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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
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统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYESIAN TEXT REGRESSION

Even though Bayesian NLP has focused mostly on unsupervised learning, Bayesian inference in general is not limited to learning from incomplete data. It is also often used for prediction problems such as classification and regression where the training examples include both the inputs and the outputs of the model.

In this section, we demonstrate Bayesian learning in the case of text regression, predicting a continuous value based on a body of text. We will continue to use the notation from Section $2.2$ and denote a document by $d$, as a set of words and word count pairs. In addition, we will assume some continuous value that needs to be predicted, denoted by the random variable $Y$. To ground the example, $D$ can be a movie review, and $Y$ can be a predicted average number of stars the movie received by critics or its revenue (Joshi et al., 2010). The prediction problem is therefore to predict the number of stars a movie receives from the movie review text.

One possible way to frame this prediction problem is as a Bayesian linear regression problem. This means we assume that we receive as input for the inference algorithm a set of examples $\left(d^{(i)}, y^{(i)}\right)$ for $i \in{1, \ldots, n}$. We assume a function $f(d)$ that maps a document to a vector in $\mathbb{R}^{K}$. This is the feature function that summarizes the information in the document as a vector, and on which the final predictions are based. For example, $K$ could be the size of the vocabulary that the documents span, and $[f(d)] j$ could be the count of the $j$ th word in the vocabulary in document $d$.

A linear regression model typically assumes that there is a stochastic relationship between $Y$ and $d:$
$$Y=\theta \cdot f(d)+\epsilon,$$
where $\theta \in \mathbb{R}^{K}$ is a set of parameters for the linear regression model and $\epsilon$ is a noise term (with zero mean), most often framed as a Gaussian variable with variance $\sigma^{2}$. For the sake of simplicity, we assume for now that $\sigma^{2}$ is known, and we need not make any inference about it. As a result, the learning problem becomes an inference problem about $\theta$.

As mentioned above, $\epsilon$ is assumed to be a Gaussian variable under the model, and as such $Y$ itself is a Gaussian with mean value $\theta \cdot f(d)$ for any fixed $\theta$ and document $d$. The variance of $Y$ is $\sigma^{2}$

In Bayesian linear regression, we assume a prior on $\theta$, a distribution $p(\theta \mid \alpha)$. Consequently, the joint distribution over $\theta$ and $Y^{(i)}$ is:
$$p\left(\theta, Y^{(1)}=y^{(1)}, \ldots, Y^{(n)}=y^{(n)} \mid d^{(1)}, \ldots, d^{(n)}, \alpha\right)=p(\theta \mid \alpha) \prod_{i=1}^{n} p\left(Y^{(i)}=y^{(i)} \mid \theta, d^{(i)}\right)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONJUGATE PRIORS

Basic inference in the Bayesian setting requires computation of the posterior distribution (Chapter 2) -the distribution over the model parameters which is obtained by integrating the information from the prior distribution together with the observed data. Without exercising caution, and putting restrictions on the prion distribution or the likelihoud function, this inference can be intractable. When performing inference with incomplete data (with latent variables), this issue becomes even more severe. In this case, the posterior distribution is defined over both of the parameters and the latent variables.

Conjugate priors eliminate this potential intractability when no latent variables exist, and also help to a large extent when latent variables do exist in the model. A prior family is conjugate to a likelihood if the posterior, obtained as a calculation of
$$\text { posterior }=\frac{\text { prior } \times \text { likelibood }}{\text { evidence }} \text {, }$$
is also a member of the prior family.
We now describe this idea in more detail. We begin by describing the use of conjugate priors in the case of having empirical observations for all random variables in the model (i.e., without having any latent variables). Let $p(\theta \mid \alpha)$ be some prior with byperparameters $\alpha$. The hyperparameters by themselves are parameters-only instead of parametrizing the likelihood function, they parametrize the prior. They can be fixed and known, or be inferred. We assume the hyperparameters are taken from a set of hyperparameters $A$. In addition, let $p(X \mid \theta)$ be a distribution function for the likelihood of the observed data. We observe an instance of the random variable $X=x$. Posterior inference here means we need to identify the distribution $p(\theta \mid x)$. We say that the prior family $p(\theta \mid \alpha)$ is a a conjugate prior with respect to the likelihood $p(X \mid \theta)$ if the following holds for the posterior:
$$p(\theta \mid x, \alpha)=p\left(\theta \mid \alpha^{\prime}\right),$$
for some $\alpha^{\prime}=\alpha^{\prime}(x, \alpha) \in A$. Note that $\alpha^{\prime}$ is a function of the observation $x$ and $\alpha$, the hyperparameter with which we begin the inference. (This means that in order to compute the posterior, we need to be able to compute the function $\alpha^{\prime}(x, \alpha)$.)

