### 统计代写|应用随机过程代写Stochastic process代考| Special topics

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|应用随机过程代写Stochastic process代考|Partially observed data

Assume now that the Markov chain is only observed at a number of finite time points. Suppose, for example, that $x_{0}$ is a known initial state and that we observe $\mathbf{x}{o}=\left(x{n_{1}}, \ldots, x_{n_{\mathrm{m}}}\right)$, where $n_{1}<\ldots<n_{m} \in N$. In this case, the likelihood function is
$$I\left(\boldsymbol{P} \mid \mathbf{x}{o}\right)=\prod{i=1}^{m} p_{\left.n_{i-1} n_{i}-t_{i-1}\right)}^{\left(l_{i}\right)}$$
where $p_{i j}^{(t)}$ represents the $(i, j)$ th element of the $t$ step transition matrix, defined in Section 1.3.1. In many cases, the computation of this likelihood will be complex. Therefore, it is often preferable to consider inference based on the reconstruction of missing observations. Let $\mathbf{x}{m}$ represent the unobserved states at times $1, \ldots$, $t{1}-1, t_{1}+1, \ldots, t_{n-1}-1, t_{n-1}+1, \ldots, t_{n}$ and let $\mathbf{x}$ represent the full data sequence. Then, given a matrix beta prior, we have that $\boldsymbol{P} \mid \mathbf{x}$ is also matrix beta. Furthermore, it is immediate that
$$P\left(\mathbf{x}{m} \mid \mathbf{x}{o}, \boldsymbol{P}\right)=\frac{P(\mathbf{x} \mid \boldsymbol{P})}{P\left(\mathbf{x}{o} \mid \boldsymbol{P}\right)} \propto P(\mathbf{x} \mid \boldsymbol{P}),$$ which is easy to compute for given $\boldsymbol{P}, \mathbf{x}{m}$. One possibility would be to set up a Metropolis within Gibbs sampling algorithm to sample from the posterior distribution of $\boldsymbol{P}$.

Such an approach is reasonable if the amount of missing data is relatively small. However, if there is much missing data, it will be very difficult to define an appropriate algorithm to generate data from $P\left(\mathbf{x}{m} \mid \mathbf{x}{o}, \boldsymbol{P}\right)$ in (3.5). In such cases, one possibility is to generate the elements of $\mathbf{x}{m}$ one by one, using individual Gibbs steps. Thus, if $t$ is a time point amongst the times associated with the missing observations, then we can generate a state $x{t}$ using
$$P\left(x_{t} \mid \mathbf{x}{-f}, \boldsymbol{P}\right) \propto p{x_{t-1} x_{t}} p_{x_{t} x_{t+1}}$$
where $\mathbf{x}_{-t}$ represents the complete sequence of states except for the state at time $t$.

## 统计代写|应用随机过程代写Stochastic process代考|Reversible Markov chains

Assume that we have a reversible Markov chain with unknown transition matrix $\boldsymbol{P}$ and equilibrium distribution $\pi$ satisfying the conditions of Definition 3.1. Then, for the standard experiment of observing a sequence of observations, $x_{0}, \ldots, x_{n}$, from the chain, where the initial state $x_{0}$ is assumed known, a conjugate prior distribution can be derived as follows.

First, the chain is represented as a graph, $G$, with vertices $V$ and edges $E$, so that two vertices $i$ and $j$ are connected by an edge, $e={i, j}$, if and only if $p_{i j}>0$ and the edges $e \in E$ are weighted so that, for $e={i, j}, w_{e} \propto \pi(i) p_{i j}=\pi(j) p_{j i}$ and $\sum_{e \in E} w_{e}=1$. Note that if $p_{i i}>0$, then there is a corresponding edge, $e={i, i}$ called a loop. The set of loops shall be denoted by $E_{\text {loop. }}$.

A conjugate probability distribution of a reversible Markov chain can now be defined as a distribution over the weights, $w$ as follows. For an edge $e \in E$, let $\bar{e}$ represent the endpoints of $e$; for a vertex $v \in V$, set $w_{v}=\sum_{e: v \in \bar{e}} w_{c}$. Also, define $\mathcal{T}$ to be the set of spanning trees of $G$, that is, the set of maximal subgraphs that contains all loops in $G$, but no cycles. For a spanning tree, $T \in \mathcal{T}$, let $E(T)$ represent the edge set of $T$. Then, a conjugate prior distribution for $w$ is given by:
$$f\left(\mathbf{w} \mid v_{0}, \mathbf{a}\right) \propto \frac{\prod_{e \in E \backslash E_{\text {losp }}} w_{e}^{a_{e}-1 / 2} \prod_{e \in E_{\text {lopp }}} w_{e}^{a_{e} / 2-1}}{w_{v_{0}}^{a_{v_{0}} / 2} \prod_{v \in V \backslash v_{0}} w_{v}^{\left(a_{v}+1\right) / 2}} \sqrt{\sum_{T \in T} \prod_{e \notin E(T)} \frac{1}{w_{c}}}$$
where $v_{0}$ represents the node of the graph corresponding to the initial state, $x_{0}$, $\mathbf{a}=\left(a_{e}\right){e \in E}$ is a matrix of arbitrary, nonnegative constants and $a{v}=\sum_{e: v \in \bar{e}} a_{e}$.
The posterior distribution is $f(\mathbf{w} \mid \mathbf{x})=f\left(\mathbf{w} \mid v_{n}, \mathbf{a}^{\prime}\right)$, where $\mathbf{a}^{\prime}=\left(a_{e}+k_{e}(\mathbf{x})\right){e \in E}$ and k{e}(\mathbf{x})=\left{\begin{aligned} \left|\left{i \in{1, \ldots, n}:\left{x_{i-1}, x_{i}\right}=e\right}\right|, & \text { for } e \in E \backslash E_{\text {loop }} \ 2\left|\left{i \in{1, \ldots, n}:\left{x_{i-1}, x_{i}\right}=e\right}\right|, & \text { for } e \in E_{\text {loop }} \end{aligned}\right.

