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计算机代写|机器学习代写Machine Learning代考|COMP5318 Bayesian Decision Theory

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计算机代写|机器学习代写Machine Learning代考|Bayesian Decision Theory

Bayesian decision theory is a fundamental decision-making approach under the probability framework. In an ideal situation when all relevant probabilities were known, Bayesian decision theory makes optimal classification decisions based on the probabilities and costs of misclassifications. In the following, we demonstrate the basic idea of Bayesian decision theory with multiclass classification.

Let us assume that there are $N$ distinct class labels, that is, $\mathcal{Y}=\left{c_1, c_2, \ldots, c_N\right}$. Let $\lambda_{i j}$ denote the cost of misclassifying a sample of class $c_j$ as class $c_i$. Then, we can use the posterior probability $P\left(c_i \mid \boldsymbol{x}\right)$ to calculate the expected loss of classifying a sample $\boldsymbol{x}$ as class $c_i$, that is, the conditional risk of the sample $x$
$$R\left(c_i \mid \boldsymbol{x}\right)=\sum_{j=1}^N \lambda_{i j} P\left(c_j \mid \boldsymbol{x}\right)$$
Our task is to find a decision rule $h: \mathcal{X} \mapsto \mathcal{Y}$ that minimizes the overall risk
$$R(h)=\mathbb{E}_{\boldsymbol{x}}[R(h(\boldsymbol{x}) \mid \boldsymbol{x})] .$$
The overall risk $R(h)$ is minimized when the conditional risk $R(h(\boldsymbol{x}) \mid \boldsymbol{x})$ of each sample $\boldsymbol{x}$ is minimized. This leads to the Bayes decision rule: to minimize the overall risk, classify each sample as the class that minimizes the conditional risk $R(c \mid \boldsymbol{x})$ :
$$h^(\boldsymbol{x})=\underset{c \in \mathcal{Y}}{\arg \min } R(c \mid \boldsymbol{x}),$$ where $h^$ is called the Bayes optimal classifier, and its associated overall risk $R\left(h^\right)$ is called the Bayes risk. $1-R\left(h^\right)$ is the best performance that can be achieved by any classifiers, that is, the theoretically achievable upper bound of accuracy for any machine learning models.

计算机代写|机器学习代写Machine Learning代考|Maximum Likelihood Estimation

A general strategy of estimating the class-conditional probability is to hypothesize a fixed form of probability distribution, and then estimate the distribution parameters using the training samples. To be specific, let $P(\boldsymbol{x} \mid c)$ denote class-conditional probability of class $c$, and suppose $P(\boldsymbol{x} \mid c)$ has a fixed form determined by a parameter vector $\boldsymbol{\theta}_c$. Then, the task is to estimate $\theta_c$ from a training set $D$. To be precise, we write $P(\boldsymbol{x} \mid c)$ as $P\left(\boldsymbol{x} \mid \boldsymbol{\theta}_c\right)$.

The training process of probabilistic models is the process of parameter estimation. There are two different ways of thinking about parameters. On the one hand, the Bayesian school thinks that the parameters are unobserved random variables following some distribution, and hence we can assume prior distributions for the parameters and estimate posterior distribution from observed data. On the other hand, the Frequentist school supports the view that parameters have fixed values though they are unknown, and hence they can be determined The remaining of this section discusses the Maximum LikeliThe remaining of this section discusses the Maximum Likelischool and is a classic method of estimating the probability distribution from samples.

Let $D_c$ denote the set of class $c$ samples in the training set $D$, and further suppose the samples are i.i.d. samples. Then, the likelihood of $D_c$ for a given parameter $\boldsymbol{\theta}c$ is $$P\left(D_c \mid \boldsymbol{\theta}_c\right)=\prod{\boldsymbol{x} \in D_c} P\left(\boldsymbol{x} \mid \boldsymbol{\theta}_c\right)$$

计算机代写|机器学习代写Machine Learning代考|Bayesian Decision Theory

$$R\left(c_i \mid \boldsymbol{x}\right)=\sum_{j=1}^N \lambda_{i j} P\left(c_j \mid \boldsymbol{x}\right)$$

$$R(h)=\mathbb{E}_{\boldsymbol{x}}[R(h(\boldsymbol{x}) \mid \boldsymbol{x})] .$$

$$\left.h^{(} \boldsymbol{x}\right)=\underset{c \in \mathcal{Y}}{\arg \min } R(c \mid \boldsymbol{x}),$$

计算机代写|机器学习代写Machine Learning代考|Maximum Likelihood Estimation

$$P\left(D_c \mid \boldsymbol{\theta}_c\right)=\prod \boldsymbol{x} \in D_c P\left(\boldsymbol{x} \mid \boldsymbol{\theta}_c\right)$$

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。