$A_{2}$ | $-$ | $-$ | 0.54 | 0.67 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.15 | 0.13 | 0.28 | 0.08
$A_{3}$ | $-$ | $-$ | $-$ | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.08 | 0.51 | 0.34 | 0.69 | 0.06
$A_{4}$ | $-$ | $-$ | $-$ | $-$ | 0.98 | 0.03 | 0.47 | 0.12 | 0.00 | 0.54 | 0.33 | 0.27 | 0.09
$A_{5}$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$A_{6}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.22 | 0.08 | 0.00 | 0.31 | 0.18 | 0.14 | 0.03
$A_{7}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.06 | 0.00 | 0.12 | 0.07 | 0.04 | 0.02
$A_{8}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.10 | 0.49 | 0.33 | 0.09 | 0.05
$A_{9}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.73 | 0.00 | 0.00 | 0.00
$A_{10}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.35 | 0.00 | 0.00
$A_{11}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.50 | 0.00
$A_{12}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.09
Table 27:  Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m({{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{\pm})>2.1$ ${\mathrm{\,Ge\kern-0.80005ptV\\!/}c^{2}}$. The resonances are: ($A_{0}$)K-matrix S-wave, ($A_{1}$) $\rho(770)$, ($A_{2}$) $\omega(782)$, ($A_{3}$) $\rho(1450)$, $(A_{4})$ $\rho(1700)$, $(A_{5})$ $f_{2}(1270)$, $(A_{6})$ ${{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{-}$ P-wave, $(A_{7})$ $D_{0}^{*}(2400)^{-}$, $(A_{8})$ $D_{2}^{*}(2460)^{-}$, $(A_{9})$ $D_{3}^{*}(2760)^{-}$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$ | $A_{9}$
---|---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 0.70 | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.28 | 0.49 | 0.43 | 0.16
$A_{1}$ | $-$ | 1.00 | 0.34 | 0.71 | 0.31 | 0.00 | 0.41 | 0.22 | 0.25 | 0.14
$A_{2}$ | $-$ | $-$ | 0.13 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
$A_{3}$ | $-$ | $-$ | $-$ | 0.42 | 0.29 | 0.00 | 0.29 | 0.23 | 0.20 | 0.07
$A_{4}$ | $-$ | $-$ | $-$ | $-$ | 0.21 | 0.00 | 0.23 | 0.16 | 0.19 | 0.04
$A_{5}$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.58 | 0.22 | 0.16 | 0.08 | 0.05
$A_{6}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.58 | 0.00 | 0.00 | 0.00
$A_{7}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.60 | 0.00 | 0.00
$A_{8}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.72 | 0.00
$A_{9}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.22
Table 28:  Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m({{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{\pm})>2.1$ ${\mathrm{\,Ge\kern-0.80005ptV\\!/}c^{2}}$. The resonances are: ($A_{0}$) K-matrix S-wave, ($A_{1}$) $\rho(770)$, ($A_{2}$) $\omega(782)$, ($A_{3}$) $\rho(1450)$, $(A_{4})$ $\rho(1700)$, $(A_{5})$ $f_{2}(1270)$, $(A_{6})$ ${{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{-}$ P-wave, $(A_{7})$ $D_{0}^{*}(2400)^{-}$, $(A_{8})$ $D_{2}^{*}(2460)^{-}$, $(A_{9})$ $D_{3}^{*}(2760)^{-}$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$ | $A_{9}$
---|---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 1.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.10 | 0.84 | 1.88 | 1.21 | 0.36
$A_{1}$ | $-$ | 2.13 | 0.06 | 1.42 | 1.02 | 0.00 | 0.87 | 0.37 | 0.14 | 0.29
$A_{2}$ | $-$ | $-$ | 0.01 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00
$A_{3}$ | $-$ | $-$ | $-$ | 0.82 | 0.73 | 0.00 | 0.13 | 0.30 | 0.10 | 0.15
$A_{4}$ | $-$ | $-$ | $-$ | $-$ | 0.61 | 0.00 | 0.88 | 0.51 | 0.89 | 0.14
$A_{5}$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.83 | 0.27 | 0.16 | 0.18 | 0.16
$A_{6}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.67 | 0.00 | 0.00 | 0.00
$A_{7}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.86 | 0.00 | 0.00
$A_{8}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.06 | 0.00
$A_{9}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.18
Table 29:  Model-dependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m({{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{\pm})>2.1$ ${\mathrm{\,Ge\kern-0.80005ptV\\!/}c^{2}}$. The resonances are: ($A_{0}$) K-matrix S-wave, ($A_{1}$) $\rho(770)$, ($A_{2}$) $\omega(782)$, ($A_{3}$) $\rho(1450)$, $(A_{4})$ $\rho(1700)$, $(A_{5})$ $f_{2}(1270)$, $(A_{6})$ ${{\kern 1.59998pt\overline{\kern-1.59998ptD}{}}{}^{0}}\pi^{-}$ P-wave, $(A_{7})$ $D_{0}^{*}(2400)^{-}$, $(A_{8})$ $D_{2}^{*}(2460)^{-}$, $(A_{9})$ $D_{3}^{*}(2760)^{-}$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.  | $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$ | $A_{8}$ | $A_{9}$
---|---|---|---|---|---|---|---|---|---|---
$A_{0}$ | 1.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.24 | 0.25 | 0.40 | 0.15
$A_{1}$ | $-$ | 0.79 | 0.02 | 0.41 | 0.25 | 0.00 | 0.26 | 0.29 | 0.19 | 0.05
$A_{2}$ | $-$ | $-$ | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$A_{3}$ | $-$ | $-$ | $-$ | 0.21 | 0.08 | 0.00 | 0.20 | 0.09 | 0.12 | 0.03
$A_{4}$ | $-$ | $-$ | $-$ | $-$ | 0.12 | 0.00 | 0.14 | 0.12 | 0.08 | 0.02
$A_{5}$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.58 | 0.14 | 0.19 | 0.08 | 0.08
$A_{6}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.75 | 0.00 | 0.00 | 0.00
$A_{7}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.52 | 0.00 | 0.00
$A_{8}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.54 | 0.00
$A_{9}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.07

##  Appendix D Results of the K-matrix parameters

The moduli and phases of the K-matrix parameters in Eq. (25) are listed in
Table 30. The break-down of systematic uncertainties are shown in Tables 31
and 32.

