<p>Let $y_t = x_t^T$. Then, $y_{t+1} = y_t P^T$. Thus, $y$ (and therefore $x$) corresponds to the original Markov Chain run backward in time.
Hope this gives some motivation towards the concept of dual chains.</p>

CC BY-SA 3.0
2012-02-24T06:14:41.327Z
2012-02-24T06:14:41.327Z
Shitikanth
<p>Let a reversible Markov process $m_{t+1}=m_t P$, where $t$ is time that has a stationary distribution $\pi$. I saw in a paper that the dual system was defined as $x_{t+1}=P x_t$. Can anyone give me some directions in order to understand how this is derived?</p>

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2012-02-22T23:34:11.520Z
That looks like the _definition_ of "dual system".  Derived from what?
Jeffε
<pr.probability><markov-chains>
Dual of a Reversible Markov Chain
Kostas
# Dual of a Reversible Markov Chain

(Asked by: username_0 on 2012-02-22)

Let a reversible Markov process $m\_{t+1}=m\_t P$, where $t$ is time that has a stationary distribution $\pi$. I saw in a paper that the dual system was defined as $x\_{t+1}=P x\_t$. Can anyone give me some directions in order to understand how this is derived?

**Tags:** pr.probability, markov-chains
> **Comments:**
> **username_1**: That looks like the _definition_ of "dual system".  Derived from what?  

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## 1 answers

### Answer by username_2 on 2012-02-24. Score: 6

Let $y\_t = x\_t^T$. Then, $y\_{t+1} = y\_t P^T$. Thus, $y$ (and therefore $x$) corresponds to the original Markov Chain run backward in time.
Hope this gives some motivation towards the concept of dual chains.


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