Detailed Balance

• To find a transition kernel T(x|y) for a homogenous, Ergodic Markov Chain
• If we pick some state x with the Probability given by g and multiply its prob g(x) with the transition Probability Density T(x|y) (weighted by Probability Density of x) then its the same as the reverse weighted transiting Probability Density from y to x
• $\forall x,y\in {\mathbb{R}}^k:T(y|x)g(x)=T(x|y)g(y)$
• If T(x|y) has detailed balance wrt g, then it is an Invariant Distribution
• ${\int }_{{\mathbb{R}}^k}T(x|y)g(y)dy={\int }_{{\mathbb{R}}^k}T(y|x)g(x)dy=g(x){\int }_{{\mathbb{R}}^k}P(y|x)dy=g(x)$