In this section, we provide brief background on information theory and details on our notation.

TabPFN is a transformer that achieves state-of-the-art performance on supervised tabular tasks by amortizing Bayesian prediction into a single forward pass. However, there is currently no method for uncertainty decomposition in TabPFN. Because it behaves, in an idealised limit, as a Bayesian in-context learner, we cast the decomposition challenge as a Bayesian predictive inference (BPI) problem. The main computational tool in BPI, predictive Monte Carlo, is challenging to apply here as it requires simulating unmodeled covariates. We therefore pursue the asymptotic alternative, filling a gap in the theory for supervised settings by proving a predictive CLT under quasi-martingale conditions. We derive variance estimators determined by the volatility of predictive updates along the context. The resulting credible bands are fast to compute, target epistemic uncertainty, and achieve near-nominal frequentist coverage. For classification, we further obtain an entropy-based uncertainty decomposition.

Can Ukraine restore its pre-war borders? Why are Tomahawk missiles for Ukraine a'red line' for Russia? Is Russia testing NATO with aerial incursions in Europe? At least 4 killed in major Russian drone, missile attack on Ukraine's Kyiv Russia's President Vladimir Putin said his forces are prevailing in what he described as a "righteous battle" in Ukraine . "Our fighters and commanders go on the attack, and the entire country, all of Russia, is waging this righteous battle and working hard," he said.

BUCHAREST - Romania scrambled fighter jets on Saturday when a drone breached the country's airspace during a Russian attack on Ukrainian infrastructure near the border, the defense ministry said. Defense Minister Ionut Mosteanu said the F-16 pilots came close to taking down the drone as it was flying very low before it left national airspace toward Ukraine. A threat of drone strikes also prompted Poland to deploy aircraft and close an airport in the eastern city of Lublin on Saturday, three days after it shot down Russian drones in its airspace with the backing of aircraft from its NATO allies. Romania, a European Union and NATO state which shares a 650-km (400-mile) border with Ukraine, has had Russian drone fragments fall onto its territory repeatedly since Russia began waging war on its neighbor. On Saturday, it scrambled two F-16 fighter jets and later two Eurofighters -- part of German air policing missions in Romania -- and warned citizens in the southeastern county of Tulcea near the Danube and its Ukrainian border to take cover, the defense ministry said in a statement.

Russia hit pipelines in Ukraine, sparking bright flames and plumes of smoke seen from Romania. Romania was forced to scramble F-16 jets after Russia carried out a strike just half a mile from the NATO nation's territory. The country's Ministry of National Defense (MApN) confirmed in a post on X that Russia carried out a drone attack near its border. "On the night of August 5-6, the Russian forces launched a massive drone attack on the civilian infrastructure in the Ismail area, Ukraine, in the vicinity of the border with Romania," Romania's defense ministry wrote in a post on X. "The radar systems of the MApN detected air targets in Ukrainian space, close to Tulcea County. At 1:10a.m., the population in the north of the county was warned via RO-Alert," the ministry added.

We introduce a novel topology, called Kernel Mean Embedding Topology, for stochastic kernels, in a weak and strong form. This topology, defined on the spaces of Bochner integrable functions from a signal space to a space of probability measures endowed with a Hilbert space structure, allows for a versatile formulation. This construction allows one to obtain both a strong and weak formulation. (i) For its weak formulation, we highlight the utility on relaxed policy spaces, and investigate connections with the Young narrow topology and Borkar (or $w^*$)-topology, and establish equivalence properties. We report that, while both the $w^*$-topology and kernel mean embedding topology are relatively compact, they are not closed. Conversely, while the Young narrow topology is closed, it lacks relative compactness. (ii) We show that the strong form provides an appropriate formulation for placing topologies on spaces of models characterized by stochastic kernels with explicit robustness and learning theoretic implications on optimal stochastic control under discounted or average cost criteria. (iii) We show that this topology possesses several properties making it ideal to study optimality, approximations, robustness and continuity properties. In particular, the kernel mean embedding topology has a Hilbert space structure, which is particularly useful for approximating stochastic kernels through simulation data.