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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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• Foundations of Data Science 数据科学基础
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## 物理代写|量子场论代写Quantum field theory代考|Fixed Points of Markov Chains

Every stochastic matrix has the eigenvalue 1 . The corresponding right eigenvector is given by $\sim(1,1, \ldots, 1)^{T}$. But since $W$ acts from the right, we are interested in the left eigenvector with eigenvalue 1 . We follow [1] and consider the series
$$\boldsymbol{p}{n}=\frac{1}{n} \sum{j=0}^{n-1} \boldsymbol{p} W^{j} .$$

Since stochastic vectors form a compact set, the series has a convergent subsequence
$$\frac{1}{n_{k}} \sum_{0}^{n_{k}-1} p W^{j} \longrightarrow \boldsymbol{P}$$
We multiply this subsequence by $W$ from the right,
$$\frac{1}{n_{k}} \sum_{1}^{n_{k}} \boldsymbol{p} W^{j} \longrightarrow \boldsymbol{P W} .$$
In the difference of the last two formulas, only two terms in the left-hand sides remain, and we obtain in the limit $n_{k} \rightarrow \infty$
$$\frac{1}{n_{k}}\left(\boldsymbol{p}-\boldsymbol{p} W^{n_{k}}\right) \longrightarrow \boldsymbol{P}-\boldsymbol{P} W$$
Since the left-hand side converges to zero, this implies the eigenvalue equation
$$\boldsymbol{P} W=\boldsymbol{P}$$
which means that every stochastic matrix $W$ has at least one fixed point $\boldsymbol{P}$, i.e., a left eigenvector with eigenvalue 1 .

We now assume that $W$ has at least one column with minimal element greater than a positive number $\delta$. This means that all configurations can jump with nonvanishing probability to at least one configuration. Stochastic matrices with this property are called attractive, and for an attractive matrix, the series $W^{n} p$ converges for any stochastic vector $\boldsymbol{p}$. Note that $W$ in (4.6) is not attractive and this explains why the associated Markov chain does not converge. For the proof of convergence, we first note that for two real numbers $p$ and $p^{\prime}$ we have
$$\left|p-p^{\prime}\right|=p+p^{\prime}-2 \min \left(p, p^{\prime}\right),$$
and this relation implies
$$\left|\boldsymbol{p}-\boldsymbol{p}^{\prime}\right|=2-2 \sum_{\omega} \min \left(p_{\omega}, p_{\omega}^{\prime}\right)$$
for two stochastic vectors. We used the $\ell_{1}$-norm $|\boldsymbol{p}|=\sum_{\omega}\left|p_{\omega}\right|$.
Next we wish to prove that an attractive $W$ acts on vectors $\Delta=\left(\Delta_{1}, \Delta_{2}, \ldots\right)$ with
$$|\Delta| \equiv \sum\left|\Delta_{\omega}\right|=2 \quad \text { and } \quad \sum \Delta_{\omega}=0$$ in a contractive way. Let us assign the stochastic vector $\boldsymbol{e}{\omega}$ to each $\omega$ which represents the probability distribution for finding $\omega$ with probability one. Thus all entries of $e{\omega}$ vanish with the exception of entry $\omega$, which is 1 . The stochastic vectors $\left{\boldsymbol{e}{\omega}\right}$ form an orthonormal basis. In a first step, we prove that $W$ is contractive for difference vectors $\boldsymbol{e}{\omega}-\boldsymbol{e}{\omega}{ }^{\prime}$. Thus we apply the identity (4.9) to the stochastic vectors $\boldsymbol{e}{\omega} W$ and $\boldsymbol{e}{\omega} W$, i.e., to rows of $W$ belonging to $\omega$ and $\omega^{\prime}$. In case of an attractive $W$, we find for $\omega \neq \omega^{\prime}$ \begin{aligned} \left|\boldsymbol{e}{\omega} W-\boldsymbol{e}{\omega^{\prime}} W\right| &=2-2 \sum{\omega^{\prime \prime}} \min \left{W\left(\omega, \omega^{\prime \prime}\right), W\left(\omega^{\prime}, \omega^{\prime \prime}\right)\right} \ & \leq 2-2 \delta=(1-\delta) \underbrace{\left|\boldsymbol{e}{\omega}-\boldsymbol{e}{\omega^{\prime}}\right|}_{=2} \text { with } 0<\delta<1 \end{aligned}

## 物理代写|量子场论代写Quantum field theory代考|Detailed Balance

The condition of detailed balance is a simple and physically well-founded constraint on a Markov chain which implies the fixed point equation (4.8). Detailed balance means a balance between any two configurations: the equilibrium probability for $\omega$, multiplied by the jump probability from $\omega$ to $\omega^{\prime}$, is equal to the equilibrium probability of $\omega^{\prime}$ multiplied by the jump probability from $\omega^{\prime}$ to $\omega$,
$$P_{\omega} W\left(\omega, \omega^{\prime}\right)=P_{\omega^{\prime}} W\left(\omega^{\prime}, \omega\right)$$
If in equilibrium the configuration $\omega$ is more likely occupied than the configuration $\omega^{\prime}$, then the transition amplitude from $\omega$ to $\omega^{\prime}$ is less than the amplitude for the reverse transition.