Table 30: The moduli and phases of the K-matrix parameters. The first uncertainty is statistical, the second the experimental systematic, and the third the model-dependent systematic. The moduli are normalised to that of the $\rho(770)$ contribution and the phase of $\rho(770)$ is set to 0∘.  Parameter | Modulus |  Phase (∘)
---|---|---
$f_{10}$ | $\kern 4.5pt17.0\pm\kern 4.5pt3.3\pm\kern 4.5pt9.5\pm\kern 4.5pt3.7$ | $347.3\pm 13.7\pm 18.7\pm\kern 4.5pt3.2$
$f_{11}$ | $\kern 4.5pt14.9\pm 17.1\pm 20.3\pm\kern 4.5pt8.0$ | $160.0\pm 70.1\pm 39.6\pm 26.2$
$f_{12}$ | $111.3\pm 23.1\pm 23.8\pm 12.8$ | $226.1\pm 12.0\pm 11.2\pm\kern 4.5pt4.9$
$f_{13}$ | $\kern 4.5pt28.7\pm 14.2\pm\kern 4.5pt8.1\pm\kern 4.5pt5.3$ | $186.5\pm 30.0\pm 30.4\pm\kern 4.5pt8.6$
$f_{14}$ | $\kern 4.5pt31.0\pm 12.8\pm 13.4\pm\kern 4.5pt8.8$ | $10.61\pm 25.9\pm 15.2\pm\kern 4.5pt2.9$
$\beta_{0}$ | $\kern 4.5pt\kern 4.5pt9.5\pm\kern 4.5pt1.8\pm\kern 4.5pt2.9\pm\kern 4.5pt1.1$ | $\kern 4.5pt20.7\pm 15.2\pm 13.5\pm 10.4$
$\beta_{1}$ | $\kern 4.5pt17.2\pm\kern 4.5pt6.4\pm\kern 4.5pt6.2\pm\kern 4.5pt4.8$ | $\kern 4.5pt19.6\pm 19.4\pm 14.4\pm\kern 4.5pt3.7$
$\beta_{2}$ | $\kern 4.5pt34.9\pm\kern 4.5pt7.6\pm 14.3\pm\kern 4.5pt3.1$ | $128.3\pm 12.1\pm\kern 4.5pt2.1\pm\kern 4.5pt1.9$
$\beta_{3}$ | $\kern 4.5pt53.5\pm 14.3\pm\kern 4.5pt9.2\pm\kern 4.5pt4.2$ | $138.7\pm 15.5\pm\kern 4.5pt7.2\pm\kern 4.5pt3.9$
$\beta_{4}$ | $\kern 4.5pt52.5\pm 10.2\pm 22.4\pm\kern 4.5pt5.9$ | $305.0\pm 10.5\pm 13.5\pm\kern 4.5pt2.2$
Table 31: Systematic uncertainties on the moduli of the K-matrix parameters. The moduli are normalised to that of $\rho(770)$.  Source | $f_{10}$ | $f_{11}$ | $f_{12}$ | $f_{13}$ | $f_{14}$ | $\beta_{0}$ | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$
---|---|---|---|---|---|---|---|---|---|---
PID | $8.1$ | $5.5$ | $22.5$ | $3.2$ | $13.2$ | $2.5$ | $3.6$ | $10.7$ | $8.0$ | $18.0$
Trigger | $0.4$ | $1.1$ | $2.5$ | $1.0$ | $1.2$ | $0.2$ | $0.7$ | $0.6$ | $1.1$ | $1.0$
Reconstruction | $0.1$ | $0.9$ | $0.4$ | $0.5$ | $0.7$ | $0.1$ | $0.4$ | $0.4$ | $0.6$ | $0.5$
Simulation statistic | $0.3$ | $4.6$ | $2.8$ | $2.7$ | $1.6$ | $0.1$ | $1.8$ | $1.0$ | $0.6$ | $0.7$
Background model | $0.6$ | $0.5$ | $6.5$ | $<0.1$ | $0.2$ | $0.1$ | $0.7$ | $1.7$ | $2.4$ | $2.7$
$D^{*}(2010)^{-}$ veto | $4.9$ | $18.9$ | $1.9$ | $6.8$ | $0.6$ | $1.4$ | $4.6$ | $9.2$ | $3.7$ | $13.1$
Total (experiment) | $9.5$ | $20.3$ | $23.8$ | $8.1$ | $13.4$ | $2.9$ | $6.2$ | $14.3$ | $9.2$ | $22.4$
Additional resonances | $3.5$ | $8.0$ | $12.6$ | $3.8$ | $7.7$ | $1.1$ | $4.6$ | $3.0$ | $3.5$ | $5.6$
RBW parameters | $1.2$ | $0.4$ | $2.3$ | $3.3$ | $4.0$ | $0.2$ | $1.1$ | $0.5$ | $2.2$ | $1.9$
$\pi\pi$ res. mass, width | $0.3$ | $0.7$ | $0.1$ | $1.5$ | $1.7$ | $0.2$ | $0.6$ | $0.4$ | $0.7$ | $0.3$
Total (model) | $3.7$ | $8.0$ | $12.8$ | $5.3$ | $8.8$ | $1.1$ | $4.8$ | $3.1$ | $4.2$ | $5.9$
Total (all) | $10.2$ | $21.8$ | $27.0$ | $9.6$ | $16.0$ | $3.1$ | $7.8$ | $14.6$ | $10.1$ | $23.2$
Table 32: Systematic uncertainties on the phases (∘) of the K-matrix parameters. The phase of $\rho(700)$ is set to 0∘ as the reference.  Source | $f_{10}$ | $f_{11}$ | $f_{12}$ | $f_{13}$ | $f_{14}$ | $\beta_{0}$ | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$
---|---|---|---|---|---|---|---|---|---|---
PID | $18.0$ | $35.0$ | $9.6$ | $25.9$ | $5.8$ | $10.4$ | $10.4$ | $1.7$ | $1.4$ | $4.3$
Trigger | $0.5$ | $0.2$ | $0.3$ | $0.2$ | $<0.1$ | $0.5$ | $0.3$ | $0.1$ | $0.5$ | $0.1$
Reconstruction | $0.3$ | $2.4$ | $0.6$ | $0.5$ | $0.6$ | $0.6$ | $0.4$ | $0.1$ | $0.6$ | $0.2$
Simulation statistic | $4.0$ | $1.0$ | $0.2$ | $2.0$ | $7.1$ | $5.2$ | $4.6$ | $0.8$ | $1.9$ | $1.3$
Background model | $3.1$ | $8.2$ | $0.4$ | $3.9$ | $0.4$ | $1.9$ | $2.5$ | $0.1$ | $0.3$ | $0.4$
$D^{*}(2010)^{-}$ veto | $0.3$ | $16.4$ | $5.8$ | $15.2$ | $12.1$ | $6.5$ | $8.5$ | $1.0$ | $6.7$ | $12.7$
Total (experiment) | $18.7$ | $39.6$ | $11.2$ | $30.4$ | $15.2$ | $13.5$ | $14.4$ | $2.1$ | $7.2$ | $13.5$
Additional resonances | $0.4$ | $5.0$ | $4.1$ | $2.1$ | $1.0$ | $3.6$ | $1.9$ | $0.2$ | $3.7$ | $1.9$
RBW parameters | $3.2$ | $25.2$ | $2.6$ | $8.3$ | $0.6$ | $9.2$ | $3.2$ | $1.9$ | $1.0$ | $1.1$
$\pi\pi$ res. mass, width | $0.3$ | $5.0$ | $0.5$ | $1.0$ | $2.6$ | $3.2$ | $0.4$ | $0.2$ | $0.7$ | $0.3$
Total (model) | $3.2$ | $26.2$ | $4.9$ | $8.6$ | $2.9$ | $10.4$ | $3.7$ | $1.9$ | $3.9$ | $2.2$
Total (all) | $19.0$ | $47.5$ | $12.3$ | $31.6$ | $15.5$ | $17.0$ | $14.9$ | $2.9$ | $8.1$ | $13.7$

