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Gravitational waves data analysis - Walter Del Pozzo

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Waves on the lake: the astrophysics behind gravitational waves
May 28 – June 1, 2018

Published in: Science
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Gravitational waves data analysis - Walter Del Pozzo

  1. 1. Walter Del Pozzo Waves on the Lake, Como, 2018 Gravitational waves data analysis Walter Del Pozzo 1
  2. 2. Walter Del Pozzo Waves on the Lake, Como, 2018 Outline • Part 1: fundaments of probability theory • Part 2: stochastic processes • Part 3: (modelled) detection of gravitational wave signals • Part 4: inference of gravitational wave physics • Part 5: overview of (selected) results 2
  3. 3. Walter Del Pozzo Waves on the Lake, Como, 2018 What is “data analysis”? 3
  4. 4. Walter Del Pozzo Waves on the Lake, Como, 2018 What is “data analysis”? 4 p(H|DI) = p(H|I) p(D|HI) p(D|I)
  5. 5. Walter Del Pozzo Waves on the Lake, Como, 2018 Why Bayes’ theorem? • Gravitational wave events are rare • Noise dominated detectors • Need to know what we are looking for VERY well to detect it/measure its properties • Matched filtering 5
  6. 6. Walter Del Pozzo Waves on the Lake, Como, 20186 Why Bayes’ theorem? • Gravitational wave events are rare • Noise dominated detectors • Need to know what we are looking for VERY well to detect it/measure its properties • Matched filtering
  7. 7. Walter Del Pozzo Waves on the Lake, Como, 2018 Why Bayes’ theorem? • Need for accurate predictions to dig into the noise floor (see Damour & Buonanno lectures) 7
  8. 8. Walter Del Pozzo Waves on the Lake, Como, 2018 Part 1 Fundaments of probability theory 8 “Probability is a property of the observer”
  9. 9. Walter Del Pozzo Waves on the Lake, Como, 2018 Fundaments • Logical propositions: • Statements to which one can assign a True (1) or False (0) value • A = “The sun is a star” • B = “The total mass of GW150914 was 70 Msun” 9
  10. 10. Walter Del Pozzo Waves on the Lake, Como, 2018 Logical operations • Arbitrarily complex sentences constructed via basic operations • Negation (NOT): • Conjunction (AND): • Disjunction (OR): 10 ¯A A · B ⌘ AB A + B
  11. 11. Walter Del Pozzo Waves on the Lake, Como, 2018 Inference • Deductive inference • Strong syllogism: • major premise: if A is true then B is true • minor premise: A is true • conclusion: B is true 11 • Inductive inference • Weak syllogism: • major premise: if A is true then B is true • minor premise: B is true • conclusion: A is more plausible (A = AB) ⌘ (A =) B)
  12. 12. Walter Del Pozzo Waves on the Lake, Como, 2018 Deductive inference • Major premise (assumption): • Minor premise (observation): • particle x is a neutron • Conclusion (inference): • Deductive inference is based on the certainty that a cause implies an effect and that one observes the cause 12 Cause =) E↵ect n0 ! p+ + e + ¯⌫e x ! p+ + e + ¯⌫e x ⌘ n0 observation
