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数学代写|统计机器学习作业代写Statistical Machine Learning代考|Genomic-Enabled Prediction BayesB and BayesC Models

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数学代写|统计机器学习作业代写Statistical Machine Learning代考|BayesC is the same as the BRR

For this model, the full conditional distributions of $\mu$ and $\sigma_{\beta}^{2}$ are the same as the model described before, that is, $\mu \mid-\sim N\left(\widetilde{\mu}, \tilde{\sigma}{\mu}^{2}\right)$ and $\sigma^{2} \mid-\sim \chi{\sim, s}^{-2}$. However, for the rest of the parameters, this does not have a known form and is not easy to simulate from them. A solution is to introduce a latent variable to represent the prior distribution of each $\beta_{j}$, and compute all the conditional distributions in this augmented scheme, including the distribution corresponding to the latent variable. To do this, note that this prior can be specified by assuming that conditional to a binary latent variable $Z_{j}$,
$$\beta_{j} \mid \sigma_{\beta}^{2}, Z_{j}=z \sim\left{\begin{array}{l} N\left(0, \sigma_{\beta}^{2}\right), \ \mathrm{DG}(0) \end{array}\right.$$

数学代写|统计机器学习作业代写STATISTICAL MACHINE LEARNING代考|Simulate from the full conditional

where $Z_{j}$ is a Bernoulli random variable with parameter $\pi_{p}\left(Z_{j} \sim \operatorname{Ber}\left(\pi_{p}\right)\right)$. With this introduced latent variable, all the full conditionals can be derived, as is described next.
If the current value of $z_{j}$ is 1 , the full conditional posterior of $\beta_{j}$ is

\begin{aligned} f\left(\beta_{j} \mid-\right) & \propto L\left(\mu, \boldsymbol{\beta}{0}, \sigma^{2} ; \boldsymbol{y}\right) f\left(\beta{j} \mid \sigma_{\beta}^{2}, z_{j}\right) \ & \propto \exp \left(-\frac{1}{2 \sigma^{2}}\left|\boldsymbol{y}-1_{n} \mu-\boldsymbol{X}{1} \boldsymbol{\beta}{0}\right|\right) \frac{1}{\sqrt{2 \pi \sigma_{\beta}^{2}}} \exp \left(-\frac{\beta_{j}^{2}}{2 \sigma_{\beta}^{2}}\right) \ & \propto \exp \left(-\frac{1}{2 \sigma^{2}} \sum_{i=1}^{n}\left(y_{i j}-x_{i j} \beta_{j}\right)^{2}-\frac{1}{2 \sigma_{\beta}^{2}} \beta_{j}^{2}\right) \ & \propto \exp \left{-\frac{1}{2}\left[\left(\sigma_{\beta}^{-2}+\sigma^{-2} \sum_{i=1}^{n} x_{i j}^{2}\right) \beta_{j}^{2}-2 \sigma^{-2} \sum_{i=1}^{n} x_{i j} y_{i j} \beta_{j}+\frac{1}{\sigma^{2}} \sum_{i=1}^{n} y_{i j}^{2}\right]\right} \ & \propto \exp \left[-\frac{\left.\left(\beta_{j}-\tilde{\beta}{j}\right)^{2}\right]}{2 \tilde{\sigma}{j}^{2}}\right], \end{aligned}
where $\quad y_{i j}=y_{i}-\sum_{k=1}^{p} x_{i k} \beta_{k} \quad, \quad \tilde{\sigma}{j}^{2}=\left(\sigma{\beta}^{-2}+\sigma^{-2} \sum_{i=1}^{n} x_{i j}^{2}\right)^{-1}, \quad$ and $\quad \tilde{\beta}{j}=$ $k \neq j$ $\sigma^{-2} \tilde{\sigma}{j}^{2} \sum_{i=1}^{n} x_{i j} y_{i j}$. That is, when the current value of $z_{j}$ is $1, \beta_{j} \mid-\sim N\left(\tilde{\beta}{j}, \tilde{\sigma}{j}^{2}\right)$. However, if $z_{j}=0$, the full conditional posterior of $\beta_{j}$ is a degenerate random variable at 0 , that is, $\beta_{j} \mid-\sim \mathrm{DG}(0)$.