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LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle53, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, P.
d’Argent11, A. Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M.
Baalouch5, S. Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W.
Baldini16, R.J. Barlow54, C. Barschel38, S. Barsuk7, W. Barter38, V.
Batozskaya28, V. Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F.
Bedeschi23, I. Bediaga1, L.J. Bel41, S. Belogurov31, I. Belyaev31, E. Ben-
Haim8, G. Bencivenni18, S. Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40,
A. Bertolin22, M.-O. Bettler47, M. van Beuzekom41, A. Bien11, S. Bifani45, T.
Bird54, A. Birnkraut9, A. Bizzeti17,i, T. Blake48, F. Blanc39, J. Blouw10, S.
Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38, W. Bonivento15, S. Borghi54,
A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, D.
Brett54, M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46, A.
Bursche40, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M.
Calvo Gomez36,p, P. Campana18, D. Campora Perez38, L. Capriotti54, A.
Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, P. Carniti20,
L. Carson50, K. Carvalho Akiba2,38, R. Casanova Mohr36, G. Casse52, L.
Cassina20,k, L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, G. Cavallero19,
R. Cenci23,t, M. Charles8, Ph. Charpentier38, M. Chefdeville4, S. Chen54,
S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, X. Cid Vidal38, G.
Ciezarek41, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J. Closier38, V.
Coco38, J. Cogan6, E. Cogneras5, V. Cogoni15,e, L. Cojocariu29, G.
Collazuol22, P. Collins38, A. Comerma-Montells11, A. Contu15,38, A. Cook46, M.
Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f, I. Counts56, B.
Couturier38, G.A. Cowan50, D.C. Craik48, A. Crocombe48, M. Cruz Torres60, S.
Cunliffe53, R. Currie53, C. D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y.
David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, C.-T. Dean51, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori38, B. Dey40, A. Di Canto38, F. Di Ruscio24, H.
Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, K. Dreimanis52, G. Dujany54, F. Dupertuis39, P.
Durante6, R. Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S. Easo49,38, U.
Egede53, V. Egorychev31, S. Eidelman34, S. Eisenhardt50, U. Eitschberger9, R.
Ekelhof9, L. Eklund51, I. El Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.M.
Evans47, T. Evans55, A. Falabella14, C. Färber11, C. Farinelli41, N. Farley45,
S. Farry52, R. Fay52, D. Ferguson50, V. Fernandez Albor37, F. Ferrari14, F.
Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M.
Fiorini16,f, M. Firlej27, C. Fitzpatrick39, T. Fiutowski27, P. Fol53, M.
Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D.
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Gascon36, C. Gaspar38, U. Gastaldi16, R. Gauld55, L. Gavardi9, G. Gazzoni5, A.
Geraci21,v, D. Gerick11, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Gianì39, V. Gibson47, L. Giubega29, V.V. Gligorov38,
C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C. Gotti20,k, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
E. Graverini40, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg63, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
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Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E. van
Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54, W.
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Jawahery58, F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B.
Jost38, N. Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,†,
S. Karodia51, M. Kelsey59, I.R. Kenyon45, M. Kenzie38, T. Ketel42, B.
Khanji20,38,k, C. Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O.
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Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk20,k, V. Kudryavtsev34,
K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54,
A. Lai15, D. Lambert50, R.W. Lambert42, G. Lanfranchi18, C. Langenbruch48, B.
Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P.
Lees4, R. Lefèvre5, A. Leflat32, J. Lefrançois7, O. Leroy6, T. Lesiak26, B.
Leverington11, Y. Li7, T. Likhomanenko64, M. Liles52, R. Lindner38, C. Linn38,
F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51, J.H. Lopes2, D.
Lucchesi22,r, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7,
I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, A. Malinin64, G.
Manca15,e, G. Mancinelli6, P. Manning59, A. Mapelli38, J. Maratas5, J.F.
Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,t, R. Märki39, J.
Marks11, G. Martellotti25, M. Martinelli39, D. Martinez Santos42, F. Martinez
Vidal66, D. Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C.
Matteuzzi20, A. Mauri40, B. Maurin39, A. Mazurov45, M. McCann53, J.
McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B. Meadows57, F. Meier9,
M. Meissner11, M. Merk41, D.A. Milanes62, M.-N. Minard4, D.S. Mitzel11, J.
Molina Rodriguez60, S. Monteil5, M. Morandin22, P. Morawski27, A. Mordà6, M.J.
Morello23,t, J. Moron27, A.B. Morris50, R. Mountain59, F. Muheim50, J.
Müller9, K. Müller40, V. Müller9, M. Mussini14, B. Muster39, P. Naik46, T.
Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21, S. Neubert11,
N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, M.
Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, D.P.
O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S. Ogilvy51, O.
Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater67, B. Osorio Rodrigues1, J.M.
Otalora Goicochea2, A. Otto38, P. Owen53, A. Oyanguren66, B.K. Pal59, A.
Palano13,c, F. Palombo21,u, M. Palutan18, J. Panman38, A. Papanestis49, M.
Pappagallo51, L.L. Pappalardo16,f, C. Parkes54, C.J. Parkinson9,45, G.
Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A. Pearce54,49, A.
Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S. Perazzini14,d, P. Perret5,
L. Pescatore45, K. Petridis46, A. Petrolini19,j, M. Petruzzo21, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, I. Polyakov31, E. Polycarpo2,
A. Popov35, D. Popov10, B. Popovici29, C. Potterat2, E. Price46, J.D. Price52,
J. Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig
Navarro39, G. Punzi23,s, W. Qian4, R. Quagliani7,46, B. Rachwal26, J.H.
Rademacker46, B. Rakotomiaramanana39, M. Rama23, M.S. Rangel2, I. Raniuk43, N.
Rauschmayr38, G. Raven42, F. Redi53, S. Reichert54, M.M. Reid48, A.C. dos
Reis1, S. Ricciardi49, S. Richards46, M. Rihl38, K. Rinnert52, V. Rives
Molina36, P. Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S.
Roiser38, V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T.
Ruf38, H. Ruiz36, P. Ruiz Valls66, J.J. Saborido Silva37, N. Sagidova30, P.
Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo66, B.
Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37, E.
Santovetti24,l, A. Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, D.
Savrina31,32, M. Schiller38, H. Schindler38, M. Schlupp9, M. Schmelling10, T.
Schmelzer9, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25,m, A. Semennikov31, I. Sepp53, N.
Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko64,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, I.
Skillicorn51, T. Skwarnicki59, E. Smith55,49, E. Smith53, J. Smith47, M.
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T. Szumlak27, S. T’Jampens4, T. Tekampe9, M. Teklishyn7, G. Tellarini16,f, F.
Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V. Tisserand4, M.
Tobin39, J. Todd57, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, K. Trabelsi39, M.T.
Tran39, M. Tresch40, A. Trisovic38, A. Tsaregorodtsev6, P. Tsopelas41, N.
Tuning41,38, M. Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin65,64, U. Uwer11,
C. Vacca15,e, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P.
Vazquez Regueiro37, C. Vázquez Sierra37, S. Vecchi16, J.J. Velthuis46, M.
Veltri17,h, G. Veneziano39, M. Vesterinen11, B. Viaud7, D. Vieira2, M. Vieites
Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40, D. Volyanskyy10, D. Voong46,
A. Vorobyev30, V. Vorobyev34, C. Voß63, J.A. de Vries41, R. Waldi63, C.
Wallace48, R. Wallace12, J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47,
N.K. Watson45, D. Websdale53, M. Whitehead48, D. Wiedner11, G. Wilkinson55,38,
M. Wilkinson59, M.P. Williams45, M. Williams56, F.F. Wilson49, J. Wimberley58,
J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S.
Wright47, K. Wyllie38, Y. Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan34, O.
Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, L. Zhang3, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
vPolitecnico di Milano, Milano, Italy
†Deceased


