### 统计代写|机器学习代写machine learning代考|Kernel Classifiers from a Bayesian Perspective

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|机器学习代写machine learning代考|The Bayesian Framework

This chapter presents the probabilistic, or Bayesian approach to learning kernel classifiers. It starts by introducing the main principles underlying Bayesian inference both for the problem of learning within a fixed model and across models. The first two sections present two learning algorithms, Gaussian processes and relevance vector machines, which were originally developed for the problem of regression estimation. In regression estimation, one is given a sample of real-valued outputs rather than classes. In order to adapt these methods to the problem of classification we introduce the concept of latent variables which, in the current context, are used to model the probability of the classes. The chapter shows that the principle underlying relevance vector machines is an application of Bayesian model selection to classical Bayesian linear regression. In the third section we present a method which directly models the observed classes by imposing prior knowledge only on weight vectors of unit length. In general, it is impossible to analytically compute the solution to this algorithm. The section presents a Markov chain Monte Carlo algorithm to approximately solve this problem, which is also known as Bayes point learning. Finally, we discuss one of the earliest approaches to the problem the kernel trick to all these algorithms thus rendering them powerful tools in the application of kernel methods to the problem of classification learning.

In the last chapter we saw that a learning problem is given by the identification of an unknown relationship $h \in \mathcal{Y}^{\mathcal{X}}$ between objects $x \in \mathcal{X}$ and classes $y \in \mathcal{Y}$ solely on the basis of a given iid sample $z=(\boldsymbol{x}, \boldsymbol{y})=\left(\left(x_{1}, y_{1}\right), \ldots,\left(x_{m}, y_{m}\right)\right) \in$ $(\mathcal{X} \times \mathcal{Y})^{m}=\mathcal{Z}^{m}$ (see Definition 2.1). Any approach that deals with this problem starts by choosing a hypothesis space ${ }^{1} \mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ and a loss function $l: \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R}$ appropriate for the task at hand. Then a learning algorithm $\mathcal{A}: \cup_{m=1}^{\infty} \mathcal{Z}^{m} \rightarrow \mathcal{H}$ aims to find the one particular hypothesis $h^{*} \in \mathcal{H}$ which minimizes a pre-defined risk determined on the basis of the loss function only, e.g., the expected risk $R[h]$ of the hypothesis $h$ or the empirical risk $R_{\text {emp }}[h, z]$ of $h \in \mathcal{H}$ on the given training sample $z \in \mathcal{Z}^{m}$ (see Definition $2.5$ and 2.11). Once we have learned a classifier $\mathcal{A}(z) \in \mathcal{H}$ it is used for further classification on new test objects. Thus, all the information contained in the given training sample is summarized in the single hypothesis learned.

## 统计代写|机器学习代写machine learning代考|The Power of Conditioning on Data

From a purely Bayesian point of view, for the task of learning we are finished as soon as we have updated our prior belief $\mathbf{P}{\mathrm{H}}$ into the posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ using equation (3.1). Nonetheless, our ultimate goal is to find one (deterministic) function $h \in \mathcal{Y} \mathcal{X}^{\mathcal{X}}$ that best describes the relationship objects and classes, which is implicitly

expressed by the unknown measure $\mathbf{P}{Z}=\mathbf{P}{Y \mid X} \mathbf{P}{X}$. In order to achieve this goal, Bayesian analysis suggests strategies based on the posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{m}}=z^{*}$ :

• If we are restricted to returning a function $h \in \mathcal{H}$ from a pre-specified hypothesis space $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$ and assume that $\mathbf{P}_{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ is highly peaked around one particular function then we determine the classifier with the maximum posterior belief.

Definition 3.6 (Maximum-a-posteriori estimator) For a given posterior belief $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}$ over a hypothesis space $\mathcal{H} \subseteq \mathcal{Y}^{\mathcal{X}}$, the maximum-a-posteriori estimator is defined by ${ }^{5}$ $\mathcal{A}{\mathrm{MAP}}(z) \stackrel{\text { def }}{=} \underset{h \in \mathcal{H}}{\operatorname{argmax}} \mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{\mathrm{m}}=z}(h)$ If we use the inverse loss likelihood and note that the posterior $\mathbf{P}{\mathrm{H} \mid \mathrm{Z}^{m}=z}$ is given by the product of the likelihood and the prior we see that this scheme returns minimizer of the training error and our prior belief, which can be thought of as a regularizer (see also Subsection 2.2.2). The drawback of the MAP estimator is that it is very sensitive to the training sample if the posterior measure is multi modal. Even worse, the classifier $\mathcal{A}_{\text {MAP }}(z) \in \mathcal{H}$ is, in general, not unique, for example if the posterior measure is uniform.

