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Continuity of the Solution of a Non-Parametric Bayesian Statistical Calibration Procedure

Akshay Prasadan, Donald Estep, Derek Bingham

Abstract

Recent work has developed a non-parametric Bayesian approach to the calibration of a computer model, which abstractly amounts to the inversion of a pushforward of stochastic input parameters by a smooth map. The framework has been used in several complex scientific applications, motivating our investigation on the continuity of the solution operator with respect to the distribution on the input parameters. We demonstrate that the solution operator for this approach is uniformly continuous in the total variation metric and weakly continuous for a broad class of distributions.

Continuity of the Solution of a Non-Parametric Bayesian Statistical Calibration Procedure

Abstract

Recent work has developed a non-parametric Bayesian approach to the calibration of a computer model, which abstractly amounts to the inversion of a pushforward of stochastic input parameters by a smooth map. The framework has been used in several complex scientific applications, motivating our investigation on the continuity of the solution operator with respect to the distribution on the input parameters. We demonstrate that the solution operator for this approach is uniformly continuous in the total variation metric and weakly continuous for a broad class of distributions.
Paper Structure (11 sections, 10 theorems, 48 equations, 4 figures, 1 table)

This paper contains 11 sections, 10 theorems, 48 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of the SCP
  3. Total Variation Continuity
  4. Weak Continuity
  5. Examples
  6. Gaussian Mixture
  7. A Concrete Application
  8. Discussion
  9. Proofs for section \ref{['section:total:variation']}
  10. Proofs for section \ref{['section:weak:continuity']}
  11. Surface Integral Representations

Key Result

Theorem 1

Let $P_{\Lambda}^t$ and $R_{\Lambda}^t$ be two measures in $\mathscr{P}_{\Lambda,Q}$ with pushforward densities $p_{\mathcal{D}}=\frac{dQP_{\Lambda}^t}{d\mu_{\mathcal{D}}}$ and $r_{\mathcal{D}}=\frac{dQR_{\Lambda}^t}{d\mu_{\mathcal{D}}}$. Let $P_p$ be a prior distribution on $\Lambda$ with density $

Figures (4)

  • Figure 1: We illustrate an example of distribution $P_{\Lambda}^t$ on a set $\Lambda\subseteq\mathbb{R}^2$, and the pushforward distribution $QP_{\Lambda}^t$ by a map $Q\colon \Lambda\to\mathcal{D}$. We set $Q(\lambda_1,\lambda_2)=\lambda_1^2+3\lambda_2^2$ and $P_{\Lambda}^t$ to be an unequally weighted mixture of two bivariate Gaussians. Left: Kernel density estimate (KDE) fitted to data drawn from $P_{\Lambda}^t$. The solid black line indicates the contour $Q^{-1}(20)$. Right: KDE of the pushforward distribution on $\mathcal{D}$ generated by applying $Q$ to points $\lambda$ drawn from $P_{\Lambda}^t$. The dotted line corresponds to $Q=20$.
  • Figure 1: The top row and bottom left plot show a heatmap of the estimated SCP solutions $\hat{P}_{\Lambda}^i$ on $\Lambda=[-5,5]\times[-5,5]$ with output map $Q(\lambda_1,\lambda_2)=\lambda_1^2+3\lambda_2^2$. Over each heatmap, we overlay as points a subsample of the respective choice of TGD $P_{\Lambda}^{t,i}$. The bottom-right plot shows a subsample of the generated prior datapoints in $\Lambda$.
  • Figure 2: Top left: Scatterplot of points drawn from (truncated) Gaussian prior on $\Lambda=\{(a,b):a\in[4,14], b\in[-3,-0.1]\}$. A point $(a,b)\in \Lambda$ corresponds to a power law $y=aR^b$ for $R>0$. Top right: Compressive strength (megapascals) versus water to binder ratio $R$, with power law fits sampled from prior. Bottom left: SCP solution for $\Lambda$ using Gaussian prior from top left panel and strength values in the vertical strip $R\in[0.2,0.4]$ displayed in the top right panel. Bottom right: Same plot as top right except we plot power laws sampled from the SCP solution, whose corresponding points are also indicated on the bottom left panel.
  • Figure 3: We display the equivalent of the bottom left and bottom right panels of \ref{['fig:example_three']}, except now we use concrete data that is aged between 25 and 50 days. We plot those strength and water to binder ratio $R$ on the right. The vertical segment $R\in[0.2,0.4]$ depicts the data used in the SCP estimation, whose solution is plotted on the left. We then pick 30 points from this SCP solution (shown as points on the left plot) and graph the corresponding power laws on the right.

Theorems & Definitions (20)

  • Theorem 1: TV Stability
  • Theorem 2: Local Limit Theorem
  • Proof 1
  • Theorem 1: Weak Convergence
  • Theorem 2: Constructive Weak Convergence
  • Theorem 3: Constructive Weak Convergence with Dirac Measure
  • Corollary 4: Constructive Weak Convergence of Mixtures
  • Proof 2: Proof of \ref{['theorem:stability']}
  • Lemma 1
  • Proof 3
  • ...and 10 more