# [Re] Badder Seeds: Reproducing the Evaluation of Lexical Methods for Bias
Measurement

Jille van der Togt
University of Amsterdam
Amsterdam, the Netherlands
<EMAIL_ADDRESS>
Lea Tiyavorabun††footnotemark:
University of Amsterdam
Amsterdam, the Netherlands
<EMAIL_ADDRESS>
Matteo Rosati††footnotemark:
University of Amsterdam
Amsterdam, the Netherlands
<EMAIL_ADDRESS>
Giulio Starace††footnotemark:
University of Amsterdam
Amsterdam, the Netherlands
<EMAIL_ADDRESS>Equal ContributionCorresponding Author

##  1 Reproducibility Summary

### Scope of Reproducibility

Combating bias in NLP requires bias measurement. Bias measurement is almost
always achieved by using _lexicons of seed terms_ , i.e. sets of words
specifying stereotypes or dimensions of interest. This reproducibility study
focuses on Antoniak and Mimno [1]’s main claim that the rationale for the
construction of these lexicons needs thorough checking before usage, as the
seeds used for bias measurement can themselves exhibit biases. The study aims
to evaluate the reproducibility of the quantitative and qualitative results
presented in the paper and the conclusions drawn thereof.

### Methodology

We re-implement the entirety of the approaches outlined in the original paper.
We train a skip-gram word2vec model with negative sampling to obtain
embeddings for four corpora. This does not require particular computing
requirements beyond standard consumer personal computers. Additional code
details can be found in our linked repository.

### Results

We reproduce most of the results supporting the original authors’ general
claim: seed sets often suffer from biases that affect their performance as a
baseline for bias metrics. Generally, our results mirror the original paper’s.
They are slightly different on select occasions, but not in ways that
undermine the paper’s general intent to show the fragility of seed sets.

### What was difficult

The significant difficulties encountered were due to a lack of publicly
available code and documentation to clarify missing information in the paper.
For this reason, many algorithms that ultimately turned out to be quite simple
required lengthy clarifications with authors or trial and error. Lastly, the
research was quite data-intensive, which caused some implementations to be
non-trivial to account for memory management.

### What was easy

Once understood, the methods proposed by the authors were relatively easy to
implement. The mathematics involved is quite straightforward. Communication
was also reasonably accessible. The authors’ emails were readily available,
and the responses came quickly and were always helpful.

### Communication with original authors

We maintained a lengthy email correspondence throughout the replication of the
paper with one author, Maria Antoniak. We contacted her to clarify extensive
aspects of the paper’s methodology. Specifically, this concerned summarizing
the data processing approach, explaining missing hyperparameters, and
outlining the aggregation of metrics across different bootstrapped models.
None of the original code was disclosed.

##  2 Introduction

The emergence of bias quantification in Natural Language Processing (NLP)
methods has given rise to two use cases, referred to as _downstream_ and
_upstream_. In the former, bias measurements are used to debias or correct
biases in word representations to avoid encoded biases trickling down when
applying these NLP models [2, 3]. In the latter, bias measurements are used on
models trained on small corpora to quantify the bias present and compare them.
This use case has endowed social scientists with the quantitative foundation
to answer political and social questions about bias across corpora in an
empirical manner [11, 6]. Crucially, most bias quantification methods depend
on lexicons of seed terms that specify the bias dimensions of interest. The
selection of seed terms varies considerably across the literature, and seed
sets themselves may exhibit social and cognitive biases [1]. It is not clear
whether it is possible to re-use seed set across corpora (thereby interfering
with _upstream_ use cases), and elements such as seed term frequency have been
shown to affect bias measurements, and thus _downstream_ uses [4].

We seek to replicate the Antoniak and Mimno [1] paper, hereafter referred to
as ”the original paper/work”. In it, the authors seek to 1) qualitatively
explore seed selection and their sources, 2) demonstrate that features of seed
sets such as pairing order, set similarity, and frequency can cause
instability in bias measurements, and 3) make recommendations for the testing
and justifying of seed sets in future work. We have replicated the experiments
showing the fragility of seed sets, thus verifying the claims of a need for
better justification and analysis of them in future literature. We have also
built a public toolkit to reproduce these measures on arbitrary seed sets and
trained embeddings.

##  3 Scope of reproducibility

This reproducibility study focuses on the authors’ main claim that seed
lexicons need thorough checking before usage to measure bias, as seeds
themselves can be biased and induce instabilities in measurement. The authors
conducted a literature review on prior works to gather many seed sets. They
subsequently evaluated the gathered seed sets with a series of bias
measurement metrics proposed by Bolukbasi et al. [2], Caliskan et al. [3], and
themselves.

Our work consists of two interconnected efforts: code replication, given the
absence of pre-existing code for the original paper, and reproducing the main
results. The latter goal is the main focus of our work and entails reproducing
the outcomes that support the paper’s central claims, which can be summarized
as follows:

  1. 1.

Bias subspaces generated from common bias subspace metrics (e.g., WEAT, PCA)
can help capture the difference represented by the seed set pairs.

  2. 2.

Bias subspaces suffer from instability due to the following factors:

    1. (a)

The ordering and pairing of the seed sets.

    2. (b)

The selection of seeds that are members of the seed sets.

    3. (c)

The degree of semantic similarity between seeds.

  3. 3.

Methods of sourcing seed sets are inconsistent, with disparate strategies
being used across NLP literature.

##  4 Methodology

The code from the original paper was not made publicly available. We,
therefore, re-implemented the entire approach from the description in the
original paper. The following section will summarize the resources and
methodology used to reproduce the original paper accurately.

###  4.1 Code

As mentioned above, the code from the original paper is not publicly
available. We fully re-implement all the code, which can be found on
GitHub111https://github.com/thesofakillers/badder-seeds. We closely follow the
original paper’s methodology to achieve accurate reproduction. The
reproduction is performed step by step, from downloading and preprocessing the
data to training the models and visualizing the results.

