Group invariance applications in statistics

By M. L. Eaton. 1989

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A theory of independent mechanisms for extrapolation in generative models (2020, background): “The authors can refer for example to (Tung 1985; Wijsman 1990; Eaton 1989) for more details.”
Group invariance principles for causal generative models (2017, background): “A key result regarding topological groups is the existence and uniqueness up to a positive constant of the Haar measure [Eaton, 1989].”
Measuring and testing dependence by correlation of distances (2007, background): “ =|ρ|, R(2)n = |ρˆ| and Vn (2) =2|ˆσxy|, where ˆσxy is the maximum likelihood estimator of Cov(X,Y). 3.2. Aﬃne invariance. Group invariance is an important concept in statistical inference (see Eaton [3] or Giri [4]), particularly when any transformation of data and/or parameters by some group eleme”
A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants (2011, unknown): “Also, let R + = (0 , ∞ ). for Any continuous function χ : G → R + such that χ ( g 1 g 2 ) = χ ( g 1 ) χ ( g 2 ) all g 1 , g 2 ∈ G is called a multiplier [Eaton (1989)].”
BAYES FACTORS AND MARGINAL DISTRIBUTIONS IN INVARIANT SITUATIONS (2016, methodology): “For the situation in section 1, let T(y) be a maximal invariant function acting on Y, with respect to G. (Again, see Eaton (1989) for definitions.) Let f? j (¢) denote the density of T(y) under Model j, and with respect to the measure „T induced by „.”
Invariant tests for multivariate normality: a critical review (2002, background): “data AXj + b (j = 1,..., n). Eaton (1989) , page 46, calls this a soft argument to sup-”
A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms (2008, methodology): “ote that F(g,y) is playing the role of t g(y) from Section 1.] As is typically done, we will abbreviate F(g,y) with gyso, for example, the second condition is written (g 1g 2)y=g 1(g 2y). As in Eaton [3], we use the term multiplier to describe a continuous homomorphism of Ginto the multiplicative gr”
Sure independence screening for ultrahigh dimensional feature space (2006, methodology): “† First, carry out the procedure in ( 17 ) to the full model f M0‐ b = f1;¢¢¢ ;pg and get a submodel f”
Safe Testing (2019, background): “The reason this is possible is that w H coincides with the right-Haar prior for this problem (Eaton, 1989; Berger et al., 1998), about which we will say more below.”
Mixtures of g Priors for Bayesian Variable Selection (2008, background): “This, along with arguments based on orthogonal parameterizations and invariance to scale and location transformations (Jeffreys 1961; Eaton 1989; Berger, Pericchi, and Varshavsky 1998), has led to the adoption of p(α,φ|Mγ ) = 1 φ , (3) βγ |φ,Mγ ∼ N ( 0, g φ (XTγ Xγ ) −1 ) , (4) as a default prior…”
Probabilistic Symmetries and Invariant Neural Networks (2020, methodology): “See, for example, Andersson (1982); Eaton (1989); Wijsman (1990); Schindler (2003); and especially Kallenberg (2017).”
The affinely invariant distance correlation (2012, background): “y invariant distance correlation and its sample version are invariant under the group of invertible ane transformations, and in addition to satisfying this often-desirable group invariance property (Eaton, 1989), they inherit the desirable properties of the standard distance dependence measures. In ”
Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem (2010, background): “This seems to run afoul of the general injunction against improper priors in model selection problems, but can nonetheless be formally justified by arguments appealing to othogonality and group invariance; see, for example, Berger, Pericchi and Varshavsky (1998) and Eaton (1989).”
Maxima of entries of Haar distributed matrices (2005, background): “To check these details readers are referred to [16], [ 17 ] or [30].”
Lossy Compression for Lossless Prediction (2021, background): “(3)This extends the definition of maximal invariants [19] beyond invariances to group actions.”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “y U := U(M,N). To get such a U, we can generate in Matlab an M × M Gaussian matrix G with i.i.d. N(0,1) entries, and let U =G(G∗G)−1/2 which is well deﬁned with probability 1; refer to Section 7.1 of [11] for instance. Then we will ﬁx this U for each pair of (M,N) as a deterministic orthogonal matri”
On likelihood ratio tests (2006, methodology): “(2.3) We shall call the test (2.3) the average likelihood ratio test and from now on shall refer to (1.1) as the maximum likelihood ratio test.”