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The detailed balance condition requires that for any two states $x$ and $y$, the probability flow from $x$ to $y$ exactly equals the flow from $y$ to $x$. This bidirectional equilibrium guarantees that the target distribution is stationary for the Markov chain.

Conditions: Must hold for all pairs of states $x$ and $y$ in the state space.; $P(x)$ and $P(y)$ are probabilities under the target distribution.; $T(y|x)$ and $T(x|y)$ are transition probabilities.

The detailed balance condition requires that for any two states $x$ and $y$, the probability flow from $x$ to $y$ exactly equals the flow from $y$ to $x$. This bidirectional equilibrium guarantees that the target distribution is stationary for the Markov chain.

Conditions: Must hold for all pairs of states $x$ and $y$ in the state space.; $P(x)$ and $P(y)$ are probabilities under the target distribution.; $T(y|x)$ and $T(x|y)$ are transition probabilities.