partial y_n} \end{array}\right]$$
According to Bertsekas [21], the joint distribution of the new random variables can be derived as
$$p(\mathbf{y})=|\mathbf{J}(\mathbf{y})| p(\mathbf{x})=|\mathbf{J}(\mathbf{y})| p\left(f^{-1}(\mathbf{y})\right),$$
where $|\mathbf{J}(\mathbf{y})|$ denotes the determinant of the Jacobian matrix.

计算机代写|机器学习代写machine learning代考|Information and Entropy

The first fundamental problem in information theory is how to quantitatively measure information. The most significant progress to address this issue is attributed to Shannon’s brilliant idea of using probabilities. The amount of information that a message delivers solely depends on the probability of observing this message rather than its real content or anything else. This treatment allows us to establish a general mathematical framework to handle information independent of application domains. According to Shannon, if the probability of observing an event $A$ is $\operatorname{Pr}(A)$, the amount of information delivered by this event $A$ is calculated as follows:
$$I(A)=\log _2\left(\frac{1}{\operatorname{Pr}(A)}\right)=-\log _2(\operatorname{Pr}(A)) .$$
When we use the binary logarithm $\log _2(\cdot)$, the unit of the calculated information is the bit. Shannon’s definition of information is intuitive and consistent with our daily experience. A small-probability event will surprise us because it contains more information, whereas a common event that happens every day is not telling us anything new.

Shannon’s idea can be extended to measure information for random variables. As we know, a random variable may take different values in different probabilities, and we can define the so-called entropy for a discrete random variable $X$ as the expectation of the information for it to take different values:
$$H(X)=\mathbb{E}\left[-\log _2 \operatorname{Pr}(X=x)\right]=-\sum_x p(x) \log _2 p(x),$$
where $p(x)$ is the p.m.f. of $X$. Intuitively speaking, the entropy $H(X)$ represents the amount of uncertainty associated with the random variable $X$, namely, the amount of information we need to fully resolve this random variable.

机器学习代考

计算机代写|机器学习代写machine learning代考|Transformation of Random Variables

$$Y_1=f_1\left(X_1, X_2, \cdots, X_n\right) Y_2 \quad=f_2\left(X_1, X_2, \cdots, X_n\right) \vdots Y_n \quad=f_n\left(X_1, X_2, \cdots, X_n\right)$$

$$\mathbf{y}=f(\mathbf{x}) \quad\left(\mathbf{x} \in \mathbb{R}^n, \mathbf{y} \in \mathbb{R}^n\right)$$

$$p(\mathbf{y})=|\mathbf{J}(\mathbf{y})| p(\mathbf{x})=|\mathbf{J}(\mathbf{y})| p\left(f^{-1}(\mathbf{y})\right),$$

计算机代写|机器学习代写machine learning代考|Information and Entropy

$$I(A)=\log _2\left(\frac{1}{\operatorname{Pr}(A)}\right)=-\log _2(\operatorname{Pr}(A))$$

$$H(X)=\mathbb{E}\left[-\log _2 \operatorname{Pr}(X=x)\right]=-\sum_x p(x) \log _2 p(x),$$

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