wit ith the generalized benford s s law
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wit ith the Generalized Benford's 's Law Arnaud Soulet, Arnaud - PowerPoint PPT Presentation

Jun 01, 2023 β€’235 likes β€’530 views

Representativeness of Knowledge Bases wit ith the Generalized Benford's 's Law Arnaud Soulet, Arnaud Giacometti, Batrice Markhoff and Fabian M. Suchanek University of Tours Telecom ParisTech Reliability of f queries on Knowledge Bases

  1. Representativeness of Knowledge Bases wit ith the Generalized Benford's 's Law Arnaud Soulet, Arnaud Giacometti, BΓ©atrice Markhoff and Fabian M. Suchanek University of Tours Telecom ParisTech

  2. Reliability of f queries on Knowledge Bases statistical query [Auer et al., 2007] How many cities are small (<1k inhabitants) in France/Yemen? 2 ISWC 2018 - Monterey, CA

  3. Reliability of f queries on Knowledge Bases statistical query [Auer et al., 2007] Does Yemen really not have any small cities? 3 ISWC 2018 - Monterey, CA

  4. Reliability of f queries on Knowledge Bases statistical crowdsourcing query Voluntary bias [Callahan and Herring, 2011;Wagner et al., 2015] 4 ISWC 2018 - Monterey, CA

  5. Reliability of f queries on Knowledge Bases statistical crowdsourcing query Voluntary bias [Callahan and Herring, 2011;Wagner et al., 2015] We do not know the KB biases, but the statistics can give us a hint? 5 ISWC 2018 - Monterey, CA

  6. Missing facts Yemeni Sanaa Aden Taiz city: [Darari et al., 2016; Population: Galarraga et al., 2017; Lajus and Suchanek, 2018; missing 1,937,451 760,923 615,222 Razniewski et al., 2015; Razniewski et al., 2016] Several methods for estimating the completeness for facts 6 ISWC 2018 - Monterey, CA

  7. Missing facts β‰  Missing entities + + missing facts Yemeni Haid al- Sanaa Aden Taiz missing city: Jazil Population: missing missing 1,937,451 760,923 615,222 few Missing facts due to missing entities are ignored! 7 ISWC 2018 - Monterey, CA

  8. Completeness = #present facts / / (#present facts + + #mis issin ing facts) Assuming that 𝒧 βˆ— is an ideal KB (= correct + complete): 𝒧 βˆ— 𝒧 βˆ— 𝒧 1 𝒧 2 Small cities Small cities Big cities Big cities What is the best KB between 𝓛 𝟐 and 𝓛 πŸ‘ for statistical queries? 8 ISWC 2018 - Monterey, CA

  9. Completeness β‰  Representativeness Assuming that 𝒧 βˆ— is an ideal KB (= correct + complete): 𝒧 βˆ— 𝒧 βˆ— 𝒧 1 𝒧 2 Small cities Small cities Big cities Big cities More complete, less representative Less complete, more representative  οƒΌ Representativeness is more important than completeness for statistics! 9 ISWC 2018 - Monterey, CA

  10. Representativeness of f Knowledge Bases A KB 𝒧 is representative of 𝒧 βˆ— iff the distribution remains the same for all uniform-sampling invariant measures. 𝒧 … <1k inhab. β‰₯1k inhab. ∼ ∼ ∼ ∼ 𝒧 βˆ— … <1k inhab. β‰₯1k inhab . 10 ISWC 2018 - Monterey, CA

  11. Representativeness of f Knowledge Bases A KB 𝒧 is representative of 𝒧 βˆ— iff the distribution remains the same for all uniform-sampling invariant measures. 𝒧 … <1k inhab. β‰₯1k inhab. ∼ ∼ ∼ ∼ The ideal knowledge base 𝒧 βˆ— is unknown! 𝒧 βˆ— … <1k inhab. β‰₯1k inhab . Challenge: How to estimate the representativeness? 11 ISWC 2018 - Monterey, CA