###  4.2 Documentation

Unfortunately, there was little to no documentation in the original work
besides the content of the original paper. This occasionally lacked crucial
information to reproduce the results or was vague on implementation details.
In addition to the original paper, Antoniak and Mimno [1] published a Github
repository that contained a JSON with the metadata on seed sets gathered from
prior works222https://github.com/maria-antoniak/bad-seeds.

###  4.3 Model descriptions

We train several bootstrapped skip-gram word2vec models with negative sampling
on unigrams on each dataset. This model attempts to predict whether a
particular word is a valid context (where the context window size is a
hyperparameter) for a given other word using a single fully connected hidden
layer. The first step in training this model is creating a vocabulary of the
entire training dataset. With this vocabulary, each word can be represented as
a one-hot vector. The network output is then a measure of the probability that
the word is a valid context. The trained weights from this hidden layer are
then used to obtain word embedding vectors for each term in the training set
vocabulary.

###  4.4 Datasets

The original paper used four datasets and one pretrained model: New York Times
articles from April 15th-June 30th,
2016333https://www.kaggle.com/nzalake52/new-york-times-articles; high-quality
WikiText articles, using the complete WikiText-103 training set [7]; Goodreads
book reviews for the romance and history and biography genres sampled from the
UCSD book Graph [12, 13]; and the pretrained word2vec GoogleNews
model444https://github.com/mmihaltz/word2vec-GoogleNews-vectors. We use these
same corpora for our research, preprocessing them as closely as possible to
the original paper. This consists of grouping the text into documents,
filtering relevant documents, lowercasing and removing special characters. We
then use spaCy [5] for tokenization and POS-tagging. Because the work is not
concerned with model performance, this study makes no use of train/dev/test
splits. The WikiText-103 dataset, however, is pre-split, so like in the
original work, we work with the training split. Links to all these datasets
can be found in our Github repository.

Preprocessing statistics of our work and the original paper can be found in
Table A.1. We find general agreement in our numbers regarding the total number
of documents per dataset. There are minor discrepancies in the Goodreads
datasets, most likely due to implementation differences. We also count
slightly fewer total words than the original paper in all cases, but the
orders of magnitude generally match. We are, however, unable to reproduce
vocabulary size accurately. We tried many strategies in the replication
process to obtain these numbers, but none were successful. Furthermore,
looking at the official dataset statistics, for example for WikiText [7], it
is clear that our reproduced vocabulary size is a lot closer to the ground
truth than the one by Antoniak and Mimno [1]. Lastly, mean document length
values of each dataset are accurately reproduced, with the WikiText values
suffering the most. The subsections below will discuss each dataset in more
detail.

#### New York Times

This dataset contains 165,900 paragraphs from 8,888 articles from the New York
Times published between April 15th and June 30th 2016. The articles cover a
broad range of sections, including but not limited to movies, sports,
technology, business, books, science, and fashion.

#### WikiText-103

This dataset contains 28,472 manually verified articles from Wikipedia.org.
The entire training dataset is used, in which lists, HTML errors, math, and
code have already been removed. Furthermore, we removed all formulas still
present in the text.

#### Goodreads

The entire Goodreads dataset contains millions of reviews. This study uses
just the Romance and the History/Biography genres. Five hundred book reviews
per book are sampled for each genre while filtering out all books with fewer
than 500 reviews and all reviews containing fewer than 20 characters.

#### GoogleNews

Google’s pretrained word2vec model is trained on ca.100 billion words from the
GoogleNews dataset (4). Our use of this model was limited to replicating the
results outlined below for additional robustness.

#### Seed Set Dataset

Part of the contributions of the original work was creating a catalogue of 178
seed sets gathered from eighteen highly-cited prior works on bias
measurements. We refer to this catalogue as the _gathered seeds_. Each element
of the catalogue comprises a seed set, the category it represents, a
justification, the source categorization, a link, and a unique ID. It is
readily available on the original author’s GitHub. A brief statistical
overview can be found in Fig. A.1. We process the catalogue by lower-casing
the seeds and removing bigrams to use them with our models. We also filter
seed sets containing less than two seeds as we argue that a single seed would
not be sufficient to form a set.

###  4.5 Experimental setup and code

An environment containing all necessary packages is included in the publicly
available repository and can be quickly set up. To mirror the original paper’s
setup, we used the _gensim_ [10] implementation of skip-gram with negative
sampling [8] to train the vector embeddings for all datasets. We used this
library to train our models as that is the framework used by the original
paper and to avoid noise due to different implementations (the investigation
of which would be outside the scope of this paper). Several PyTorch [9]
implementations are also available on GitHub if that is
preferred555https://github.com/theeluwin/pytorch-
sgns,666https://github.com/ddehueck/skip-gram-negative-sampling.

We reproduce the original paper’s results by focusing on two popular seed-
based bias metrics to measure bias in corpus-derived embeddings: WEAT and PCA.
These metrics are used to produce a _bias subspace_ vector given a pair of
seed sets that specifies a bias dimension of interest. The WEAT method,
introduced in Caliskan et al. [3], produces a vector based on the difference
between the mean vectors of the two target sets. The PCA method, described in
Bolukbasi et al. [2], instead requires that each seed term in one of the seed
sets be paired with one seed term from the other seed set. The subspace vector
is then the first principal component resulting from the PCA of a matrix
constructed by, for each pair of seeds, taking the two half vectors from the
pair’s mean to the two pair members and using them as two columns of the
matrix.

We also reproduce the original paper’s coherence metric, which aims to
quantify the robustness of the bias subspace. This metric is calculated as the
absolute value of the difference in mean ranks of the terms in two seed sets
when all the model’s vocabulary is ranked by cosine similarity to the bias
subspace. Another metric used is set similarity, the cosine similarity between
the average vectors of two seed sets.

Finally, when aggregating embeddings of a specific word across bootstrapped
models, we take the average of the embedding vectors in each model that
includes the word. Given a particular pair of seed sets for coherence
aggregation, we only average coherence scores for models containing every seed
term in the two sets to avoid aggregating coherence based on different seed
sets.

###  4.6 Hyperparameters

100-dimensional embeddings were trained for five epochs on all four datasets,
with a five-word negative context sampling rate and a window size of five. We
trained embeddings with a minimum word count of 0, 10, and 100 due to
variation in the original paper. This process was repeated for 20 bootstrapped
samples of each dataset (with the sample size equal to the number of documents
in the dataset), resulting in 20 separate models. The bootstrapping provided
the stochasticity required for robustness. To ensure this reproducibility, we
use a random seed of 42 throughout.

###  4.7 Computational requirements

The execution of the reproduced code does not take excessive computing power.
This study used no GPUs or computing clusters. We ran the experiments on an
Intel I9 9900k and 32GB of 3200MHz RAM running Ubuntu 20.04.3 LTS. Table A.3
shows peak RAM usage and time in seconds to completion for every subprocess of
the replication.

