Adaptive finite-candidate Gaussian-process search uses the correct sequential posterior

At the initial history, no adaptive observations have yet been added; if the supplied fixed initial list is empty, the conditional law is the given GP prior, and otherwise [F1] gives its posterior whenever its observed covariance is positive definite. The algorithm computes that initial state before any acquisition, and the all-data-vector posterior version in [F1] fixes the kernel for every possible observed value.

Assume as the induction hypothesis that the posterior formula is valid after a finite history $H_t$. By [F3], the next candidate index is a measurable function of $H_t$, so conditioning on the selected site adds no information beyond $H_t$. Conditional on $H_t$, the candidate vector has the finite Gaussian posterior from the induction hypothesis, the selected site is fixed, and the fresh observation error is independent of the process and previous errors by [A2]. Observation availability is already determined by $H_t$ and that site, so a successful response adds no further selection information. Thus the new noisy observation is a Gaussian coordinate of this conditional vector plus independent Gaussian noise. The observed covariance of the augmented list is positive definite; when $t\ge 1$, its scalar conditional variance is the positive Schur complement of $C_t$ in $C_{t+1}$, and when $t=0$ it is the positive observed variance in $C_1$. The finite Gaussian conditioning formula therefore gives the one-observation update. Since $H_{t+1}$ consists of $H_t$ and this new observation, with its site already determined by $H_t$, this is conditioning on the augmented data list.

Applying [F1] to that augmented list yields the stated posterior after the new observation for every observed value. The algorithm computes and stores this state immediately after appending the observation, including when the history length reaches $N$. This proves the induction step, including repeated selected sites whenever the augmented covariance remains positive definite. If the solve is unavailable, no posterior state for the augmented history is asserted.

Induction over the finite successful history proves the sequential posterior formula at every history with a computed state, including the terminal horizon state. The argument uses only conditional Gaussian updating and the history-measurable finite selection rule, so it gives no regret or global-optimality conclusion.