What is the mathematical formula for the detailed balance condition?

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.

Reasoning, step by step

1. Identify any two states $x$ and $y$.
2. Calculate the probability flow from $x$ to $y$ as $P(x)T(y|x)$.
3. Calculate the reverse probability flow from $y$ to $x$ as $P(y)T(x|y)$.
4. Set the two flows equal to each other.
5. Verify that this equality holds for all state pairs.

Example

The video displays the equation $P(x)T(y|x) = P(y)T(x|y)$ alongside a diagram of two circles labeled $x$ and $y$ connected by bidirectional yellow arrows representing probability flows.

Common misconceptions

• Believing that detailed balance only requires the total flow into a state to equal the total flow out (global balance), rather than pairwise equality.
• Assuming the condition applies only to adjacent states in a specific network topology, rather than all pairs of states.

Watch the explanation

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