Marco Rubio warns China of'repercussions' as he reveals what really happened during closed-door Trump and Xi meeting Ex-Yankees star Carl Pavano'peed in shampoo bottles and soiled the bed,' ex-wife claims as bitter prenup feud takes disgusting twist Fury as Kash Patel SNORKELS at sacred war tomb where 900 sailors still lie... then jets off to Las Vegas Glamorous Texas Democrat's secret KINK exposed: Congressional candidate's past life returns to haunt her After theater groping shame, Lauren Boebert is being bankrolled by America's cringiest ex-congressman... and it exposes a MASSIVE hypocrisy Horrifying final days of killer dad Chris Watts' pregnant wife before she was slaughtered alongside their daughters. Read all the chilling texts and receipts in full for first time: 'My eyes burn from crying' RHOBH star Diana Jenkins denies claims she put Hayden Panettiere in bed with'undressed man' when she was 18 Trump reveals Xi's offer to break Iran's Hormuz chokehold... as China's price for the rescue looms Mystery blonde Trump aide with unfettered access to President's phone sparks White House friction: Real reason his posts contain random capital letters... and shadowy team behind them unmasked Despicable crimes paid for couple's lavish lifestyle that they flaunted online while gold chain-wearing husband fleeced $1BILLION from taxpayers New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history Bitter cat fight erupts over DHS'sugar baby' scandal: Veteran female intelligence officer launches explosive new accusations that go right to top of counterterror HQ I lost 9lb in two weeks by making one simple tweak to my lifestyle. I didn't use Mounjaro, diet or change how I exercise and I couldn't believe the results... anyone can do it too I'm godfather to Candace Owens' daughter and Charlie Kirk was my friend... so I know the real reason she's attacking Erika - and I'll never publicly condemn her Britney Spears seen'barking and carrying knife' during chaotic restaurant visit I've had acid reflux all my life. Target customers threaten to boycott store after controversial'upgrade' to shopping cart New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history A new DNA analysis of remains belonging to several direct descendants of Christopher Columbus may have uncovered a history-changing truth about the explorer's origins. For centuries, historians have believed the explorer was born in Genoa, Italy, rising from humble beginnings to persuade the Catholic Monarchs to finance what many considered an impossible voyage across the Atlantic.

Stabilized regression aims to identify a set of predictors whose conditional relationship with a response variable remains invariant across different environments. Existing graphical characterizations of the stable blanket are mainly developed for structural causal models (SCMs) without hidden variables or causal cycles. However, latent variables and feedback relationships naturally arise in many applications, and they can change both the Markov blanket and the set of predictors that remain stable under interventions. This paper studies stable blankets in graphical causal models with hidden variables, causal cycles, and both features simultaneously. For models with hidden variables, we use acyclic directed mixed graphs (ADMGs) and $m$-separation to characterize the Markov blanket and to construct intervention-stable predictor sets. We introduce the notion of an intervened sub-district and use it to describe how interventions may affect districts connected to the response. For models with cycles, we work with directed graphs (DGs) and directed mixed graphs (DMGs) together with $σ$-separation, treating strongly connected components (SCCs) as the basic graphical units. We then combine these ideas to analyze models with both hidden variables and cycles. The main results give graphical characterizations of Markov blankets, stable frontiers, and stable blankets in these generalized settings. In particular, we identify conditions under which the response is conditionally independent of intervention variables given a suitable predictor set, and we describe when such sets are minimal or unique. These results extend the graphical interpretation of stabilized regression beyond acyclic fully observed models.

Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.

In this section, we first prove the proposed graphical representation for a removable variable in a MAGM (Theorem 1). Then, we discuss how this representation reduces to Theorem 5 of [11] in the case of DAGs. Throughout our proofs, we say a path between X and Y is blocked by a set Wif it is not m-connecting relative to W. In this case, there exists a non-collider W on the path which is a member of W, or there exists a collider W on the path such that W/2 Anc({X,Y }[ W). In both cases we say W blocks this path with respect to W, or W blocks the path in short when W is clear from the context. We say X is a descendant of Y if Y 2Anc(X), and we denote by DeM(X) the set of descendants of X in the MAGM, and De(X) whenever the graph is clear from the context. A.1 Graphical representation Theorem 1. Vertex X is removable in a MAGM over the variables V, if and only if 1. for any Y 2Adj(X) and Z 2Ch(X)[N(X)\{Y}, Y and Z are adjacent, and 2. for any collider path u =( X,V1,...,V m,Y) and Z 2 V\{X,Y,V1,...,V m} such that {X,V1,...,V m} Pa(Z), Y and Z are adjacent. Let H denote the induced subgraph of M over V\{X}. For any W V\{X,Y,Z}, (Z,X,Y) is an m-connecting path relative to W in M, as X is a non-collider and X/2W. That is, no such W can m-separate Y and Z. Since X is removable in M, by definition of removability, (Y?Z|W)M ()(Y?Z|W)H. Again for any W V\{X,Y,Z}, (Z,X,V1,...,V m,Y) is an m-connecting path relative to W in M since I) every collider on this path is a parent (and therefore an ancestor) of Z, and II) X/2W and X is the only non-collider on this path. That is, no such W can m-separate Y and Z. Since X is removable in M, Equation 8 implies that Y and Z have no m-separating sets in H. Hence, Y is adjacent to Z in H, and therefore, in M. if part: We need to prove that for any Y,Z 2V\{X} and any W V\{X,Y,Z}, (Y?Z|W)M ()(Y?Z|W)H.

Robust 6d object pose estimation by learning rgb-d features.

For each rule, if the left-hand side graph is an induced subgraph of a PDAGG, orient the undirected edge on it with the direction on the right-hand side.