◄ WORLD V · SONNY 5DART 200 · a helldive at the net

THE DETAILED BALANCE run it backwards and it looks the same

A chain is reversible when, in equilibrium, the probability flux from i to j equals the flux back: πᵢ Pᵢ⫺ = π⫺ P⫺ᵢ for every pair. This detailed balance is stronger than mere stationarity (which only balances total in-flow against out-flow at each state), and it is the trick that lets you design a chain with any target π you want — the foundation under Markov-chain Monte Carlo.

THE TECHNIQUE πᵢ Pᵢ⫺ = π⫺ P⫺ᵢ : the flux each way is equal

Build a Metropolis chain for a chosen target π, then read the flux table — each ordered pair i→j carries exactly as much probability as j→i, and the difference is zero to the last bit: live demo







  
HISTORY & CREDIT Boltzmann 1872 · used by Metropolis 1953

  
“Stationary and reversible are the same thing.” — no. Stationarity balances flow into and out of each state in aggregate; detailed balance balances it edge by edge. Every reversible chain is stationary, but a directed cycle can be stationary while flowing strictly one way — stationary with no detailed balance. cited

  
1872 · Ludwig Boltzmann — detailed balance appears in kinetic theory: at equilibrium every microscopic transition is balanced by its reverse.
1936 · Kolmogorov's criterion — a chain is reversible iff every cycle has equal forward and backward probability product.
1953–70 · Metropolis (1953) and Hastings (1970) turn it into a tool: engineer P to satisfy detailed balance for any target π you can evaluate.

  
Reversibility is a time-symmetry: a movie of the chain at equilibrium plays the same forwards and backwards. It is the self-inverse — the palindrome — showing up in probability rather than in bits. Boltzmann 1872






  
RECOMMEND FOR I-13 reversibility holds to the last bit, computed

  
On the canonical compiler a Metropolis chain built for π = (0.2, 0.3, 0.5) satisfies detailed balance exactly — every forward flux equals its reverse, difference zero:

  
$ i13 run detbal.i13   # Metropolis chain for pi=(0.2,0.3,0.5), symmetric proposal
flux 0<->1 : f01 = 0.1    f10 = 0.1    diff d01 = 0
flux 0<->2 : f02 = 0.1    f20 = 0.1    diff d02 = 0
flux 1<->2 : f12 = 0.15   f21 = 0.15   diff d12 = 0

  
Recommend: detailed balance is LIT for I-13 — verified a Metropolis-built chain for target (0.2,0.3,0.5) gives matched fluxes fᵢ⫺ = f⫺ᵢ on all three edges, with differences exactly 0. This is the reversibility that makes MCMC correct, and it is the self-inverse principle (Verlet, CRC) wearing a probabilistic coat.