where $|\cdot|$ represents the cardinality of a set. Therefore, for an edge $e$ which is not a loop, $k_{e}(\mathbf{x})$ represents the number of traversals of $e$ by the path $\mathbf{x}=\left(x_{0}, x_{1}, \ldots, x_{n}\right)$ and for a loop, $k_{e}(\mathbf{x})$ is twice the number of traversals of $e$.

The integrating constant and moments of the distribution are known and it is straightforward to simulate from the posterior distribution; for more details, see Diaconis and Rolles (2006).

## 统计代写|应用随机过程代写Stochastic process代考|Higher order chains and mixtures of Markov chains

Bayesian inference for the full $r$ th order Markov chain model can, in principle, be carried out in exactly the same way as inference for the first-order model, by expanding the number of states appropriately, as outlined in Section 3.2.2.

Example 3.11: In the Australian rainfall example, Markov chains of orders $r=2$ and 3 were considered. In each case, $\operatorname{Be}(1 / 2,1 / 2)$ priors were used for the first nonzero element of each row of the transition matrix and it was assumed that the initial $r$ states were generated from the equilibrium distribution. Then, the predictive equilibrium probabilities of the different states under each model are as follows

The log marginal likelihoods are $-30.7876$ for the second-order model and $-32.1915$ for the third-order model, respectively, which suggest that the simple Markov chain model should be preferred.

Bayesian inference for the MTD model of (3.1) is also straightforward. Assume first that the order $r$ of the Markov chain mixture is known. Then, defining an indicator variable $Z_{n}$ such that $P\left(Z_{n}=z \mid \mathbf{w}\right)=w_{z}$, observe that the mixture transition model can be represented as
$$P\left(X_{n}=x_{n} \mid X_{n-1}=x_{n-1}, \ldots, X_{n-r}=x_{n-r}, Z_{n}=z, P\right)=p_{x_{n-z} x_{n}}$$
Then, a posteriori,
$$P\left(Z_{n}=z \mid X_{n}=x_{n}, \ldots, X_{n-r}=x_{n-r}, Z_{n}=z, \boldsymbol{P}\right)=\frac{w_{z} p_{x_{n-z} x_{n}}}{\sum_{j=1}^{r} w_{j} p_{x_{n-j} x_{n}}}$$

Now, define the usual matrix beta prior for $\boldsymbol{P}$, a Dirichlet prior for $\mathbf{w}$, say $\mathbf{w} \sim \operatorname{Dir}\left(\beta_{1}, \ldots, \beta_{r}\right)$, and a probability model $P\left(x_{0}, \ldots, x_{r-1}\right)$ for the initial states of the chain. Then given a sequence of data, $\mathbf{x}=\left(x_{0}, \ldots, x_{n}\right)$, if the indicator variables are $\mathbf{z}=\left(z_{r}, \ldots, z_{n}\right)$ then
\begin{aligned} f(\boldsymbol{P} \mid \mathbf{x}, \mathbf{z}, \mathbf{w}) &=\prod_{t=r}^{n} p_{x_{t-z_{t}} x_{t}} f(\boldsymbol{P}) \ f(\mathbf{w} \mid \mathbf{z}) &=\prod_{t=r}^{n} w_{z_{t}} f(\mathbf{w}), \end{aligned}
which are matrix beta and Dirichlet distributions, respectively. Therefore, a simple Gibbs sampling algorithm can be set up to sample the posterior distribution of $\mathbf{w}, \boldsymbol{P}$ by successively sampling from (3.6), (3.7), and (3.7).

When the order of the chain is unknown, two approaches might be considered. First, models of different orders could be fitted and then Bayes factors could be used for model selection as in Section 3.3.5. Otherwise a prior distribution can be defined over the different orders and then a variable dimension MCMC algorithm such as reversible jump (Green, 1995, Richardson and Green 1997) could be used to evaluate the posterior distribution, as in the following example.

## 统计代写|应用随机过程代写Stochastic process代考|Reversible Markov chains

F(在∣在0,一种)∝∏和∈和∖和丢失 在和一种和−1/2∏和∈和种族 在和一种和/2−1在在0一种在0/2∏在∈在∖在0在在(一种在+1)/2∑吨∈吨∏和∉和(吨)1在C

## 统计代写|应用随机过程代写Stochastic process代考|Higher order chains and mixtures of Markov chains

(3.1) 的 MTD 模型的贝叶斯推理也很简单。首先假设订单r马尔可夫链混合物是已知的。然后，定义一个指示变量从n这样磷(从n=和∣在)=在和，观察混合过渡模型可以表示为

F(磷∣X,和,在)=∏吨=rnpX吨−和吨X吨F(磷) F(在∣和)=∏吨=rn在和吨F(在),

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