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Single-Current Form of the Path Likelihood Ratio

Abstract

For the single-peak parity kernel, the forward-versus-reversed log-likelihood of every finite path is one net current times the cycle affinity plus two endpoint terms.

Definition 1.1 (Regions of the single-peak kernel).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.region (✓ std3).

Source. Repository-derived.

Commentary.

The state space carries a sign function chi with values plus or minus one and a distinguished peak z with chi(z) = 1. The region H is the peak itself, B is the rest of the peak’s sign class, and Z is the opposite sign class.

Definition 1.2 (Single-peak profile).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.profile (✓ std3).

Source. Repository-derived.

Commentary.

The profile b takes the value r at the peak, the value -q on the rest of the peak’s sign class, and the value 0 on the opposite class.

Definition 1.3 (Single-peak parity kernel).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.kernel (✓ std3).

Source. Repository-derived.

Commentary.

With normalizer N, the transition weight from x to y is (1 + chi(x) chi(y) b(x)) / N. On the hypercube of dimension d with chi the product of coordinates and N = 2^d this is the parity kernel P(x,y) = (1 + a(x) chi(y)) / N with a(x) = chi(x) b(x).

Definition 1.4 (Forward path law).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.forwardLaw (✓ std3).

Source. Repository-derived.

Commentary.

The forward law of the path x_0, …, x_T starts from the uniform mass 1/N and multiplies the transition weights along the path.

Definition 1.5 (Time-reversed path law).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.reverseLaw (✓ std3).

Source. Repository-derived.

Commentary.

The reversed law evaluates the same path with every transition traversed backwards, again from the uniform mass 1/N.

Definition 1.6 (Region transition counts).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.transitions (✓ std3).

Source. Repository-derived.

Commentary.

N_UV counts the times t < T at which the path moves from region U to region V.

Definition 1.7 (Endpoint defect).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.endpointDefect (✓ std3).

Source. Repository-derived.

Commentary.

Delta_U is the indicator that the path starts in U minus the indicator that it ends in U.

Definition 1.8 (Region transition weight).

Formalization. D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.regionWeight (✓ std3).

Source. Repository-derived.

Commentary.

The weight w with parameters a, b, c assigns a to H -> B, b to B -> Z and c to Z -> H, the negatives to the reversed transitions, and 0 to every transition inside one region.

Theorem 1.9 (Single-current form of the direction log-likelihood).

Proof. Machine-checked in Lean as D5/S3/Estimation/TimeArrow/SinglePeakPathCurrent.log_forward_div_reverse_eq_current (✓ std3). ∎

Source. Repository-derived.

Commentary.

Assume chi takes only the values 1 and -1, chi(z) = 1, |r| < 1, |q| < 1 and N > 0. Then for every path x and horizon T the log of the forward law divided by the reversed law is the displayed expression, where A = log((1 + r)/(1 - q)), B_0 = log(1 + q) and C = -log(1 - r).

Every transition weight is positive. A transition inside one region has equal forward and backward weights. The ratios across regions are (1 + r)/(1 - q) from H to B, 1 + q from B to Z and 1/(1 - r) from Z to H, because chi(x) chi(y) is 1 inside a sign class and -1 across classes. The uniform initial masses cancel, the logarithm of the product of ratios is the sum of region weights w(Y_t, Y_(t+1)) with a = A, b = B_0 and c = C.

That sum is reduced to one current as follows. Each weight equals a potential difference plus the affinity a + b + c times the signed indicator of the edge between Z and H, with potential 0 on H, a on B and a + b on Z. Summing along the path telescopes the potentials, and the three endpoint defects add to zero, which turns the potential difference into a Delta_H - b Delta_Z. Consequently the triple (J, Y_0, Y_T) with J = N_ZH - N_HZ determines the likelihood ratio of the two time directions.

References