统计代写| Stationary distribution stat代写
统计代考
11.3 Stationary distribution
The concepts of recurrence and transience are important for understanding the longrun behavior of a Markov chain. At first, the chain may spend time in transient states. Eventually though, the chain will spend all its time in recurrent states. But what fraction of the time will it spend in each of the recurrent states? This question is answered by the stationary distribution of the chain, also known as the steadystate distribution. We will learn in this section that for irreducible and aperiodic
Markov chains
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Markov chains, the stationary distribution describes the long-run behavior of the chain, regardless of its initial conditions. It will tell us both the long-run probability of being in any particular state, and the long-run proportion of time that the chain spends in that state.
Definition 11.3.1 (Stationary distribution). A row vector $\mathrm{s}=\left(s_{1}, \ldots, s_{M}\right)$ such that $s_{i} \geq 0$ and $\sum_{i} s_{i}=1$ is a stationary distribution for a Markov chain with transition matrix $Q$ if
$$
\sum_{i} s_{i} q_{i j}=s_{j}
$$
for all $j$. This system of linear equations can be written as one matrix equation:
$$
\mathrm{s} Q=\mathrm{s}
$$
Recall that if $\mathrm{s}$ is the distribution of $X_{0}$, then $\mathrm{s} Q$ is the marginal distribution of $X_{1}$. Thus the equation $\mathrm{s} Q=\mathrm{s}$ means that if $X_{0}$ has distribution $\mathrm{s}$, then $X_{1}$ also has distribution $\mathbf{s}$. But then $X_{2}$ also has distribution $\mathbf{s}$, as does $X_{3}$, etc. That is, a Markov chain whose initial distribution is the stationary distribution s will stay in the stationary distribution forever.
One way to think about the stationary distribution of a Markov chain intuitively is to imagine a large number of particles, each independently bouncing from state to state according to the transition probabilities. After a while, the system of particles will approach an equilibrium where, at each time period, the number of particles leaving a state will be counterbalanced by the number of particles entering that state, and this will be true for all states. At this equilibrium, the system as a whole will appear to be stationary, and the proportion of particles in each state will be distributions more after Definition 11.4.1.
11.3.2 (Stationary distribution is marginal, not conditional). When a Markov chain is at the stationary distribution, the unconditional PMF of $X_{n}$ equals s for all $n$, but the conditional PMF of $X_{n}$ given $X_{n-1}=i$ is still encoded by the $i$ th row of the transition matrix $Q$.
If a Markov chain starts at the stationary distribution, then all of the $X_{n}$ are identically distributed (since they have the same marginal distribution s), but they are not necessarily independent, since the conditional distribution of $X_{n}$ given $X_{n-1}=i$ is, in general, different from the marginal distribution of $X_{n}$.
11.3.3 (Sympathetic magic). If a Markov chain starts at the stationary distribution, then the marginal distributions of the $X_{n}$ are all equal. This is not the same as saying that the $X_{n}$ themselves are all equal; confusing the random variables $X_{n}$ with their distributions is an example of sympathetic magic.
For very small Markov chains, we may solve for the stationary distribution by hand, using the definition. The next example illustrates this for a two-state chain.
统计代考
11.3 平稳分布
递归和瞬态的概念对于理解马尔可夫链的长期行为很重要。起初，链条可能会在瞬态中花费时间。但最终，链条将把所有时间都花在循环状态中。但是它会在每个循环状态中花费多少时间？这个问题由链的平稳分布来回答，也称为稳态分布。我们将在本节中学习对于不可约和非周期
马尔可夫链
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马尔可夫链，平稳分布描述了链的长期行为，无论其初始条件如何。它将告诉我们处于任何特定状态的长期概率，以及链在该状态中花费的长期时间比例。
定义 11.3.1（平稳分布）。一个行向量 $\mathrm{s}=\left(s_{1}, \ldots, s_{M}\right)$ 使得 $s_{i} \geq 0$ 和 $\sum_{i} s_{i }=1$ 是具有转移矩阵 $Q$ 的马尔可夫链的平稳分布，如果
$$
\sum_{i} s_{i} q_{i j}=s_{j}
$$
对于所有 $j$。这个线性方程组可以写成一个矩阵方程：
$$
\mathrm{s} Q=\mathrm{s}
$$
回想一下，如果 $\mathrm{s}$ 是 $X_{0}$ 的分布，那么 $\mathrm{s} Q$ 是 $X_{1}$ 的边际分布。因此方程 $\mathrm{s} Q=\mathrm{s}$ 意味着如果 $X_{0}$ 有分布 $\mathrm{s}$，那么 $X_{1}$ 也有分布 $\mathbf{ s}$。但是 $X_{2}$ 也有分布 $\mathbf{s}$，$X_{3}$ 等也有。也就是说，初始分布为平稳分布 s 的马尔可夫链将保持在平稳分布中永远。
直观地考虑马尔可夫链的平稳分布的一种方法是想象大量粒子，每个粒子根据转移概率独立地从一个状态跳到另一个状态。一段时间后，粒子系统将接近平衡，在每个时间段，离开一个状态的粒子数量将与进入该状态的粒子数量相平衡，这对所有状态都是如此。在这种平衡状态下，整个系统看起来是静止的，每个状态下的粒子比例在定义 11.4.1 之后会更加分布。
11.3.2（平稳分布是边际的，不是有条件的）。当马尔可夫链处于平稳分布时，$X_{n}$ 的无条件 PMF 对所有 $n$ 都等于 s，但 $X_{n}$ 的条件 PMF 给定 $X_{n-1}=i$仍然由转移矩阵 $Q$ 的第 $i$ 行编码。
如果马尔可夫链从平稳分布开始，那么所有的 $X_{n}$ 都是同分布的（因为它们具有相同的边际分布 s），但它们不一定是独立的，因为 $X_{n 的条件分布}$ 给定 $X_{n-1}=i$ 通常不同于 $X_{n}$ 的边际分布。
11.3.3（同情魔法）。如果马尔可夫链从平稳分布开始，则 $X_{n}$ 的边际分布都是相等的。这不等于说 $X_{n}$ 本身都是平等的；将随机变量 $X_{n}$ 与其分布混淆是同情魔法的一个例子。
对于非常小的马尔可夫链，我们可以使用定义手动求解平稳分布。下一个示例说明了这一点，适用于两个状态的链。
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