A sequential experiment chooses how much information to collect as data arrive. Its procedure consists of a stopping rule and a terminal action, and its loss may include both decision error and sampling cost. Likelihood-ratio martingales make change of measure compatible with stopping.
Stopped observations
Definition. Let \((\mathcal F_t)_{t\geq0}\) be a filtration generated by observations. A stopping time \(\tau\) satisfies \[ \{\tau\leq t\}\in\mathcal F_t. \] The stopped \(\sigma\)-field is \[ \mathcal F_\tau = \{A:A\cap\{\tau\leq t\}\in\mathcal F_t \text{ for every }t\}. \] A sequential terminal decision must be \(\mathcal F_\tau\)-measurable.
The full action can be written \((\tau,a)\), with loss \[ L(\theta,\tau,a) =L_0(\theta,a)+c\tau \] for per-observation cost \(c\), or with a more general time-dependent cost.
Likelihood-ratio martingales
Theorem. Let \(Q\ll P\) on \(\mathcal F_\infty\), and define \[ L_t=E_P\!\left( \frac{\mathrm{d}Q}{\mathrm{d}P} \,\middle|\,\mathcal F_t \right). \] Then \((L_t)\) is a nonnegative \(P\)-martingale and \[ Q(A)=E_P(L_\tau1_A) \] for every bounded stopping time \(\tau\) and \(A\in\mathcal F_\tau\). Thus \(L_\tau\) is the likelihood ratio of the stopped experiment.
Proof. The martingale property follows from iterated conditional expectation. For bounded \(\tau\), write \(A=\bigcup_t(A\cap\{\tau=t\})\) in discrete time. Each piece belongs to \(\mathcal F_t\), and \[ Q(A\cap\{\tau=t\}) = E_P\{L_t1_{A\cap\{\tau=t\}}\}. \] Summing gives \(Q(A)=E_P(L_\tau1_A)\). Continuous time follows by approximation. \(\square\)
For unbounded stopping times, optional stopping requires uniform integrability or localization. A nonnegative likelihood martingale may lose mean at an unbounded stopping time, which corresponds to loss of absolute continuity in the stopped limit.
Sequential probability ratio test
Definition. For simple \(P_0\) versus \(P_1\), let \[ \Lambda_t = \frac{\mathrm{d}P_1|_{\mathcal F_t}} {\mathrm{d}P_0|_{\mathcal F_t}}. \] The SPRT stops at \[ \tau=\inf\{t:\Lambda_t\notin(B,A)\}, \qquad0<B<1<A, \] rejects \(H_0\) at the upper boundary, and accepts it at the lower boundary.
Proposition. Ignoring overshoot, choices \[ A\approx\frac{1-\beta}{\alpha}, \qquad B\approx\frac{\beta}{1-\alpha} \] produce type-I error \(\alpha\) and type-II error \(\beta\). Rigorous inequalities follow from the likelihood martingale: \[ P_0(\text{reject})\leq A^{-1}, \qquad P_1(\text{accept})\leq B. \]
Proof. On rejection, \(\Lambda_\tau\geq A\). Since \(E_0\Lambda_\tau\leq1\) for a nonnegative stopped martingale, \[ 1\geq E_0\Lambda_\tau \geq A\,P_0(\text{reject}). \] Under \(P_1\), \(1/\Lambda_t\) is the reverse likelihood martingale, and on acceptance \(1/\Lambda_\tau\geq1/B\), yielding the second bound. \(\square\)
Expected sample size
For i.i.d. log likelihood increments \[ Y_i=\log\frac{p_1(X_i)}{p_0(X_i)}, \] Wald's identity gives, under integrability, \[ E_j\log\Lambda_\tau=E_j\tau\,E_jY_1. \] Since \[ E_1Y_1=D_{\mathrm{KL}}(P_1\|P_0)>0, \qquad E_0Y_1=-D_{\mathrm{KL}}(P_0\|P_1)<0, \] the expected sample sizes are approximately boundary distances divided by the corresponding KL information per observation.
Brownian motion with drift
Theorem (local sequential limit). Under LAN and a functional central limit theorem, the partial log likelihood process for a local alternative \(h\) converges to \[ t\mapsto h^\top I^{1/2}W_t-\frac12t\,h^\top Ih, \] where \(W_t\) is standard Brownian motion. Stopping rules that are continuous functionals of the path transfer to stopping times in this Brownian drift experiment.
The limit experiment observes information continuously in time. Boundary-crossing probabilities and expected stopping times become diffusion calculations rather than fixed-sample Gaussian tail calculations.
Sequential optimality
Wald and Wolfowitz showed that, for simple hypotheses and specified error probabilities, the SPRT minimizes expected sample size under both hypotheses among broad classes of sequential tests. In Bayesian sequential decision problems, dynamic programming compares the immediate posterior loss with the continuation value plus sampling cost; the optimal stopping region is described in posterior or likelihood-ratio coordinates.
Stopping rules and likelihood principles
If the stopping rule is known and parameter independent, the likelihood for the observed stopped path is the ordinary product likelihood evaluated at the realized sample size. Nevertheless, frequentist operating characteristics depend on the stopping rule because the distribution over possible paths and sample sizes changes. Likelihood proportionality and experiment equivalence are therefore distinct claims.