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计算机代写|图形模型代考Graphical Models代写|CS228 Basic Rules

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计算机代写|图形模型代考Graphical Models代写|Basic Rules

The probability of the disjunction (logical sum) of two propositions is given by the sum rule: $P(A+B \mid C)=P(A \mid C)+P(B \mid C)-P(A, B \mid C)$; if propositions $A$ and $B$ are mutually exclusive given $C$, we can simplify it to: $P(A+B \mid C)=P(A \mid$ $C)+P(B \mid C)$. This can be generalized for $N$ mutually exclusive propositions to:
$$P\left(A_{1}+A_{2}+\cdots A_{N} \mid C\right)=P\left(A_{1} \mid C\right)+P\left(A_{2} \mid C\right)+\cdots+P\left(A_{N} \mid C\right)$$
In the case that there are $N$ mutually exclusive and exhaustive hypothesis, $H_{1}, H_{2}, \ldots, H_{N}$, and if the evidence $B$ does not favor any of them, then according to the principle of indifference: $P\left(H_{i} \mid B\right)=1 / N$.

According to the logical interpretation there are no absolute probabilities, all are conditional on some background information ${ }^{1} . P(H \mid B)$ conditioned only on the background $B$ is called a prior probability; once we incorporate some additional information $D$ we call it a posterior probability, $P(H \mid D, B)$. From the product rule we obtain:
$$P(D, H \mid B)=P(D \mid H, B) P(H \mid B)=P(H \mid D, B) P(D \mid B)$$
From which we obtain:
$$P(H \mid D, B)=\frac{P(H \mid B) P(D \mid H, B)}{P(D \mid B)}$$
This last equation is known as the Bayes rule and the term $P(D \mid H, B)$ as the likelihood, $L(D)$.

计算机代写|图形模型代考Graphical Models代写|Random Variables

If we consider a finite set of exhaustive and mutually exclusive propositions ${ }^{2}$, then a discrete variable $X$ can represent this set of propositions, such that each value $x_{i}$ of $X$ corresponds to one proposition. If we assign a numerical value to each proposition $x_{i}$, then $X$ is a discrete random variable. For example, the outcome of the toss of a die is a discrete random variable with 6 possible values $1,2, \ldots, 6$. The probabilities for all possible values of $X, P(X)$ is the probability distribution of $X$. Considering the die example, for a fair die the probability distribution will be:

$$\begin{array}{lcccccc} x & 1 & 2 & 3 & 4 & 5 & 6 \ P(x) & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 \end{array}$$
This is an example of a uniform probability distribution. There are several probability distributions which have been defined. Another common distribution is the binomial distribution. Assume we have an urn with $N$ colored balls, red and black, of which $M$ are red, so the fraction of red balls is $\pi=M / N$. We draw a ball at random, record its color, and return it to the urn, mixing the balls again (so that, in principle, each draw is independent from the previous one). The probability of getting $r$ red balls in $n$ draws is:
$$P(r \mid n, \pi)=\left(\begin{array}{l} n \ r \end{array}\right) \pi^{r}(1-\pi)^{n-r},$$
where $\left(\begin{array}{l}n \ r\end{array}\right)=\frac{n !}{r !(n-r) !}$.
This is an example of a binomial distribution which is applied when there are $n$ independent trials, each with two possible outcomes (success or failure), and the probability of success is constant over all trials. There are many other distributions, we refer the interested reader to the additional reading section at the end of the chapter.

图形模型代写

计算机代写|图形模型代考Graphical Models代写|Basic Rules

$$P\left(A_{1}+A_{2}+\cdots A_{N} \mid C\right)=P\left(A_{1} \mid C\right)+P\left(A_{2} \mid C\right)+\cdots+P\left(A_{N} \mid C\right)$$

$$P(D, H \mid B)=P(D \mid H, B) P(H \mid B)=P(H \mid D, B) P(D \mid B)$$

$$P(H \mid D, B)=\frac{P(H \mid B) P(D \mid H, B)}{P(D \mid B)}$$

计算机代写图形模型代考Graphical Models代写|Random Variables

$$\begin{array}{lllllllllllll} x & 1 & 2 & 3 & 4 & 5 & 6 P(x) & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 & 1 / 6 \end{array}$$

$$P(r \mid n, \pi)=(n r) \pi^{r}(1-\pi)^{n-r},$$

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。