Bayes-assisted anytime-valid $t$-tests for Gaussian means with unknown variance

We propose an anytime-valid test of $H_0:\mu=0$ versus $H_1:\mu\neq 0$ for i.i.d.\ normal data with unknown mean $\mu$ and unknown variance $\sigma^2$, using prior information on the standardized effect size $\theta^\star=\mu/\sigma$. The test mixes a scale-invariant likelihood ratio over a prior on the effect size and applies the extended Ville inequality to obtain a valid $p$-process under arbitrary stopping. When the prior is well specified, the method improves power for alternatives favoured by the prior. For every proper prior the reciprocal-Ville $p$-value degenerates to one. If in addition the prior has polynomial tails, the extended-Ville $p$-value converges to a non-informative benchmark. Thus the procedure gains power when the prior is accurate and reverts to the non-informative benchmark under extreme prior-data conflict.