On the limits of topological data analysis for statistical inference

Figure 1.  Illustration of different asymptotic regimes

Figure 2.  Scatterplot and Betti curve for point clouds $ \mathbb{X}_n $ obtained from two different distributions in $ \{f_{\boldsymbol{\theta }}: \boldsymbol{\theta } \in \Theta\} $. (Left) $ \mathbb{X}_n \sim f_{\boldsymbol{\theta }} $ for $ \boldsymbol{\theta } = (0.46, 0.47, 0.03, 0.04) $ in blue, (Center) $ \mathbb{X}_n \sim f_{\boldsymbol{\theta }} $ for $ \boldsymbol{\theta } = (0.17, 0.29, 0.21, 0.24) $ in orange, and (Right) the (normalized) Betti curve $ r \mapsto \beta_1(\mathbb{X}_n, r)/n $ for the two point clouds

Figure 3.  Betti curves and the Betti numbers in the thermodynamic regime for $ \{f_{\boldsymbol{\theta }}: \boldsymbol{\theta } \in \Theta\} $ from Example 3.7

Figure 4.  Illustration $ f_{\theta }(x, y) $ from Example 4.2 for $ \theta \in \{\pi/15, \pi/4, 2\pi/3\} $

Figure 5.  The sets $ \left\{\boldsymbol{x} \in \mathbb{R}^2: f_\rho(\boldsymbol{x}) \geq t\right\} $ for $ \rho \in \{-0.99, -0.5, 0.99\} $ respectively. For a fixed level $ t $, all three of them have the same mass. In general, $ \mathbb{P}_{\rho}\big({\{\boldsymbol{x} \in \mathbb{R}^2 : f_\rho(\boldsymbol{x}) \ge t}\}) $ is the same for each $ |\rho|<1 $

Figure 6.  Superlevel sets of the probability density function $ f_{\alpha } $ on $ \mathcal{X} $ from Example 5.9 when $ \alpha \in \{0, -5, 8\} $. For a fixed level $ t \ge 0 $, the mass of the superlevel set $ \{\boldsymbol{x} \in \mathcal{X} : f_{\alpha }(\boldsymbol{x}) \ge t\} $ is the same for each $ \alpha $

Figure 7.  Illustration of $ \mathcal{F}$-equivalence when $ g \sim 0.5 \Gamma(10, 5) + 0.5 \Gamma(10, 2) $ is a mixture of Gamma distributions. For different values of $ \theta \in \Theta $, the excess mass functions $ \hat{f}_{\theta}(t) $ are identical, as shaded in orange for $ t = 0.006 $

Figure 8.  Illustration of the family of distributions from Example 5.9

Figure 9.  Superlevel sets of the density function $ f_{\alpha } $ on $ \mathcal{X}= \mathbb{S}^2 \setminus \{\boldsymbol{p}, -\boldsymbol{p}\} $ from Example A.3 when $ g \sim \rm{Beta}(10, 10) $ is the density function of a Beta distribution on $ (0, 1) $, and for $ \alpha\in \{0, 1, -2, -5\} $