  13. 13. Walter Del Pozzo Waves on the Lake, Como, 2018 Inductive inference • Major premise (assumption): • Minor premise (observation): • Conclusion (inference): • x is plausibly a neutron • Inductive inference is based on observation of an effect and tries to reconstruct the cause 13 Cause =) E↵ect n0 ! p+ + e + ¯⌫e observation x ! p+ + e + ¯⌫e
  14. 14. Walter Del Pozzo Waves on the Lake, Como, 2018 Bayesian theory: desiderata • Bayesian theory is based on a set of “desiderata”: • Degrees of plausibility are represented by real numbers • Plausibility must be in qualitative agreement with rationality and common sense • Consistency: • if a conclusion can be reached in more than one way, every possible way must lead to the same result • the theory must account for all relevant information available • equivalent states of knowledge must lead to the same degree of plausibility assignment • Probability is a map of degree of plausibility to [0,1] 14
  15. 15. Walter Del Pozzo Waves on the Lake, Como, 2018 Information based inference • Probabilities are always conditional • Probability assignments depend on the “background information” 15 I: The die is unbiased p(6|I) = 1 6<latexit sha1_base64="799YthDpgeRAW5vCILHRq+UXuQg=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyURqW6EohvdVbAPaEuZTCft0MkkzEyEGoO/4saFIm79D3f+jdM2C209cOFwzr3ce48Xcaa043xbC4tLyyurubX8+sbm1ra9s1tXYSwJrZGQh7LpYUU5E7Smmea0GUmKA4/Thje8GvuNeyoVC8WdHkW0E+C+YD4jWBupa+9HxfLjzTG6QG1fYpK4aVJOu3bBKTkToHniZqQAGapd+6vdC0kcUKEJx0q1XCfSnQRLzQinab4dKxphMsR92jJU4ICqTjK5PkVHRukhP5SmhEYT9fdEggOlRoFnOgOsB2rWG4v/ea1Y++edhIko1lSQ6SI/5kiHaBwF6jFJieYjQzCRzNyKyACbFLQJLG9CcGdfnif1k5LrlNzb00LlMosjBwdwCEVw4QwqcA1VqAGBB3iGV3iznqwX6936mLYuWNnMHvyB9fkDhiqT+w==</latexit><latexit sha1_base64="799YthDpgeRAW5vCILHRq+UXuQg=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyURqW6EohvdVbAPaEuZTCft0MkkzEyEGoO/4saFIm79D3f+jdM2C209cOFwzr3ce48Xcaa043xbC4tLyyurubX8+sbm1ra9s1tXYSwJrZGQh7LpYUU5E7Smmea0GUmKA4/Thje8GvuNeyoVC8WdHkW0E+C+YD4jWBupa+9HxfLjzTG6QG1fYpK4aVJOu3bBKTkToHniZqQAGapd+6vdC0kcUKEJx0q1XCfSnQRLzQinab4dKxphMsR92jJU4ICqTjK5PkVHRukhP5SmhEYT9fdEggOlRoFnOgOsB2rWG4v/ea1Y++edhIko1lSQ6SI/5kiHaBwF6jFJieYjQzCRzNyKyACbFLQJLG9CcGdfnif1k5LrlNzb00LlMosjBwdwCEVw4QwqcA1VqAGBB3iGV3iznqwX6936mLYuWNnMHvyB9fkDhiqT+w==</latexit><latexit sha1_base64="799YthDpgeRAW5vCILHRq+UXuQg=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyURqW6EohvdVbAPaEuZTCft0MkkzEyEGoO/4saFIm79D3f+jdM2C209cOFwzr3ce48Xcaa043xbC4tLyyurubX8+sbm1ra9s1tXYSwJrZGQh7LpYUU5E7Smmea0GUmKA4/Thje8GvuNeyoVC8WdHkW0E+C+YD4jWBupa+9HxfLjzTG6QG1fYpK4aVJOu3bBKTkToHniZqQAGapd+6vdC0kcUKEJx0q1XCfSnQRLzQinab4dKxphMsR92jJU4ICqTjK5PkVHRukhP5SmhEYT9fdEggOlRoFnOgOsB2rWG4v/ea1Y++edhIko1lSQ6SI/5kiHaBwF6jFJieYjQzCRzNyKyACbFLQJLG9CcGdfnif1k5LrlNzb00LlMosjBwdwCEVw4QwqcA1VqAGBB3iGV3iznqwX6936mLYuWNnMHvyB9fkDhiqT+w==</latexit><latexit sha1_base64="799YthDpgeRAW5vCILHRq+UXuQg=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyURqW6EohvdVbAPaEuZTCft0MkkzEyEGoO/4saFIm79D3f+jdM2C209cOFwzr3ce48Xcaa043xbC4tLyyurubX8+sbm1ra9s1tXYSwJrZGQh7LpYUU5E7Smmea0GUmKA4/Thje8GvuNeyoVC8WdHkW0E+C+YD4jWBupa+9HxfLjzTG6QG1fYpK4aVJOu3bBKTkToHniZqQAGapd+6vdC0kcUKEJx0q1XCfSnQRLzQinab4dKxphMsR92jJU4ICqTjK5PkVHRukhP5SmhEYT9fdEggOlRoFnOgOsB2rWG4v/ea1Y++edhIko1lSQ6SI/5kiHaBwF6jFJieYjQzCRzNyKyACbFLQJLG9CcGdfnif1k5LrlNzb00LlMosjBwdwCEVw4QwqcA1VqAGBB3iGV3iznqwX6936mLYuWNnMHvyB9fkDhiqT+w==</latexit> I’: