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# 统计代写| Reversibility stat代写

## 统计代考

11.4 Reversibility
We have seen that the stationary distribution of a Markov chain is extremely useful for understanding its long-run behavior. Unfortunately, in general it may be computationally difficult to find the stationary distribution when the state space is large. This section addresses an important special case where working with eigenvalue equations for large matrices can be avoided.
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Definition 11.4.1 (Reversibility). Let $Q=\left(q_{i j}\right)$ be the transition matrix of a Markov chain. Suppose there is $\mathrm{s}=\left(s_{1}, \ldots, s_{M}\right)$ with $s_{i} \geq 0, \sum_{i} s_{i}=1$, such that
$$s_{i} q_{i j}=s_{j} q_{j i}$$
for all states $i$ and $j$. This equation is called the reversibility or detailed balance condition, and we say that the chain is reversible with respect to $\mathrm{s}$ if it holds.

The term “reversible” comes from the fact that a reversible chain, started according to its stationary distribution, behaves in the same way regardless of whether time is run forwards or backwards. If you record a video of a reversible chain, started according to its stationary distribution, and then show the video to a friend, either in the normal way or with time reversed, your friend will not be able to determine from watching the video whether time is running forwards or backwards.

As discussed after Definition 11.3.1, we can think about the stationary distribution of a Markov chain intuitively in terms of a system consisting of a large number of particles independently bouncing around according to the transition probabilities. In the long run, the proportion of particles in any state $j$ is the stationary probability of state $j$, and the flow of particles out of state $j$ is counterbalanced by the flow of sarticles into state $j .$ fo see this in more detail, let $n$ be the number of particles and $j$. By definition, $\mathbf{s}$ is the stationary distribution of the chain if and only if
$$s_{j}=\sum_{i} s_{i} q_{i j}=s_{j} q_{j j}+\sum_{i: i \neq j} s_{i} q_{i j}$$
for all states $j$. This equation can be rewritten as
$$n s_{j}\left(1-q_{j j}\right)=\sum_{i: i \neq j} n s_{i} q_{i j}$$
The left-hand side is the approximate number of particles that will exit from state $j$ on the next step, since there are $n s_{j}$ particles at state $j$, each of which will stay at $j$ with probability $q_{j j}$ and leave with probability $1-q_{j j} .$ The right-hand side is the approximate number of particles that will enter state $j$ on the next step, since for each $i \neq j$ there are $n s_{i}$ particles at state $i$, each of which will enter state $j$ with probability $q_{i j}$. So there is a balance between particles leaving state $j$ and particles entering state $j$.

The reversibility condition imposes a much more stringent form of balance, in which for each pair of states $i, j$ with $i \neq j$, the flow of particles from state $i$ to state $j$ is counterbalanced by the flow of particles from state $j$ to state $i$. To see this, write the reversibility equation for states $i$ and $j$ as
$$n s_{i} q_{i j}=n s_{j} q_{j i}$$
The left-hand side is the approximate number of particles that will go from state $i$ to state $j$ on the next step, since there are $n s_{i}$ particles at state $i$, each of which

## 统计代考

11.4 可逆性

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$$s_{i} q_{i j}=s_{j} q_{j i}$$

“可逆”一词源于这样一个事实，即根据其平稳分布开始的可逆链无论时间是向前还是向后运行，都以相同的方式运行。如果你录制一个可逆链的视频，按照它的平稳分布开始，然后以正常方式或时间倒转的方式将视频显示给朋友，你的朋友将无法通过观看视频来确定时间是否正在向前或向后运行。

$$s_{j}=\sum_{i} s_{i} q_{i j}=s_{j} q_{j j}+\sum_{i: i \neq j} s_{i} q_{i j}$$

$$n s_{j}\left(1-q_{j j}\right)=\sum_{i: i \neq j} n s_{i} q_{i j}$$

$$n s_{i} q_{i j}=n s_{j} q_{j i}$$