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A coupling of probability distributions $\mu ,\nu$ is a joint probability distribution of random variables $(X,Y)$ whose marginal distributions are $\mu ,\nu$. It lets us choose the statistical dependence between two random constructions while preserving the individual laws. This choice can make a comparison much easier: giving two simulations the same random input may keep their paths close, even though separately generating both paths would hide that relationship.^[ref]

For example, two fair coin tosses can be generated independently, with mismatch probability $1/2$, or by tossing once and setting $Y=X$, with mismatch probability zero. Both constructions have the same two Bernoulli distributions as marginals. Their joint laws differ. A maximal coupling makes the values agree as often as possible; a coalescing coupling arranges permanent agreement of two processes after a random time. Minimizing expected geometric separation instead leads to Wasserstein distance.

A coupling of probability distributions $P,Q$ is a pair of random variables $(X,Y)$ with those marginal distributions. On a finite set, and more generally on a standard Borel space, the least achievable mismatch isA maximal coupling attains the minimum; maximal here refers to agreement. An arbitrary coupling can have a much larger mismatch.^[ref]

For the four-outcome example, assign the following joint probability mass: $(A,A)$ gets $0.1$, $(B,B)$ gets $0.1$, $(C,C)$ gets $0.2$, and $(D,D)$ gets $0.3$. The remaining $0.3$ must have $X=A$; place $0.2$ on $(A,B)$ and $0.1$ on $(A,C)$. Row sums give $P$, column sums give $Q$, and the diagonal contains $0.7$. Agreement can reach 70%; mismatch cannot fall below 30%.

To see the lower bound, for any event $E$,The two indicator functions agree whenever $X=Y$. To attain the bound on a finite set, keep mass $\min \{P(x),Q(x)\}$ at $(x,x)$ and couple the remaining excesses. The two residual probability distributions live on disjoint subsets, so their part necessarily mismatches. When the probability distributions are identical, the residual part disappears entirely.

This is a choice of joint construction, not a claim that independently drawn observations are usually equal. Two independent random variables with the same continuous probability distribution can disagree with probability one even though their total variation distance is zero. The construction of a maximal coupling deliberately introduces statistical dependence to reveal how similar the marginal laws are.

On a finite state space $F$ with finite objective function values, the fixed-$T$ target is a Boltzmann distribution:The partition function $Z_T$ normalizes the weights but cancels from the Metropolis–Hastings acceptance probability. A short detailed balance proof explains the target. With symmetric $q$ and $x\ne y$,Summing over $x$ gives ${\pi }_T{P}_T={\pi }_T$, so this is a stationary distribution. On a finite state space, a connected graph of positive symmetric proposals makes the accepted Markov chain an irreducible Markov chain. Convergence from any starting state additionally requires an aperiodic Markov chain; one positive self-transition suffices under irreducibility. Replacing $q$ by $(I+q)/2$ ensures this. For asymmetric $q$, require connectivity through bidirectional proposals: one-way proposals are rejected.^[ref]

In the five-state barrier example for simulated annealing, $T_{\ast }=1/\ln 2$ gives exact Boltzmann weights $2^{-E(x)}$. Multiplying by $16$ givesFor the uphill edge, detailed balance is visible numerically:Thus the global minimum has the largest individual stationary probability, $16/33$, but is far from certain at this positive $T_{\ast }$. At $T_{\ast }/2$ the exact stationary distribution is $(16,64,1,16,256)/353$.

For any finite $F$, let $M$ be the set of global minima and $E_{\min }$ the smallest objective function value. Cancelling $e^{-E_{\min }/T}$ from numerator and denominator shows that each nonminimal state's weight tends to zero as $T\downarrow 0$, while every state in $M$ has equal weight. Therefore ${\pi }_T$ tends to the uniform distribution on $M$. This is a statement about a family of equilibrium probability distributions; it does not prove that a cooling Markov chain follows that family.

Simulated annealing accepts worse feasible solutions to escape a local minimum. Minimize an objective function $E(x)$, analogous to energy. Effective temperature $T>0$ has the same units as $E$, absorbing the Boltzmann constant; it controls exploration, rather than a material's thermodynamic temperature.^[ref]

Given current feasible point $x$, draw $y$ from a proposal distribution $q(x,y)$. For a symmetric proposal distribution, $q(x,y)=q(y,x)$, use the Metropolis–Hastings acceptance probabilityAccept with this conditional probability; on rejection retain $x$. Every downhill or equal-cost proposal is accepted. Draw a fresh uniform random variable $U$ on $(0,1)$ and accept if $U<a_T(x,y)$. The off-diagonal transition probability $P_T(x,y)=q(x,y)a_T(x,y)$ includes proposing $y$.^[ref]

For a maximization problem with objective function $f$, set $E=-f$: every improving proposed neighbor is accepted, and a worsening move of loss $\Delta =f(x)-f(y)>0$ has conditional probability $\exp (-\Delta /T)$ of acceptance, under the same symmetry assumption. Likelihood-power annealing uses precisely this sign convention for a log-likelihood. Retain the best feasible solution seen, since the current state can worsen even after a global minimum has been visited.

If the proposal distribution is asymmetric, use the full Metropolis–Hastings acceptance probability, for $q(x,y)>0$,A missing reverse proposal gives zero acceptance. Even a downhill proposal need not be accepted with probability one in this case. Choosing uniformly among a variable number of neighbors is generally asymmetric: $q(x,y)=1/d(x)$ and $q(y,x)=1/d(y)$ give the correction $d(x)/d(y)$, where $d$ is vertex degree.^[ref]

The useful idea is to separate exploring from insisting on improvement: an uphill move can open a route to a better feasible solution. Its usefulness depends on the proposal distribution, the scale of the objective function, and the cooling schedule for simulated annealing. Finite practical schedules carry no general guarantee of finding a global minimum.

Total variation distance asks how much two probability distributions can disagree about the same event. For probability measures $P,Q$ on one measurable space $(S,\mathcal{F})$, its definition isThe supremum ranges over all measurable sets, so $d_{\mathrm{TV}}(P,Q)\le \epsilon$ guarantees that every yes-or-no prediction has probability error at most $\epsilon$. This metric takes values between zero and one; it is zero exactly when the two probability measures agree. This makes it useful for comparing a sampling algorithm's output with its target probability distribution, and for defining the mixing time of a Markov chain.

On a finite set, add the excess probability mass wherever $P$ exceeds $Q$. That excess is also half the sum of all absolute differences:The last expression reveals the common probability mass that the two probability distributions can share.^[ref] The coupling characterization of total variation distance turns this overlap into an actual joint construction; minimum sum of errors in a simple hypothesis test gives it a statistical interpretation. One quantity connects probability bookkeeping, comparison of sampling methods, and the limits of a statistical test.