The condition of detailed balance guarantees that $\boldsymbol{P}$ is a fixed point of the chain,
$$\sum_{\omega} P_{\omega} W\left(\omega, \omega^{\prime}\right)=\sum_{\omega} P_{\omega^{\prime}} W\left(\omega^{\prime}, \omega\right)=P_{\omega^{\prime}}$$ but it does not fix $W$ uniquely. We may use the residual freedom to choose simple and efficient algorithms. In particular the fast Metropolis and heat bath algorithms are universally applicable and are often used. For statistical systems which can be dualized, there exist the more efficient cluster algorithms, which do not suffer from the problem of the so-called critical slowing down. These and other Monte Carlo algorithms will be introduced and applied in this book. More material can be found in the textbooks [2-5].

## 物理代写|量子场论代写Quantum field theory代考|Acceptance Rate

The probabilities $W(\omega, \omega)$ of not jumping are not constrained by the condition of detailed balance. Thus we may change the probabilities $W\left(\omega, \omega^{\prime}\right)$ for the transition between different configurations without violating the sum rule (4.1) if we only readjust the unconstrained $W(\omega, \omega)$.

We factorize the transition probability into the product of a test probability and an acceptance rate
$$W\left(\omega, \omega^{\prime}\right)=T\left(\omega, \omega^{\prime}\right) A\left(\omega, \omega^{\prime}\right)$$
where $T\left(\omega, \omega^{\prime}\right)$ is the probability of testing the new configuration $\omega^{\prime}$ with given initial configuration $\omega$. If $\omega^{\prime}$ is tested, then the quantity $0 \leq A\left(\omega, \omega^{\prime}\right) \leq 1$ corresponds to the probability that the transition to $\omega^{\prime}$ is accepted. Note that the conditions
$$\frac{T\left(\omega, \omega^{\prime}\right) A\left(\omega, \omega^{\prime}\right)}{T\left(\omega^{\prime}, \omega\right) A\left(\omega^{\prime}, \omega\right)}=\frac{P_{\omega^{\prime}}}{P_{\omega}}$$
do not fix the ratios of acceptance rates. A good Monte Carlo algorithm requires the best possible choice for these rates. For too small rates, only a tiny fraction of jumps is accepted, and the system is stuck in its initial configuration. We waste valuable computing time without passing through the configuration space. Thus, in many cases, one sets the greater of two acceptance rates $A\left(\omega, \omega^{\prime}\right)$ and $A\left(\omega^{\prime}, \omega\right)$ equal to 1. The smaller rate is chosen such that the condition $(4.24)$ is satisfied.

## 物理代写|量子场论代写Quantum field theory代考|Fixed Points of Markov Chains

pn=1n∑j=0n−1p在j.

1nķ∑0nķ−1p在j⟶磷

1nķ∑1nķp在j⟶磷在.

1nķ(p−p在nķ)⟶磷−磷在

|p−p′|=p+p′−2分钟(p,p′),

|p−p′|=2−2∑ω分钟(pω,pω′)

|Δ|≡∑|Δω|=2 和 ∑Δω=0以一种收缩的方式。让我们分配随机向量和ω每个ω表示找到的概率分布ω概率为一。因此所有条目和ω除进入外消失ω，即 1 。随机向量\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别形成一个标准正交基。第一步，我们证明在对于差分向量是收缩的和ω−和ω′. 因此我们将恒等式（4.9）应用于随机向量和ω在和和ω在，即，到行在属于ω和ω′. 在有吸引力的情况下在, 我们发现ω≠ω′\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

## 物理代写|量子场论代写Quantum field theory代考|Detailed Balance

∑ω磷ω在(ω,ω′)=∑ω磷ω′在(ω′,ω)=磷ω′但它不能解决在独一无二。我们可以利用剩余自由度来选择简单有效的算法。特别是快速 Metropolis 和热浴算法普遍适用并且经常使用。对于可以对偶的统计系统，存在更有效的聚类算法，这些算法不存在所谓的临界减速问题。本书将介绍和应用这些和其他蒙特卡罗算法。更多的材料可以在教科书中找到[2-5]。