##  5 Results

###  5.1 Quantitative Results

We started by confirming that the bias subspace does capture the difference or
bias that the seed pairs are intended to represent. For this, we reproduced an
experiment by Antoniak and Mimno [1] ranking the cosine similarity between the
first Principal Component (PC) of the bias subspace and all words in the
corpus. The top and bottom ten words for each bias subspace are shown in Fig.
1(a). In the shown words of the gender pair subspace and the shuffled gender
pair subspace gender-related words are found, whereas none are present in the
random pair subspace. However, only the gender pair subspace divides nicely
between male and female terms. We extended this by calculating the cosine
similarity of the top and bottom ten words from the ordered bias subspace for
the shuffled bias subspace. The results in Fig. A.2 show she and his as the
two highest-ranked words, which are not split along the intended bias
subspace.

(a) Compares the top and bottom ten words of each bias subspace ranked by
cosine similarity out of all words in the corpus.

(b) Top & bottom ten words of ordered subspace ranked for the shuffled
subspace.

Figure 1: Replication of Fig. 4 of the original paper. Ranks words from corpus
by cosine similarity against different bias subspaces (first principal
component), with NYT frequency threshold 100.

Fig. 2 shows that the first PC has almost always a very high explained
variance ratio for the bias subspace of ordered pairs, which drops off quickly
for the subsequent PCs. Instead, the explained variance ratio per PC drops
more smoothly for the shuffled pairs. Fig. A.2 shows this behavior by
computing the top and bottom ten words by cosine similarity against the second
PC of the gender subspace. We can observe that the bias subspace of the
ordered pairs does not contain gender words anymore. In contrast, the shuffled
subspace does have gender words such as her, thereby replicating the trend
observed in Fig. 2. It is also important to note that in Fig. 2 there are
exceptional cases where shuffled seed sets produce the first PC with a higher
explained variance than the ordered seed sets. In general, these results
replicate the trends of the original experiments.

Figure 2: Replication of Fig. 3. The first ten principal components of the
bias subspace for different seed pairs on the NYT corpus with a minimum
frequency of 0.

Fig 3 shows that bias measurement is highly inconsistent across seed sets with
the same seed category sourced from different papers. We used the cosine
similarity between female seed sets and the word unpleasantness as a bias
measurement. The cosine similarity varies greatly between seed sets,
replicating the same trends as the original paper.

Figure 3: Reproduction of Fig. 2. Displaying the cosine similarity between the
averaged vector of unpleasantness across all 20 bootstrapped models and
different seeds sets of the category female.

Fig. 4 explores the relationship between set similarity and the robustness of
the bias subspace. The relationship between set similarity and the explained
variance of the PCA-derived bias subspace vector is plotted for each dataset
and frequency thresholds. The original paper shows this relationship only for
the WikiText dataset, and we find a similar negative correlation between set
similarity and explained variance for that dataset.

Table A.2 qualitatively explores this relationship, ranking both gathered and
generated sets by coherence. More semantically dissimilar seed sets score
higher in coherence than more similar sets. In the gathered sets, seed sets
related to names have extremely low coherence due to their semantics being
very similar and the set pairs containing duplicate terms (see "names black"
and "names white"). In the generated sets, we see that very different terms
(such as those relating to careers and those related to lower body
clothing/parts) have high coherence. In contrast, sets such as food terms
score much lower. We observe a similar pattern when using the PCA algorithm as
a basis for coherence. These results show the replicability of the original
paper, as they are almost identical.

Figure 4: Replication of Fig. 5 from the original paper, displaying Explained
Variance Ratio (top) and Coherence (bottom) vs Set Similarity across the four
datasets. We highlight two pairs of gathered seed sets, Black vs White roles
and names. For some corpora, seed terms were not found in the embeddings,
causing the highlighted pair to be missing.

###  5.2 Qualitative Results

The original paper gathered 178 seed sets of eighteen highly-cited prior work
on bias measurement. These seeds are both embedding-based and non-embedding-
based bias detection methods, often overlapping. The seeds are chosen in a
multitude of ways. Only unigram seeds are selected, and words that do not
appear in the training corpus are omitted. We have validated the accuracy of
Table 3 in the original paper by reviewing each of the eighteen papers and
determining which methods the authors used. We briefly summarize them below:

#### Borrowed from social sciences

Select seed sets are borrowed from prior psychology and other social sciences
work.

#### Crowd-Sourced

Crowd-based annotation can create custom seed sets. This method can aid in
gathering contemporary associations and stereotypes. However, controlling
crowd demographics often poses a problem. This can lead to stereotypes being
hard-coded into the seeds.

#### Population-Derived

Seeds can also be derived from government-collected population datasets. These
datasets are usually names and occupations common to specific demographic
groups. A significant problem with this method is that the data tends to be
often US-centric and thus gives a distorted view of the rest of the world.

#### Adapted from Lexical Resources

Researchers can also draw seeds from existing dictionaries, lexicons and other
public resources. The advantage is that these seeds have already undergone a
round of validation.

#### Corpus-Derived

This quantitative method is used to extract seeds terms from a corpus. It has
the advantage of ensuring high-frequency words are selected but suffers from
similar risks as crowd-sourced seeds.

#### Curated

Seed hand-selection by authors often yields high precision seeds but is slow
and relies on unbiased authors.

#### Re-used

The last method relies on prior bias measurement research for seed terms. The
advantage is that the seeds have already been used, but researchers should not
use them without validation.

###  5.3 Results beyond original paper

#### Set Similarity and Bias Subspace in Additional Datasets

We extended the original paper’s set similarity versus bias subspace explained
variance analysis to cover all datasets (beyond WikiText) in Fig. 4. The
negative trend is still present with the NYT corpus, but not in the Goodreads
corpora, where the trend is almost absent or slightly positive. In addition,
the positions of the highlighted seed set pairs are variable across corpora.
We also extended this work to examine the relationship between seed pair
coherence versus set similarity, where the inverse relationship is present in
all datasets. Notice that the requirement that coherence is calculated only
for models that contain all seed terms (as described in Section 4.5) makes
specific pairs of seed sets be ignored, as seen from the lack of the two
highlighted set pairs for select datasets.

#### Testing Minimum Frequency Filter

Due to inconsistencies both in the paper and in communication with the author
in the reported minimum frequency filter for the skip-gram models, we
experimented with minimum frequencies $\mu\in\\{0,10,100\\}$. These enabled us
to see results across the whole vocabulary in the case of $\mu=0$ and reduce
noise from rare words in the case of $\mu=10$. We also used $\mu=100$ to
generate Fig. 1 as the original paper.

#### Seed Toolkit and Pairing Seed Set Data.

Other than extending the experiments of the original paper, we have two
additional contributions. For the sake of reproducibility, we make our code
publicly available and design our repository as an open Python package that
can be used to obtain bias subspace vectors and assess seed set robustness.
This toolkit can help future researchers who aim to evaluate their seeds
carefully. Our second contribution is an augmentation of the seed dataset
provided by Antoniak and Mimno [1]. We provided additional annotations
regarding pairing, i.e. we identify which seeds to pair together along
standard bias dimensions in a queriable .csv format.

