### 统计代写|应用随机过程代写Stochastic process代考|Advantages of the Bayesian approach

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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|应用随机过程代写Stochastic process代考|Advantages of the Bayesian approach

Obviously, a Bayesian approach using a prior distribution for $\mathbf{P}$ with mass on irreducible, aperiodic chains eliminates the possible problems associated with classical inference. Another, more theoretical justification of the use of a Bayesian approach to inference for Markov chains can be based on de Finetti type theorems.

The well-known de Finetti (1937) theorem states that for an infinitely exchangeable sequence, $X_{1}, X_{2}, \ldots$ of zero-one random variables with probability measure $P$, there exists a distribution function $F$ such that the joint mass function is
$$p\left(x_{1}, \ldots x_{n}\right)=\int_{\theta} \theta \sum_{i=1}^{n} x_{i}(1-\theta)^{n-\sum_{i=1}^{n} x_{i}} \mathrm{~d} F(\theta) .$$
Obviously, observations from a Markov chain cannot generally be regarded as exchangeable and so the basic de Finetti theorem cannot be applied. However, an appropriate definition of exchangeability is to say that a probability measure $P$ defined on recurrent Markov chains is partially exchangeable if it gives equal probability to all sequences $X_{1}, \ldots, X_{n}$ (assuming some fixed $x_{0}$ ) with the same transition count matrix. Given this definition of exchangeability, it can be shown that for a finite sequence, say $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$, there exists a distribution function $F$ so that
$$p\left(\mathbf{x} \mid x_{0}\right)=\int_{P} p_{i j}^{n_{0}} \mathrm{~d} F(\boldsymbol{P})$$
where $n_{i j}$ are the transition counts. Similar to the standard de Finetti theorem, the distribution $F$ may be interpreted as a Bayesian prior distribution for $\boldsymbol{P}$.

## 统计代写|应用随机过程代写Stochastic process代考|Conjugate prior distribution and modifications

Given the experiment of this Section, a natural conjugate prior for $\boldsymbol{P}$ is defined by letting $\mathbf{p}{i}=\left(p{i l}, \ldots, p_{i K}\right)$ have a Dirichlet distribution, say
$\mathbf{p}{i} \sim \operatorname{Dir}\left(\boldsymbol{\alpha}{i}\right), \quad$ where $\boldsymbol{\alpha}{i}=\left(\alpha{i l}, \ldots, \alpha_{i K}\right)$ for $i=1, \ldots, K .$
This defines a matrix beta prior distribution. Given this prior distribution and the likelihood function of (3.3), the posterior distribution is also of the same form, so that
$$\mathbf{p}{i} \mid \mathbf{x} \sim \operatorname{Dir}\left(\alpha{i}^{\prime}\right) \quad \text { where } \alpha_{i j}^{\prime}=\alpha_{i j}+n_{i j} \text { for } i, j=1, \ldots, K .$$
When little prior information is available, a natural possibility is to use the Jeffreys prior, which is a matrix beta prior with $\alpha_{i j}=1 / 2$ for all $i, j=1, \ldots, K$. An

alternative, improper prior distribution along the lines of the Haldane (1948) prior for binomial data is to set
$$f\left(\mathbf{p}{i}\right) \propto \prod{j=1}^{K} \frac{1}{p_{i j}},$$
which can be thought of as the limit of a matrix beta prior, setting $\alpha_{i j} \rightarrow 0$ for all $i, j=1, \ldots, K$. In this case, the posterior distribution is $\mathbf{p}{i} \mid \mathbf{x} \sim \operatorname{Dir}\left(n{i l}, \ldots, n_{i k}\right)$ so that, for example, the posterior mean of the $i j$ th element of the transition matrix is $E\left[p_{i j} \mid \mathbf{x}\right]=n_{i j} / n_{i-}$, equal to the maximum likelihood estimate. However, this approach cannot be recommended, as if any $n_{i j}=0$, which may often be the case for chains with a relatively large number of states, then the posterior distribution is improper.

## 统计代写|应用随机过程代写Stochastic process代考|Forecasting short-term behavior

Suppose that we wish to predict future values of the chain. For example, we can predict the next value of the chain, at time $n+1$ using
\begin{aligned} P\left(X_{n+1}=j \mid \mathbf{x}\right) &=\int P\left(X_{n+1}=j \mid \mathbf{x}, \boldsymbol{P}\right) f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P} \ &=\int p_{x_{n j} j} f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P}=\frac{\alpha_{x_{z} j}+n_{x_{n_{n} j}}}{\alpha_{x_{n^{}}}+n_{x_{n^{}}}}, \end{aligned}
where $\alpha_{i}=\sum_{j=1}^{K} \alpha_{i j}$.
Prediction of the state at $t>1$ steps is slightly more complex. For small $t$, we can use
$$P\left(X_{n+t}=j \mid \mathbf{x}\right)=\int\left(\boldsymbol{P}^{t}\right){x{n} j} f(\boldsymbol{P} \mid \mathbf{x}) \mathrm{d} \boldsymbol{P},$$
which gives a sum of Dirichlet expectation terms. However, as $t$ increases, the evaluation of this expression becomes computationally infeasible. A simple alternative is to use a Monte Carlo algorithm based on simulating future values of the chain as follows:
For $s=1, \ldots, S$ :
Generate $\boldsymbol{P}^{(s)}$ from $f(\boldsymbol{P} \mid \mathbf{x})$.
Generate $x_{n+1}^{(s)}, \ldots, x_{n+t}^{(s)}$ from the Markov chain with $\boldsymbol{P}^{(s)}$ and initial state $x_{n}$.

Then, $P\left(X_{n+t}=j \mid \mathbf{x}\right) \approx \frac{1}{5} \sum_{s=1}^{S} I_{x_{n+1}^{(n)}=j}$ where $I$ is an indicator function and $E\left[X_{n+t} \mid \mathbf{x}\right] \approx \frac{1}{s} \sum_{s=1}^{s} X_{n+t^{*}}^{(s)}$

Example 3.4: Assume that it is now wished to predict the Sydney weather on March 21 and 22 . Given that it did not rain on March 20 , then immediately, we have
\begin{aligned} P(\text { no rain on March } 21 \mid \mathbf{x}) &=E\left[p_{11} \mid \mathbf{x}\right]=0.823, \ P(\text { no rain on March } 22 \mid \mathbf{x}) &=E\left[p_{11}^{2}+p_{12} p_{21} \mid \mathbf{x}\right]=0.742, \ P(\text { no rain on both }) &=E\left[p_{11}^{2} \mid \mathbf{x}\right]=0.681 . \end{aligned}

## 统计代写|应用随机过程代写Stochastic process代考|Advantages of the Bayesian approach

p(X1,…Xn)=∫θθ∑一世=1nX一世(1−θ)n−∑一世=1nX一世 dF(θ).

p(X∣X0)=∫磷p一世jn0 dF(磷)

## 统计代写|应用随机过程代写Stochastic process代考|Conjugate prior distribution and modifications

p一世∼目录⁡(一种一世),在哪里一种一世=(一种一世l,…,一种一世ķ)为了一世=1,…,ķ.

p一世∣X∼目录⁡(一种一世′) 在哪里 一种一世j′=一种一世j+n一世j 为了 一世,j=1,…,ķ.

F(p一世)∝∏j=1ķ1p一世j,

：s=1,…,小号:

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