数学代写|统计机器学习作业代写STATISTICAL MACHINE LEARNING代考|BAYESC IS THE SAME AS THE BRR

$\mu$ and $\sigma_{\beta}^{2}$ are the same as the model described before, that is, $\mu \mid-\sim N\left(\widetilde{\mu}, \tilde{\sigma}{\mu}^{2}\right)$ and $\sigma^{2} \mid-\sim \chi{\tilde{v}, S}^{-2}$. However, for the rest of the parameters, this does not have a known form and is not easy to simulate from them. A solution is to introduce a latent variable to represent the prior distribution of each $\beta_{j}$, and compute all the conditional distributions in this augmented scheme, including the distribution corresponding to the latent variable. To do this, note that this prior can be specified by assuming that conditional to a binary latent variable $Z_{j}$,
$$\beta_{j} \mid \sigma_{\beta}^{2}, Z_{j}=z \sim\left{\begin{array}{l} N\left(0, \sigma_{\beta}^{2}\right) \ \mathrm{DG}(0) \end{array}\right.$$
where $Z_{j}$ is a Bernoulli random variable with parameter $\pi_{p}\left(Z_{j} \sim \operatorname{Ber}\left(\pi_{p}\right)\right)$.

数学代写|统计机器学习作业代写STATISTICAL MACHINE LEARNING代考|SIMULATE FROM THE FULL CONDITIONAL

\begin{aligned} f\left(\beta_{j} \mid-\right) & \propto L\left(\mu, \boldsymbol{\beta}{0}, \sigma^{2} ; \boldsymbol{y}\right) f\left(\beta{j} \mid \sigma_{\beta}^{2}, z_{j}\right) \ & \propto \exp \left(-\frac{1}{2 \sigma^{2}}\left|\boldsymbol{y}-1_{n} \mu-\boldsymbol{X}{1} \boldsymbol{\beta}{0}\right|\right) \frac{1}{\sqrt{2 \pi \sigma_{\beta}^{2}}} \exp \left(-\frac{\beta_{j}^{2}}{2 \sigma_{\beta}^{2}}\right) \ & \propto \exp \left(-\frac{1}{2 \sigma^{2}} \sum_{i=1}^{n}\left(y_{i j}-x_{i j} \beta_{j}\right)^{2}-\frac{1}{2 \sigma_{\beta}^{2}} \beta_{j}^{2}\right) \ & \propto \exp \left{-\frac{1}{2}\left[\left(\sigma_{\beta}^{-2}+\sigma^{-2} \sum_{i=1}^{n} x_{i j}^{2}\right) \beta_{j}^{2}-2 \sigma^{-2} \sum_{i=1}^{n} x_{i j} y_{i j} \beta_{j}+\frac{1}{\sigma^{2}} \sum_{i=1}^{n} y_{i j}^{2}\right]\right} \ & \propto \exp \left[-\frac{\left.\left(\beta_{j}-\tilde{\beta}{j}\right)^{2}\right]}{2 \tilde{\sigma}{j}^{2}}\right] \end{aligned}
where $\quad y_{i j}=y_{i}-\sum_{k=1}^{p} x_{i k} \beta_{k} \quad, \quad \widetilde{\sigma}{j}^{2}=\left(\sigma{\beta}^{-2}+\sigma^{-2} \sum_{i=1}^{n} x_{i j}^{2}\right)^{-1} \quad, \quad$ and $\quad \widetilde{\beta}{j}=$ $k \neq j$ $\sigma^{-2} \tilde{\sigma}{j}^{2} \sum_{i=1}^{n} x_{i j} y_{i j}$. That is, when the current value of $z_{j}$ is $1, \beta_{j} \mid-\sim N\left(\tilde{\beta}{j}, \tilde{\sigma}{j}^{2}\right)$. However, if $z_{j}=0$, the full conditional posterior of $\beta_{j}$ is a degenerate random variable at 0 , that is, $\beta_{j} \mid-\sim \mathrm{DG}(0)$.
The full conditional distribution of $Z_{j}$ is

广义线性模型代考

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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。