##  6 Discussion

Overall, our results replicate the data reported in the original paper. This
replication lends strong support to the general claim of the original paper
that seed sets incorporate strong inductive biases that affect their
performance as grounding for bias metrics and that researchers should be more
cognizant of these limitations.

Instability in bias subspaces can be introduced by selecting seeds in seed
sets, as stated in claim 2b. Our results in Fig. 3 support this as they
reproduce the original work. The same bias measurement varies across seed sets
selected by different authors who assigned it to the same category. In
addition, the dependence of the bias subspace on seed set selection is further
supported by Fig. 4. The two highlighted seed sets (black vs white
roles/names) are generally distinct in position for each corpus, despite
theoretically attempting to define similar bias dimensions.

Another source of instability claimed by Antoniak and Mimno [1] is the
ordering and pairing of seed sets. In Fig. 2 we show that the explained
variance ratio for the ordered bias subspaces can behave very differently from
the shuffled bias subspaces, supporting claim 2a. Our work in Fig. 1(a) also
supports this claim. While the ordered subspace successfully splits the top
words along the intended subspace of male and female, the first PC of the
shuffled bias subspace has words such as mother and boyfriend both ranked on
the same end. This shows that while the subspace still picks up on gender
words, it does not represent the intended subspace. Supporting claim 2a that
bias subspaces can become less meaningful with a shuffled seed pairing. We
could further confirm this behavior by calculating the cosine similarity of
the top words of the ordered subspace for the shuffled subspace in Fig. 1(b).
These results show that she and his are ranked next to each other at the top
and not split along the intended bias subspace. These experiments lend strong
support to claim 2a that the order of seed pairs can substantially influence
the meaningfulness of the bias subspace and, consequently, the bias metrics.

Finally, bias subspaces suffer instability due to semantically overlapping
seeds being less distinguishable in the bias subspace, as stated in claim 2c.
Our results in Table A.2 and Fig. 4 demonstrate that bias subspace vectors are
less robust when the seed sets are semantically similar or overlapping. This
relationship lends strong credence to claim 2c. However, our results did show
that this inverse relationship is not conserved across a minority of corpora
(e.g., the Goodreads datasets) for the explained variance metric. More
broadly, however, this still shows that the reliability of seed selection is
quite variable. While similar seed sets may generate robust bias subspaces for
more semantically equivalent seed pairs for some corpora, that is not
guaranteed. Therefore, while this inverse relationship may be minimized for
specific corpora, extensive corpus-specific seed set investigations are still
required.

#### What was easy.

The original paper clearly described the algorithms used to obtain bias
metrics. Additionally, it carefully cited the papers that first proposed them,
which specified further details. This aided our understanding of the
underlying concepts and accelerated the implementation of the frameworks.
Model training and embedding generation was also facilitated by the pre-
existing _gensim_ framework. This permitted greater focus on reproducing the
details of the experiments than choosing between alternative implementations
of skip-gram word2vec. In addition, responsive authors permitted quick
clarifications through email communication when important details were not
clear.

#### What was difficult.

The original paper did not make code publicly available and largely lacked
documentation. Only the gathered seeds were provided via GitHub (2). This made
it necessary to reproduce all the code from scratch.

In select instances, the paper crucially omitted important information, making
us reliant on communication with the authors. This was most pronounced when
aggregating embeddings or other metrics across the bootstrapped model
sampling, where vocabulary sizes were different. This meant that not all
models had good embeddings for all seed terms. We had to consider several
different approaches before settling on the averaging criteria described in
Section 4.5.

Finally, preprocessing the data was more difficult than initially imagined.
The tokenization pipeline in the original paper was vaguely specified, and
differences in our implementation caused the slight discrepancies in Table
A.3. The POS tagging with spaCy was imperfect, resulting in the incorrect
tagging of several proper nouns as common nouns, making it hard to control for
POS in random seed generation.

#### Communication with original authors.

While the authors did not disclose any code, we maintained a lengthy email
correspondence with them. One author, Maria Antoniak, was contacted to clarify
hyperparameters of the word2vec model, the methodology for generating random
seeds across bootstrapped models, and which bias metrics (PCA or WEAT) were
used for different results. She also described her dataset processing
pipeline, as there were many alternate ways to process the corpora before
training.

##  7 Conclusion

Overall, our results replicate the ones reported in the original paper. This
lends strong support to the general claim of the original paper that seed sets
incorporate significant inductive biases that affect their performance as
grounding for bias metrics and that researchers should be more cognizant of
these limitations. Aside from confirming the danger of blindly using seed
sets, we also provide additional contributions. First of all, all code used to
replicate the original paper is publicly available. This code can obtain bias
subspace vectors and assess seed set robustness. Secondly, we extended the
original paper’s set similarity versus bias subspace explained variance
analysis to cover all datasets. Furthermore, we implement multiple numbers of
minimum frequencies that further enable results across the entire vocabulary.
Lastly, we provide an additional annotation pairing of the original seed
dataset.

We have highlighted a need for carefully justifying the use of particular sets
through empirical means, but a theoretically sound and systematic method for
doing so is still in its infancy. Further work may explore what criteria seed
sets should satisfy to demonstrate robustness. In addition, future researchers
may want to extend this work to bigram seed terms and embeddings to explore
the limitations of more expressive seeds and bias dimensions.

## References

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##  Appendix A Appendix

Figure A.1:  Replication of Fig. 1 from the original paper, illustrating basic
statistics of the gathered seeds. Table A.1: Comparing corpora summary
statistics after preprocessing (original paper statistics obtained from Table
2).

Dataset |  | Total
---
Documents
Total Words | Vocabulary Size |  | Mean Document
---
Length
| original | ours | original | ours | original | ours | original | ours
NYT | 8,888 | 8,888 | 7,244,457 | 7,217,851 | 162,998 | 109,713 | 815 | 812
WikiText | 28,472 | 28,472 | 99,197,146 | 87,077,718 | 546,828 | 228,318 | 3,484 | 3,058
Goodreads (Romance) | 197,000 | 194,500 | 24,856,924 | 24,695,141 | 214,572 | 249,114 | 126 | 127
Goodreads (History/Biog) | 136,000 | 135,000 | 14,324,947 | 14,168,742 | 163,171 | 193,012 | 105 | 105

Table A.2:  Replication of Table 4 from the original paper. Seeds that are
more semantically similar have lower coherence scores. We use the WEAT metric
(the difference between the mean vectors of the seed sets) to generate the
subspace and the NYT dataset embeddings for this data. We average coherence
scores across the $n$ models (out of $20$) that contain the paired seed sets
and round to 3 decimal places. Unfortunately, while we tried to limit
generated sets to only common nouns, proper nouns and, more rarely, verbs
appeared in the sets due to issues with the spaCy POS tagger.

