Notes on: Chwialkowski, K., Strathmann, H., & Gretton, A. (2016): A kernel test of goodness of fit

Table of Contents

Notation

  • chwialkowski16_kernel_test_goodn_fit_f5137804281d3ed956eb8079171af344aa367cfc.png is the RKHS of real-valued functions on chwialkowski16_kernel_test_goodn_fit_de0b57391ed8b0d9becf399c343de2bf83467ce1.png with reproducing kernel chwialkowski16_kernel_test_goodn_fit_094b02afce734f4ce51933d0093ef3d2da9f8123.png
  • chwialkowski16_kernel_test_goodn_fit_09ce3636afb059b763243ddcd85077f4c36ce3d8.png the product RKHS consisting of elements chwialkowski16_kernel_test_goodn_fit_60919f9a5e3a83535856a2848d8a092a37f6ddd3.png with chwialkowski16_kernel_test_goodn_fit_3becaa79911af5d67f74ca356416f5aa9416cbae.png, and standard inner product chwialkowski16_kernel_test_goodn_fit_fc5ff1ea6333608ed0cba642b26042013a80cf9f.png
  • chwialkowski16_kernel_test_goodn_fit_0aa0470120f07b61ef9e985225cf9bbd6b79177c.png is rv. distributed according to measure chwialkowski16_kernel_test_goodn_fit_ab437e1f9b3376761b155efe111c9860607c4b86.png
  • chwialkowski16_kernel_test_goodn_fit_0207be880056b9a69e22e729dd37bced29cd174a.png is rv. distributed according to the target measure chwialkowski16_kernel_test_goodn_fit_fefe9e556d399665a26a37824ec578cbffb0cabe.png

Test Definition: Statistic and Threshold

Defining Stein operator on RKHS

  • Define the Stein operator chwialkowski16_kernel_test_goodn_fit_21b0c8a5ebaf36d2a85aef6af6dea09b5bf25774.png acting on chwialkowski16_kernel_test_goodn_fit_ec7a870491deef44843e61e4e9bdf40dd250a1ad.png

    chwialkowski16_kernel_test_goodn_fit_9a51aaf1d6fe90c4ca4cea067b0cae7dbb20b7a5.png

  • Operator can be expressed using the function:

    chwialkowski16_kernel_test_goodn_fit_68ac61f1bd140b35b8dd1e4381b35b14652517c2.png

    whose expected inner product wiht chwialkowski16_kernel_test_goodn_fit_cdd1cc131da6040eca078917132a377727053c44.png vies expected value of the Stein operator:

    chwialkowski16_kernel_test_goodn_fit_510ce2c060d612ad8899b31912f92adad73cd330.png

    where chwialkowski16_kernel_test_goodn_fit_ea4da14b3a87efab2a73b9eebd4e4d10118ba623.png is the i-th component of chwialkowski16_kernel_test_goodn_fit_c931a645f7ab69dd1f7b96566b67cc614606a58e.png.

  • Stein discrepancy in the RKHS:

    chwialkowski16_kernel_test_goodn_fit_17d21b11f60500e2941af0a2fba81ee6a13016a5.png

Wild Boostrap Testing

  • Method for estimating the squared Stein discrepancy chwialkowski16_kernel_test_goodn_fit_c3d005b661d83465c8f03531b2b374cdeba17a7c.png from samples chwialkowski16_kernel_test_goodn_fit_d506c8d8ab74626125de7f455224913ea28f9632.png