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# 计算机代写|机器学习代写Machine Learning代考|ACDL2022 Hyperparameters

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## 计算机代写|机器学习代写Machine Learning代考|Hyperparameters

There are often implicit parameters in our model that we hold fixed, such as the covariance constants in linear regression, or the parameters that govern the prior distribution over the weights. These are usually called “hyperparameters.” For example, in the RBF model, the hyperparameters constitute the parameters $\alpha, \sigma^2$, and the parameters of the basis functions (e.g., the width of the basis functions). Thus far we have assumed that the hyperparameters were “known” (which means that someone must set them by hand), or estimated by cross-validation (which has a number of pitfalls, including long computation times, especially for large numbers of hyperparameters). Instead of either of these approaches, we may apply the Bayesian approach in order to directly estimate these values as well.

To find a MAP estimate for the $\alpha$ parameter in the above linear regression example we compute:
$$\alpha^*=\arg \max \ln p\left(\alpha \mid x_{1: N}, y_{1: N}\right)$$
Where
$$p\left(\alpha \mid x_{1: N}, y_{1: N}\right)=\frac{p\left(y_{1: N} \mid x_{1: N}, \alpha\right) p(\alpha)}{p\left(y_{1: N} \mid x_{1: N}\right)}$$
and
\begin{aligned} p\left(y_{1: N} \mid x_{1: N}, \alpha\right) & =\int p\left(y_{1: N}, \mathbf{w} \mid x_{1: N}, \alpha\right) d \mathbf{w} \ & =\int p\left(y_{1: N} \mid x_{1: N}, \mathbf{w}, \alpha\right) p(\mathbf{w} \mid \alpha) d \mathbf{w} \ & =\int\left(\prod_i p\left(y_i \mid x_i, \mathbf{w}, \alpha\right)\right) p(\mathbf{w} \mid \alpha) d \mathbf{w} \end{aligned}

## 计算机代写|机器学习代写Machine Learning代考|Bayesian Model Selection

How do we choose which model to use? For example, we might like to automatically choose the form of the basis functions or the number of basis functions. Cross-validation is one approach, but it can be expensive, and, more importantly, inaccurate if small amounts of data are available. In general one intuition is that we want to choose simple models over complex models to avoid overfitting,insofar as they provide equivalent fits to the data. Below we consider a Bayesian approach to model selection which provides just such a bias to simple models.

The goal of model selection is to choose the best model from some set of candidate models $\left{\mathcal{M}i\right}{i=1}^L$ based on some observed data $\mathcal{D}$. This may be done either with a maximum likelihood approach (picking the model that assigns the largest likelihood to the data) or a MAP approach (picking the model with the highest posterior probability). If we take a uniform prior over models (i.e. $p\left(\mathcal{M}_i\right)$ is a constant for all $\left.i=1 \ldots L\right)$ then these approaches can be seen to be equivalent since:
\begin{aligned} p\left(\mathcal{M}_i \mid \mathcal{D}\right) & =\frac{p\left(\mathcal{D} \mid \mathcal{M}_i\right) p\left(\mathcal{M}_i\right)}{p(\mathcal{D})} \ & \propto p\left(\mathcal{D} \mid \mathcal{M}_i\right) \end{aligned}
In practice a uniform prior over models may not be appropriate, but the design of suitable priors in these cases will depend significantly on one’s knowledge of the application domain. So here we will assume a uniform prior over models and focus on $p\left(\mathcal{D} \mid \mathcal{M}_i\right)$.

## 计算机代写|机器学习代写Machine Learning代考|Hyperparameters

$$\alpha^*=\arg \max \ln p\left(\alpha \mid x_{1: N}, y_{1: N}\right)$$

$$p\left(\alpha \mid x_{1: N}, y_{1: N}\right)=\frac{p\left(y_{1: N} \mid x_{1: N}, \alpha\right) p(\alpha)}{p\left(y_{1: N} \mid x_{1: N}\right)}$$

$$p\left(y_{1: N} \mid x_{1: N}, \alpha\right)=\int p\left(y_{1: N}, \mathbf{w} \mid x_{1: N}, \alpha\right) d \mathbf{w} \quad=\int p\left(y_{1: N} \mid x_{1: N}, \mathbf{w}, \alpha\right) p(\mathbf{w} \mid \alpha) d \mathbf{w}=\int\left(\prod_i p\left(y_i \mid x_i, \mathbf{w}, \alpha\right)\right) p(\mathbf{w} \mid \alpha) d \mathbf{w}$$

## 计算机代写|机器学习代写Machine Learning代考|Bayesian Model Selection

$$p\left(\mathcal{M}_i \mid \mathcal{D}\right)=\frac{p\left(\mathcal{D} \mid \mathcal{M}_i\right) p\left(\mathcal{M}_i\right)}{p(\mathcal{D})} \quad \propto p\left(\mathcal{D} \mid \mathcal{M}_i\right)$$

## 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。