Coherence | Generated Set A | Generated Set B
---|---|---
1.000 | know, believe, think, guess, mean | governor, mayor, legislature, senator, democrat
1.000 | foot-8, foot-7, foot-3, foot-5, to-4 | rousteing, atkins, cornejo, ehrenreich, yorke
0.999 | associate, assistant, economist, engineer, accountant | heels, shoes, pants, legs, fingers
… | … | …
0.062 | hertl, agnieszka, goran, brouwer, koivu | bases, wings, outs, scoreless, rockies
0.059 | molina, glasser, pitney, darren, mackenzie | carver, mina, boyce, curator, deputy
0.053 | lime, juice, lemon, potato, garlic | combo, bodysuit, raisin, koji, mango
Coherence | Gathered Set A | Gathered Set B
0.999 | CAREER: executive, management, professio… | FAMILY: home, parents, children, famil…
0.968 | MALE: brother, father, uncle, grandfat… | FEMALE: sister, mother, aunt, grandmot…
0.942 | TERRORISM: terror, terrorism, violence,… | OCCUPATIONS: banker, carpenter, doctor,…
… | … | …
0.093 | MALE NAMES: john, paul, mike, kevin, … | FEMALE NAMES: amy, joan, lisa, sarah,…
0.053 | NAMES BLACK: harris, robinson, howard, … | NAMES WHITE: harris, nelson, robinson, …
0.026 | NAMES ASIAN: cho, wong, tang, huang, … | NAMES CHINESE: chung, liu, wong, huang…

Table A.3: Computing power needed for each action in the replication process. Action | Time (s) | RAM (MB)
---|---|---
Downloading the data | 293 | 427
Preprocessing the data | 3054 | 19018
Training all models | 7806 | 21054
Table A.1 | 4274 | 661
Fig. 1 | 22 | 4363
Fig. 2 | 19 | 4370
Fig. 3 | 4 | 1510
Fig. 4 | 500 | 1610
Figure A.2: Extension on Fig. 4 from the original paper. Ranks words from the NYT corpus by cosine similarity against different bias subspaces (2nd principal component), with NYT frequency threshold 100.  Table A.4: Seeds used in various Figures Figures |  Seed ID |  Seeds
---|---|---
Fig. 3 |  female-Kozlowski_et_al_2019 |  [“woman", “women", “she", “her", “her", “hers", “girl", “girls", “female", “feminine"]
female_1-Caliskan_et_al_2017 |  [“sister", “female", “woman", “girl", “daughter", “she", “hers", “her"]
definitional_female-Bolukbasi_et_al_2016 |  [“woman", “girl", “she", “mother", “daughter", “gal", “female", “her", “herself", “mary"]
female_singular-Hoyle_et_al_2019 |  [“woman", “girl", “mother", “daughter", “sister", “wife", “aunt", “niece", “empress", “queen", “princess", “duchess", “lady", “dame", “waitress", “actress", “goddess", “policewoman", “postwoman", “heroine", “witch", “stewardess", “she"]
female_definition_words_2-Zhao_et_al_2018 |  [“lady", “saleswoman", “noblewoman", “hostess", “coquette", “nun", “heroine", “actress", “chairwoman", “businesswoman", “spokeswoman", “waitress", “councilwoman", “stateswoman", “policewoman", “countrywomen", “horsewoman", “headmistress", “governess", “widow", “witch", “fiancee"]
female_stereotype_words-Zhao_et_al_2018 |  [“baker", “counselor", “nanny", “librarians", “socialite", “assistant", “tailor", “dancer", “hairdresser", “cashier", “secretary", “clerk", “stenographer", “optometrist", “housekeeper", “bookkeeper", “homemaker", “nurse", “stylist", “receptionist"]
Fig. 2 |  definitional_female-Bolukbasi_et_al_2016 |  [“woman", “girl", “she", “mother", “daughter", “gal", “female", “her", “herself", “mary"]
definitional_male-Bolukbasi_et_al_2016 |  [“man", “boy", “he", “father", “son", “guy", “male", “his", “himself", “john"]
definitional_female-Bolukbasi_et_al_2016 shuffled |  ["herself", "woman", "daughter", "mary", "her", "girl", "mother", "she", "female", "gal"]
definitional_male-Bolukbasi_et_al_2016 shuffled |  [ "man", "his", "he", "son", "guy", "himself", "father", "boy", "male", "john"]
upperclass-Kozlowski_et_al_2019 |  [“rich", “richer", “richest", “affluence", “affluent", “expensive", “luxury", “opulent"]
lowerclass-Kozlowski_et_al_2019 |  [“poor", “poorer", “poorest", “poverty", “impoverished", “inexpensive", “cheap", “needy"]
upperclass-Kozlowski_et_al_2019 shuffled |  ["richer", "opulent", "luxury", "affluent", "rich", "affluence", "richest", "expensive" ]
lowerclass-Kozlowski_et_al_2019 shuffled |  [ "poorer", "impoverished", "poorest", "cheap", "needy", "poverty", "inexpensive", "poor"]
names_chinese-Garg_et_al_2018 |  [“chung", “liu", “wong", “huang", “ng", “hu", “chu", “chen", “lin", “liang", “wang", “wu", “yang", “tang", “chang", “hong", “li"]
names_hispanic-Garg_et_al_2018 |  [“ruiz", “alvarez", “vargas", “castillo", “gomez", “soto", “gonzalez", “sanchez", “rivera", “mendoza", “martinez", “torres", “rodriguez", “perez", “lopez", “medina", “diaz", “garcia", “castro", “cruz"]
names_chinese-Garg_et_al_2018 shuffled |  ["tang", "chang", "chu", "yang", "wu","hong", "huang", "wong", "hu", "liu", "lin", "chen", "liang", "chung", "li", "ng", "wang"]
names_hispanic-Garg_et_al_2018 shuffled |  ["ruiz", "rodriguez", "diaz", "perez", "lopez", "vargas", "alvarez", "garcia","cruz", "torres", "gonzalez", "soto", "martinez", "medina", "rivera", "castillo", "castro", "mendoza", "sanchez", "gomez"]
Fig. 1 |  definitional_female-Bolukbasi_et_al_2016 |  [“woman", “girl", “she", “mother", “daughter", “gal", “female", “her", “herself", “mary"]
definitional_male-Bolukbasi_et_al_2016 |  [“man", “boy", “he", “father", “son", “guy", “male", “his", “himself", “john"]
Fig. A.2 |  definitional_female-Bolukbasi_et_al_2016 shuffled |  ["female", "she", "woman", "gal", "her", "daughter", "girl", "herself", "mother", "mary"]
|  definitional_male-Bolukbasi_et_al_2016 shuffled |  ["john","man", "son","father", "male","himself", "guy","he", "his","boy"]
|  random seeds 1 |  [“essential", “want", “suspension", “talked", “competitive", “information", “hero", “bat", “seconds", “black"]
|  random seeds 2 |  [“derby", “passed", “achieve",, “discussed", “providing", “resulted", “inmates", “wearing", “bid", “rose"]