Alphabeta MathSmall-ball MAP and Cameron–Martin geometry — Examples and Counterexamples
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Areas of Elementary Plane Figures
- Banach Gaussian priors, RKHS support, and concentration
- Banach Valued Integration and the Radon Nikodym Property
- Bayesian models, priors, and posterior laws
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Cyclic Groups and Direct Products
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Direct Matrix Factorisations: LU, Cholesky and QR
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Domination and Likelihood Ratios
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability and the Probabilistic Method
- Finite-window spatial likelihood
- Foundations of the Real Numbers for Analysis
- Fourier Transform Convolution and Approximate Identities
- Fubini and Change of Variables
- Function-space priors and sample regularity
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hausdorff via the Diagonal
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Inverse maps, identifiability, and stability
- LAN, Contiguity and Efficiency
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Likelihood potentials and posterior propriety
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Loss, Risk and Decision Theory
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed effects and longitudinal models
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Posterior propriety and dominated Bayes formula
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Probability Spaces Random Variables and Expectation
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Radon Measures and the Riesz Markov Kakutani Theorem
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Sine, Cosine, and the Definition of Pi
- Small-ball MAP and Cameron–Martin geometry
- Splitting Fields
- Standard-Borel Real Codings and Determining Classes
- State-space model laws
- Statistical Experiments, Models and Identifiability
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Convergence Tightness and Representation
- Weak nonparametric Bernstein–von Mises limits
2 · Logical flowchart
3 · Items
Example
Finite-dimensional small-ball MAP equals the density mode
Counterexample
An infinite-dimensional Gaussian has no locally finite translation-invariant density reference
Example