📐 AI MathID: detailed-balance

Markov Chain & Detailed Balance

Q: From detailed balance $\pi(i) P(i \to j) = \pi(j) P(j \to i)$, prove $\sum_i \pi(i) P(i \to j) = \pi(j)$ — i.e. $\pi$ is stationary — justifying each step?

【Markov Chain & Detailed Balance 核心考点 1】From detailed balance $\pi(i) P(i \to j) = \pi(j) P(j \to i)$, prove $\sum_i \pi(i) P(i \to j) = \pi(j)$ — i.e. $\pi$ is stationary — justifying each step?。技术解析：Definition: Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution \pi: if the transition kernel P(i \to j) satisfies the pairwise reversibility \pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j, then \pi is a stationary di... 完整推导与工程落地见：https://talentme.airsota.com/resources/tech/roadmap/mindmap/Math/card/detailed-balance

Q: Is detailed balance sufficient or necessary for stationarity? Give an example of a stationary but non-reversible chain?

【Markov Chain & Detailed Balance 核心考点 2】Is detailed balance sufficient or necessary for stationarity? Give an example of a stationary but non-reversible chain?。技术解析：Definition: Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution \pi: if the transition kernel P(i \to j) satisfies the pairwise reversibility \pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j, then \pi is a stationary di... 完整推导与工程落地见：https://talentme.airsota.com/resources/tech/roadmap/mindmap/Math/card/detailed-balance

Q: How does the vector form $\pi^T P = \pi^T$ follow from detailed balance?

【Markov Chain & Detailed Balance 核心考点 3】How does the vector form $\pi^T P = \pi^T$ follow from detailed balance?。技术解析：Definition: Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution \pi: if the transition kernel P(i \to j) satisfies the pairwise reversibility \pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j, then \pi is a stationary di... 完整推导与工程落地见：https://talentme.airsota.com/resources/tech/roadmap/mindmap/Math/card/detailed-balance

Q: Why do MCMC constructions use detailed balance instead of solving $\pi^T P = \pi^T$ directly?

【Markov Chain & Detailed Balance 核心考点 4】Why do MCMC constructions use detailed balance instead of solving $\pi^T P = \pi^T$ directly?。技术解析：Definition: Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution \pi: if the transition kernel P(i \to j) satisfies the pairwise reversibility \pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j, then \pi is a stationary di... 完整推导与工程落地见：https://talentme.airsota.com/resources/tech/roadmap/mindmap/Math/card/detailed-balance

Q: Which regularity conditions (irreducibility, aperiodicity, positive recurrence) are additionally needed for convergence to stationarity?

【Markov Chain & Detailed Balance 核心考点 5】Which regularity conditions (irreducibility, aperiodicity, positive recurrence) are additionally needed for convergence to stationarity?。技术解析：Definition: Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution \pi: if the transition kernel P(i \to j) satisfies the pairwise reversibility \pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j, then \pi is a stationary di... 完整推导与工程落地见：https://talentme.airsota.com/resources/tech/roadmap/mindmap/Math/card/detailed-balance

马尔可夫链与细致平衡

🎯Core Definition

Detailed Balance is the key sufficient condition for a Markov chain to have stationary distribution

\pi

: if the transition kernel

P(i \to j)

satisfies the pairwise reversibility

📌Overview

\pi(i) P(i \to j) = \pi(j) P(j \to i), \quad \forall\, i, j,

📌Overview

then

\pi

is a stationary distribution of the chain. Proof: for any state

j

📌Overview

\sum_i \pi(i) P(i \to j) = \sum_i \pi(j) P(j \to i) = \pi(j) \sum_i P(j \to i) = \pi(j),

📌Overview

the first step substitutes detailed balance into each term of the sum, the last uses row-stochasticity

\sum_i P(j \to i) = 1

— one step from

\pi

stays at

\pi

, i.e.

\pi^T P = \pi^T

, exactly the definition of stationarity.

💡Use Cases

the mathematical foundation of MCMC (especially Metropolis-Hastings) — designing a transition kernel reduces to verifying detailed balance pointwise, and stationarity follows automatically; interviews often ask "how does detailed balance imply a stationary distribution".

⚡Key Problems Solved

it turns "find a stationary distribution" from solving the linear system

\pi^T P = \pi^T

(an eigenvalue problem) into checking a pairwise condition, which is far easier to construct; note that detailed balance is sufficient but not necessary — non-reversible chains can be stationary without it, yet MCMC almost always uses reversible designs (simpler and usually faster mixing).

🎯5 High-Frequency Exam Points

From detailed balance

\pi(i) P(i \to j) = \pi(j) P(j \to i)

, prove

\sum_i \pi(i) P(i \to j) = \pi(j)

— i.e.

\pi

is stationary — justifying each step?

Is detailed balance sufficient or necessary for stationarity? Give an example of a stationary but non-reversible chain?

How does the vector form

\pi^T P = \pi^T

follow from detailed balance?

Why do MCMC constructions use detailed balance instead of solving

\pi^T P = \pi^T

directly?

Which regularity conditions (irreducibility, aperiodicity, positive recurrence) are additionally needed for convergence to stationarity?

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Updated 2026-08-12

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