• If we are not confined to returning a function from the original hypothesis space $\mathcal{H}$ then we can use the posterior measure $\mathbf{P}{\mathrm{H}{\mid \mathrm{Z}^{m}}=z}$ to induce a measure $\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathrm{Z}^{m}=z}$ over classes $y \in \mathcal{Y}$ at a novel object $x \in \mathcal{X}$ by $$\mathbf{P}{\mathrm{Y} \mid \mathrm{X}=x, \mathbf{Z}^{m}=z}(y)=\mathbf{P}_{\mathrm{H} \mid \mathrm{Z}^{m}=z}({h \in \mathcal{H} \mid h(x)=y})$$
This measure can then be used to determine the class $y$ which incurs the smallest loss at a given object $x$.

## 统计代写|机器学习代写machine learning代考|Bayesian Linear Regression

In the regression estimation problem we are given a sequence $\boldsymbol{x}=\left(x_{1}, \ldots, x_{m}\right) \in$ $\mathcal{X}^{m}$ of $m$ objects together with a sequence $t=\left(t_{1}, \ldots, t_{m}\right) \in \mathbb{R}^{m}$ of $m$ real-valued outcomes forming the training sample $z=(x, t)$. Our aim is to find a functional relationship $f \in \mathbb{R}^{\mathcal{X}}$ between objects $x$ and target values $t$. In accordance with Chapter 2 we will again consider a linear model $\mathcal{F}$
$\mathcal{F}={x \mapsto\langle\mathbf{x}, \mathbf{w}\rangle \mid \mathbf{w} \in \mathcal{K}}$,
where we assume that $\mathbf{x} \stackrel{\text { def }}{=} \phi(x)$ and $\phi: \mathcal{X} \rightarrow \mathcal{K} \subseteq \ell_{2}^{n}$ is a given feature mapping (see also Definition 2.2). Note that $\mathbf{x} \in \mathcal{K}$ should not be confused with the training sequence $\boldsymbol{x} \in \mathcal{X}^{m}$ which results in an $m \times n$ matrix $\mathbf{X}=\left(\mathbf{x}{1}^{\prime} ; \ldots ; \mathbf{x}{m}^{\prime}\right)$ when $\boldsymbol{\phi}$ is applied to it.

First, we need to specify a prior over the function space $\mathcal{F}$. Since each function $f_{\mathrm{w}}$ is uniquely parameterized by its weight vector $\mathbf{w} \in \mathcal{K}$ it suffices to consider a prior distribution on weight vectors. For algorithmic convenience let the prior distribution over weights be a Gaussian measure with mean $\mathbf{0}$ and covariance $\mathbf{I}{n}$, i.e., $\mathbf{P}{\mathrm{W}}=\operatorname{Normal}\left(\mathbf{0}, \mathbf{I}{n}\right)$. Apart from algorithmical reasons such a prior favors weight vectors $\mathbf{w} \in \mathcal{K}$ with small coefficients $w{i}$ because the log-density is proportional to $-|\mathbf{w}|^{2}=$ $-\sum_{i=1}^{n} w_{i}^{2}$ (see Definition A.26). In fact, the weight vector with the highest apriori density is $\mathbf{w}=\mathbf{0}$.

Second, we must specify the likelihood model $\mathbf{P}{T^{m} \mid X^{m}=x, W=w}$. Let us assume that, for a given function $f{\mathrm{w}}$ and a given training object $x \in \mathcal{X}$, the real-valued output $\mathrm{T}$ is normally distributed with mean $f_{\mathrm{w}}(x)$ and variance $\sigma_{t}^{2}$. Using the notion of an inverse loss likelihood such an assumption corresponds to using the squared loss, i.e., $l_{2}(f(x), t)=(f(x)-t)^{2}$ when considering the prediction task under a machine learning perspective. Further, it shall be assumed that the real-valued outputs $\mathrm{T}{1}$ and $\mathrm{T}{2}$ at $x_{1}$ and $x_{2} \neq x_{1}$ are independent. Combining these two requirements results in the following likelihood model:
$$\mathbf{P}{\mathrm{T}^{m} \mid \mathbf{X}^{\mathrm{m}}=x, \mathbf{W}=\mathbf{w}}(t)=\operatorname{Normal}\left(\mathbf{X w}, \sigma{t}^{2} \mathbf{I}_{m}\right) \text {. }$$

## 统计代写|机器学习代写machine learning代考|The Power of Conditioning on Data

• 如果我们仅限于返回一个函数H∈H从预先指定的假设空间H⊆是X并假设磷H∣从米=和在一个特定函数周围高度达到峰值，然后我们确定具有最大后验置信度的分类器。

• 如果我们不局限于从原始假设空间返回一个函数H那么我们可以使用后验测度磷H∣从米=和引发措施磷是∣X=X,从米=和过课是∈是在一个新奇的物体上X∈X经过磷是∣X=X,从米=和(是)=磷H∣从米=和(H∈H∣H(X)=是)
然后可以使用此度量来确定类别是在给定对象上产生最小的损失X.

F=X↦⟨X,在⟩∣在∈ķ,

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## MATLAB代写

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