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数学代写|统计计算作业代写Statistical Computing代考|The Gibbs sampler

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数学代写|统计计算作业代写Statistical Computing代考|Description of the method

The Gibbs sampler is applicable in situations where the state space $S$ can be written as a finite product of spaces, that is where we have
$$S=S_{1} \times S_{2} \times \cdots \times S_{n} .$$
Elements of this space are vectors $x=\left(x_{i}\right){i=1,2, \ldots, n}$ where $x{i} \in S_{i}$ for every $i$. Situations where the Gibbs sampler can be applied include:

• In Bayesian parameter estimation problems, $n$ is typically small, say 2 or 3 , and often the spaces $S_{i}$ have different dimensions. We will study examples of this type in Section 4.4.2.In applications from statistical mechanics or image processing, $n$ is typically large but all spaces $S_{i}$ are the same. In this case we write
• $$• S=C^{I} •$$
• where $I={1,2, \ldots, n}$ and $C$ is the state space for the individual components.

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Application to parameter estimation

Let $k \in \mathbb{N}$ and $\sigma>0$ be fixed and let $\mu_{1}, \ldots, \mu_{k} \sim \varphi_{\mu}$ be independent random vectors in $\mathbb{R}^{d}$, where we assume the distribution $\varphi_{\mu}$ to have a density. Given $\mu=$ $\left(\mu_{1}, \ldots, \mu_{k}\right)$, let $X_{1}, \ldots, X_{n}$ be an i.i.d. sample from the mixture distribution
$$\varphi_{X \mid \mu}=\frac{1}{k} \sum_{a=1}^{k} \mathcal{N}\left(\mu_{a}, \sigma I_{d}\right)$$
where the mixture components are normal distribution with mean $\mu_{a}$ and covariance matrix $\sigma I_{d}$ and where $I_{d}$ denotes the $d$-dimensional identity matrix. In the later parts of this example we will restrict ourselves to the case $d=2$ and we will assume that, under the prior distribution, $\mu_{1}, \ldots, \mu_{k} \sim \mathcal{U}([-10,+10] \times[-10,+10])$ i.i.d., that is we will assume that the prior density for the cluster means $\mu_{a}$ is given by
$$\varphi_{\mu}\left(\mu_{a}\right)=\frac{1}{20^{2}} \mathbb{1}{[-10,10] \times[-10,10]}\left(\mu{a}\right)$$
for all $\mu_{a} \in \mathbb{R}^{2}$.

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Applications to image processing

A sample of size 100, similar to Figure $4.5$ but using a different seed of the random number generator. Different from Figure 4.5, the Markov chain here has not yet converged to the stationary distribution, despite the presence of a burn-in period of length 10000: two components of $\mu$, represented by the symbols $\times$ and $\diamond$, are concentrated in one cluster while the component $+i$ is situated in the gap between two clusters. This figure was created by selecting one of a large number of runs of the algorithm.
where $I$ is the lattice
$$I={1,2, \ldots, L} \times{1,2, \ldots, L}$$
The elements of $S$ are vectors of the form $x=\left(x_{i}\right){i \in I}$. States $x \in S$ can be visualised as square grids of small black and white dots, where the colour at location $i \in I$ in the grid encodes the possible values $-1$ and $+1$ for $x{i}$.

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|DESCRIPTION OF THE METHOD

Gibbs 采样器适用于状态空间小号可以写成空间的有限乘积，这就是我们有

• 在贝叶斯参数估计问题中，n通常很小，例如 2 或 3 ，并且通常是空格小号一世有不同的维度。我们将在 4.4.2 节研究这种类型的例子。在统计力学或图像处理的应用中，n通常很大，但所有空间小号一世是相同的。在这种情况下，我们写
• $$• S=C^{我} •$$
• 在哪里一世=1,2,…,n和C是各个组件的状态空间。

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|APPLICATION TO PARAMETER ESTIMATION

$$\varphi_{\mu}\left给出\mu_{a}\对\mu_{a}\对=\frac{1}{20^{2}} \mathbb{1} {−10,10\次−10,10}\left(\mu {a}\right)$$

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|APPLICATIONS TO IMAGE PROCESSING

where I is the lattice
I=1,2,…,L×1,2,…,L
The elements of S are vectors of the form $x=\leftx_{i}\rightx_{i}\right{i \in I}.Statesx \in Scanbevisualisedassquaregridsofsmallblackandwhitedots,wherethecolouratlocationi \in Iinthegridencodesthepossiblevalues-1and+1forx{i}$.