This is the first step of proof of MCMC in my notes

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I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The text says "we start by constructing a markov chain, making it $\pi(x)\pi(x_p\mid x) = \pi(x_p)\pi(x\mid x_p)$... But how can we prove for any any markov chains which are ergodic and aperiodic, we have $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$?


closed as off-topic by Xi'an, Michael Chernick, gung Feb 5 '18 at 18:33

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  • 2
    $\begingroup$ You're familiar with the relationship between conditional and joint density $f(x,y) = f(y|x) f(x)$? $\endgroup$ – Glen_b Jul 22 '16 at 2:27
  • $\begingroup$ @Glen_b yes, but in my case, it is a little bit different I guess. For example ,"Bowl A contains 2 red chips; bowl B contains two white chips; and bowl C contains 1 red chip and 1 white chip. A bowl is selected at random, and one chip is taken at random from that bowl.", $P(C|W)P(W)=P(W|C)P(C)=P(C\cap W)$,so $P(C\cap W)$ has a meaning beacuse $P(C|W)P(W)$ and $P(W|C)P(C)$ are the same thing , but in a the markov chain, $\pi(x\cap x_p)$ are different between $\pi(x)\pi(x_p|x)$ and $\pi(x_p)\pi(x|x_p)$ as the two starting positions are different $\endgroup$ – whoisit Jul 22 '16 at 3:02

how come $π(x)π(x_p∣x)=π(x_p)π(x∣x_p)$?

This is a consequence of the form of the transition kernel for the Metropolis-Hastings algorithm:

The Markov transition kernel associated with this algorithm is $$ \pi(y|x) = \rho(x,y) q(y|x) + (1-r(x)) \delta_x(y) \;, $$ where $q$ denotes the density of the proposal distribution, $r(x)=\int \rho(x,y) q(y|x) \text{d}y$ and $\delta_x$ denotes the Dirac mass in $x$. It is straightforward to verify that \begin{align*} \rho(x,y) q(y|x)\pi(x)&=\rho(y,x) q(x|y)\pi(y) \\ (1-r(x)) \delta_x(y) \pi(x)&=(1-r(y)) \delta_y(x)\pi(y) \;, \end{align*} which together establish detailed balance for the Metropolis-Hastings chain.

From this equality, establishing stationarity of $\pi$ is straightforward, as seen by integrating both sides in $x$.

Is it true for any Markov chains which are ergodic and aperiodic?

This is a sufficient condition to establish stationarity (and in the case of an irreducible Markov chain ergodicity) but not a necessary condition. The Gibbs sampler comes as a counter-example. MALA (Metropolis adjusted Langevin algorithm) as another one.


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