The mathematical definition of conjugate priors does not immediately shed light on why they make Bayesian inference more tractable. In fact, according to the deinfition above, the use of a conjugate prior does not guarantee computational tractability. Conjugate priors are useful when the function $\alpha^{\prime}(x, \alpha)$ can be efficiently computed, and indeed this is often the case when conjugate priors are used in practice.

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONJUGATE PRIORS AND NORMALIZATION CONSTANTS

Consider posterior inference in Equation 3.1. The key required calculation was computing the normalization constant $\int_{\theta} p(\theta \mid \alpha) p(x \mid \theta) d \theta$ in order to fully identify the posterior distribution. ${ }^{1}$ This normalization constant is also equal to $p(x \mid \alpha)$, since it is just a marginalization of $\theta$ from the joint distribution $p(\theta, x \mid \alpha)$.

Therefore, the key step required in computing the posterior is calculating $p(x \mid \alpha)$, also called “the evidence.” The posterior can be then readily evaluated at each point $\theta$ by dividing the product of the prior and the likelihood by the evidence.

The use of conjugate prior in Example $3.1$ eliminated the need to explicitly calculate this normalization constant, but instead we were able to calculate it more indirectly. Identifying that Equation $3.2$ has the algebraic form (up to a constant) of the normal distribution immediately dictates that the posterior is a normal variable with the appropriate $\alpha^{\prime}(x, \alpha)$. Therefore, explicitly computing the evidence $p(x \mid \alpha)$ is unnecessary, because the posterior was identified as a (normal) distribution, for which its density is fully known in an analytic form.

If we are interested in computing $p(x \mid \alpha)$, we can base our calculation on the well-known density of the normal distribution. Equation $3.1$ implies that for any choice of $\theta$, it holds that
$$p(x \mid \alpha)=\int_{\theta} p(\theta \mid \alpha) p(x \mid \theta) d \theta=\frac{p(\theta \mid \alpha) p(x \mid \theta)}{p(\theta \mid x, \alpha)} .$$
This is a direct result of applying the chain rule in both directions:
$$p(x, \theta \mid \alpha)=p(x \mid \alpha) p(\theta \mid \alpha, x)=p(\theta \mid \alpha) p(x \mid \theta, \alpha)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYESIAN TEXT REGRESSION

$$Y=\theta \cdot f(d)+\epsilon$$

$$p\left(\theta, Y^{(1)}=y^{(1)}, \ldots, Y^{(n)}=y^{(n)} \mid d^{(1)}, \ldots, d^{(n)}, \alpha\right)=p(\theta \mid \alpha) \prod_{i=1}^{n} p\left(Y^{(i)}=y^{(i)} \mid \theta, d^{(i)}\right)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONJUGATE PRIORS

$$\text { posterior }=\frac{\text { prior } \times \text { likelibood }}{\text { evidence }}$$

$$p(\theta \mid x, \alpha)=p\left(\theta \mid \alpha^{\prime}\right)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|CONJUGATE PRIORS AND NORMALIZATION CONSTANTS

$$p(x \mid \alpha)=\int_{\theta} p(\theta \mid \alpha) p(x \mid \theta) d \theta=\frac{p(\theta \mid \alpha) p(x \mid \theta)}{p(\theta \mid x, \alpha)} .$$

$$p(x, \theta \mid \alpha)=p(x \mid \alpha) p(\theta \mid \alpha, x)=p(\theta \mid \alpha) p(x \mid \theta, \alpha)$$

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MATLAB代写

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