  12. Example: population of f capitals Abidjan Bangkok Conakry Kingston Mogadishu Santiago Abuja Beijing Dakar Kinshasa Montevideo Seoul Accra Belgrade Damascus Kuala Lumpur Nairobi Sofia Addis Ababa Berlin Dhaka Lilongwe Niamey Taipei Algiers Bogota Doha Lima Ouagadougou Tashkent Amman Brasilia Erbil London Paris Tbilisi Ankara Brazzaville Freetown Luanda Phnom Penh Tegucigalpa Antananarivo Bucharest Havana Lusaka Prague Tokyo Ashgabat Budapest Islamabad Madrid Pyongyang Tripoli Bahawalpur Buenos Aires Jakarta Managua Quito Tunis Baku Cairo Kabul Maputo Riyadh Ulaanbaatar Bamako Caracas Khartoum Mexico City Sana'a Vienna 12 ISWC 2018 - Monterey, CA

  13. Example: population of f capitals 4 707 404 8 280 925 1 660 973 1 041 084 1 750 000 6 158 080 1 235 880 21 700 000 1 146 053 10 125 000 1 305 082 9 971 111 2 291 352 1 166 763 1 711 000 1 768 000 3 138 369 1 260 120 3 384 569 3 610 156 6 970 105 1 077 116 1 302 910 2 704 974 3 415 811 7 878 783 1 351 000 8 852 000 1 626 950 2 309 600 4 007 526 2 556 149 1 025 000 8 673 713 2 229 621 1 118 035 4 587 558 1 827 000 1 050 301 2 825 311 1 501 725 1 157 509 1 613 375 1 883 425 2 106 146 1 742 979 1 267 449 13 617 445 1 031 992 1 759 407 1 900 000 3 141 991 2 581 076 1 126 000 1 052 000 2 890 151 9 607 787 2 205 676 2 671 191 1 056 247 2 122 300 10 230 350 3 678 034 1 766 184 7 125 180 1 372 000 1 809 106 3 273 863 5 185 000 8 918 653 1 937 451 1 852 997 13 ISWC 2018 - Monterey, CA

  14. Example: population of f capitals 4 707 404 8 280 925 1 660 973 1 041 084 1 750 000 6 158 080 1 235 880 2 1 700 000 1 146 053 1 0 125 000 1 305 082 9 971 111 2 291 352 1 166 763 1 711 000 1 768 000 3 138 369 1 260 120 3 384 569 3 610 156 6 970 105 1 077 116 1 302 910 2 704 974 3 415 811 7 878 783 1 351 000 8 852 000 1 626 950 2 309 600 4 007 526 2 556 149 1 025 000 8 673 713 2 229 621 1 118 035 4 587 558 1 827 000 1 050 301 2 825 311 1 501 725 1 157 509 1 613 375 1 883 425 2 106 146 1 742 979 1 267 449 1 3 617 445 1 031 992 1 759 407 1 900 000 3 141 991 2 581 076 1 126 000 1 052 000 2 890 151 9 607 787 2 205 676 2 671 191 1 056 247 2 122 300 1 0 230 350 3 678 034 1 766 184 7 125 180 1 372 000 1 809 106 3 273 863 5 185 000 8 918 653 1 937 451 1 852 997 What is the distribution of the first significant digit of capital inhabitants? 14 ISWC 2018 - Monterey, CA

  15. Benford’s law Population of cities 0.30 0.20 Benford’s law 0.10 0.00 1 2 3 4 5 6 7 8 9 15 ISWC 2018 - Monterey, CA

  16. Benford’s law Population of cities Discharge of rivers Length of rivers 0.30 0.30 0.30 0.20 0.20 0.20 0.10 0.10 0.10 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 16 ISWC 2018 - Monterey, CA

  17. Benford’s law Population of cities Discharge of rivers Length of rivers 0.30 0.30 0.30 0.20 0.20 0.20 0.10 0.10 0.10 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 𝑄 𝑔𝑗𝑠𝑑𝑒 𝑒𝑗𝑕𝑗𝑒 π‘Œ = 𝑒 = log 1 + 1 𝑒 [Newcomb, 1881;Benford, 1938] 17 ISWC 2018 - Monterey, CA

