2

I am having trouble understanding the definition of detailed balance in the case of a continuous state space. The definition of detailed balance that I am working from is:

A pmf $\pi$ on $\mathcal{X}$ satisfies detailed balance with respect to $T$ if $$ \pi_a T_{ab} = \pi_b T_{ba} \ \ \forall a,b \in \mathcal{X} $$

As I understand it, $T$ is the transition matrix and $T_{ab}$ represents the probability of moving from state $a$ to state $b$. I am struggling with the notion of $T$ in the case where $\mathcal{X}$ is continuous. Say for example that $\mathcal{X} = \mathbb{R}$, then wouldn't $T_{ab}=0$ as a single value in $\mathbb{R}$ has measure 0? Also, what is the structure of the transition matrix $T$? Does it have an uncountably infinite number of rows and columns? How is this possible?

samvoit4
  • 185
  • 1
  • 8
  • 2
    Perhaps https://stats.stackexchange.com/questions/46389 helps. – whuber May 26 '21 at 18:55
  • 1
    Thanks @whuber . I think this is getting towards what I am struggling with although some of it is over my head. For an example, if $\pi = \mathcal{N}(\mu, \Sigma)$ what would the elements of $T$ be? I'm not sure I can answer this question based on your example? – samvoit4 May 26 '21 at 19:23

1 Answers1

4

enter image description here

Detailed balance for a continuous Markov chain with transition kernel $K(\cdot,\cdot)$ and stationary density $f(\cdot)$ writes as $$f(x)K(x,y)=f(y)K(y,x)$$ It means that, in a stationary regime, the joint distribution of $(x_t,x_{t+1})$ is the same as the joint distribution of $(x_{t+1},x_{t})$. This implies that the chain $(x_t)$ is (time-)reversible.

Xi'an
  • 90,397
  • 9
  • 157
  • 575
  • I see. So am I understanding correctly that the transition kernel $K$ is the continuous analogue to the transition matrix $T$? – samvoit4 May 26 '21 at 20:01
  • 1
    The transition kernel $K$ is used to generate the following value of the Markov chain conditional on the current value (first argument). It thus acts as a conditional density. – Xi'an May 26 '21 at 20:29