  18. The Generalized Benford’s Law Population of cities Discharge of rivers Length of rivers 0.30 0.30 0.30 Ξ± β†’ 0 Ξ± β†’ 0 Ξ± β†’ 0 0.20 0.20 0.20 0.10 0.10 0.10 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 𝑄 𝑔𝑗𝑠𝑑𝑒 𝑒𝑗𝑕𝑗𝑒 π‘Œ = 𝑒 = 1 + 𝑒 𝛽 βˆ’ 𝑒 𝛽 10 𝛽 βˆ’ 1 [HΓΌrlimann, 2014] 18 ISWC 2018 - Monterey, CA

  19. The Generalized Benford’s Law Population of cities Discharge of rivers Length of rivers 0.30 0.30 0.30 Ξ± β†’ 0 Ξ± β†’ 0 Ξ± β†’ 0 0.20 0.20 0.20 0.10 0.10 0.10 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0.75 0.75 0.75 Actors per movie Persons per birth place Out-degree of wikipedia pages Ξ± =-0.155 Ξ± =-0.149 0.50 0.50 0.50 Ξ± =-0.486 0.25 0.25 0.25 0.00 0.00 0.00 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 19 ISWC 2018 - Monterey, CA

  20. Key idea of f our method representativeness = compliance with the Generalized Benford’s Law #𝒒𝒔𝒇𝒕𝒇𝒐𝒖_π’ˆπ’ƒπ’…π’–π’• = #𝒒𝒔𝒇𝒕𝒇𝒐𝒖_π’ˆπ’ƒπ’…π’–π’•+#𝒏𝒋𝒕𝒕𝒋𝒐𝒉_π’ˆπ’ƒπ’…π’–π’•_π’ˆπ’‘π’”_π’…π’‘π’π’’π’Žπ’‹π’ƒπ’π’…π’‡ Population in France Population in Yemen DBpedia 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Representativeness = 97% Representativeness = 79% 20 ISWC 2018 - Monterey, CA

  21. Our method in superv rvised context xt facts of 𝑠 on 𝒧 distribution of the fsd facts of 𝑠 on 𝒧 βˆ— Benford’s law Using the known distribution of the first significant digit 21 ISWC 2018 - Monterey, CA

  22. Our method in superv rvised context xt facts of 𝑠 on 𝒧 distribution of the fsd 200 Population in Yemen 150 100 50 0 1 2 3 4 5 6 7 8 9 facts of 𝑠 on 𝒧 βˆ— 378 present facts 101 missing facts Benford’s law Representativeness: 101 πŸ’πŸ–πŸ— = πŸ’πŸ–πŸ— + 𝟐𝟏𝟐 = 79% 378 Computing the minimum number of facts for retrieving Benford’s law 22 ISWC 2018 - Monterey, CA

  23. Our method in unsuperv rvised context xt facts of 𝑠 on 𝒧 distribution of the fsd facts of 𝑠 on 𝒧 βˆ— GBL with Ξ± =0.12 ideal distribution is unknown! Learning the parameter Ξ± of the Generalized Benford’s Law 23 ISWC 2018 - Monterey, CA

  24. Our method in unsuperv rvised context xt facts of 𝑠 on 𝒧 distribution of the fsd 200 Population in Yemen 150 100 50 0 1 2 3 4 5 6 7 8 9 facts of 𝑠 on 𝒧 βˆ— 378 present facts 78 missing facts GBL with Ξ± =0.12 Representativeness: ideal distribution is 78 unknown! πŸ’πŸ–πŸ— = πŸ’πŸ–πŸ— + πŸ–πŸ— = 82% 378 Computing the minimum number of facts for retrieving Benford’s law 24 ISWC 2018 - Monterey, CA

  25. Experimental study  Evaluation protocol 1. Take a correct and complete relation as gold standard 2. Degrade the completeness by discarding facts 3. Approximate the representativeness  Gold standard: population in French cities according to govt statistics  Degradation:  Most-populated: remove the least populated cities  Least-populated: remove the most populated cities  Random: remove cities randomly 25 ISWC 2018 - Monterey, CA

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