The die is biased p(6|I0 ) > 1 6<latexit sha1_base64="RJN52dHNZBTgecsE3XbHfPN+Ic8=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoMQm7ArEq0kaKNdBPOA7BpmJ7PJkNnZYWZWCOsWNv6KjYUitn6EnX/j5FFo4oELh3Pu5d57AsGo0o7zbeWWlldW1/LrhY3Nre0de3evqeJEYtLAMYtlO0CKMMpJQ1PNSFtIgqKAkVYwvBz7rXsiFY35rR4J4keoz2lIMdJG6tpFUa4+XN95QtKIHMFz6IUS4dTN0mrWtUtOxZkALhJ3RkpghnrX/vJ6MU4iwjVmSKmO6wjtp0hqihnJCl6iiEB4iPqkYyhHEVF+Onkig4dG6cEwlqa4hhP190SKIqVGUWA6I6QHat4bi/95nUSHZ35KuUg04Xi6KEwY1DEcJwJ7VBKs2cgQhCU1t0I8QCYFbXIrmBDc+ZcXSfO44joV9+akVLuYxZEHRXAAysAFp6AGrkAdNAAGj+AZvII368l6sd6tj2lrzprN7IM/sD5/ABgslxk=</latexit><latexit sha1_base64="RJN52dHNZBTgecsE3XbHfPN+Ic8=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoMQm7ArEq0kaKNdBPOA7BpmJ7PJkNnZYWZWCOsWNv6KjYUitn6EnX/j5FFo4oELh3Pu5d57AsGo0o7zbeWWlldW1/LrhY3Nre0de3evqeJEYtLAMYtlO0CKMMpJQ1PNSFtIgqKAkVYwvBz7rXsiFY35rR4J4keoz2lIMdJG6tpFUa4+XN95QtKIHMFz6IUS4dTN0mrWtUtOxZkALhJ3RkpghnrX/vJ6MU4iwjVmSKmO6wjtp0hqihnJCl6iiEB4iPqkYyhHEVF+Onkig4dG6cEwlqa4hhP190SKIqVGUWA6I6QHat4bi/95nUSHZ35KuUg04Xi6KEwY1DEcJwJ7VBKs2cgQhCU1t0I8QCYFbXIrmBDc+ZcXSfO44joV9+akVLuYxZEHRXAAysAFp6AGrkAdNAAGj+AZvII368l6sd6tj2lrzprN7IM/sD5/ABgslxk=</latexit><latexit sha1_base64="RJN52dHNZBTgecsE3XbHfPN+Ic8=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoMQm7ArEq0kaKNdBPOA7BpmJ7PJkNnZYWZWCOsWNv6KjYUitn6EnX/j5FFo4oELh3Pu5d57AsGo0o7zbeWWlldW1/LrhY3Nre0de3evqeJEYtLAMYtlO0CKMMpJQ1PNSFtIgqKAkVYwvBz7rXsiFY35rR4J4keoz2lIMdJG6tpFUa4+XN95QtKIHMFz6IUS4dTN0mrWtUtOxZkALhJ3RkpghnrX/vJ6MU4iwjVmSKmO6wjtp0hqihnJCl6iiEB4iPqkYyhHEVF+Onkig4dG6cEwlqa4hhP190SKIqVGUWA6I6QHat4bi/95nUSHZ35KuUg04Xi6KEwY1DEcJwJ7VBKs2cgQhCU1t0I8QCYFbXIrmBDc+ZcXSfO44joV9+akVLuYxZEHRXAAysAFp6AGrkAdNAAGj+AZvII368l6sd6tj2lrzprN7IM/sD5/ABgslxk=</latexit><latexit sha1_base64="RJN52dHNZBTgecsE3XbHfPN+Ic8=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoMQm7ArEq0kaKNdBPOA7BpmJ7PJkNnZYWZWCOsWNv6KjYUitn6EnX/j5FFo4oELh3Pu5d57AsGo0o7zbeWWlldW1/LrhY3Nre0de3evqeJEYtLAMYtlO0CKMMpJQ1PNSFtIgqKAkVYwvBz7rXsiFY35rR4J4keoz2lIMdJG6tpFUa4+XN95QtKIHMFz6IUS4dTN0mrWtUtOxZkALhJ3RkpghnrX/vJ6MU4iwjVmSKmO6wjtp0hqihnJCl6iiEB4iPqkYyhHEVF+Onkig4dG6cEwlqa4hhP190SKIqVGUWA6I6QHat4bi/95nUSHZ35KuUg04Xi6KEwY1DEcJwJ7VBKs2cgQhCU1t0I8QCYFbXIrmBDc+ZcXSfO44joV9+akVLuYxZEHRXAAysAFp6AGrkAdNAAGj+AZvII368l6sd6tj2lrzprN7IM/sD5/ABgslxk=</latexit>
  16. 16. Walter Del Pozzo Waves on the Lake, Como, 2018 The quantitative rules • Basic rules: • Product rule • Sum rule 16 p(AB|C) = p(A|BC)p(B|C) p(A + B|C) = p(A|C) + p(B|C) p(AB|C) = p(A|C) + p(B|C) () p(AB|C) = 0 A and B are mutually exclusive (on the information C)
  17. 17. Walter Del Pozzo Waves on the Lake, Como, 2018 Bayes theorem • From the product rule • Bayes theorem: 17 p(AB|C) = p(A|BC)p(B|C) = p(B|AC)p(A|C) p(A|BC) = p(B|AC)p(A|C) p(B|C)
  18. 18. Walter Del Pozzo Waves on the Lake, Como, 2018 Bayes theorem • Define the following statements: • A = H: hypothesis • B = D: observation • C = I: whatever relevant information 18 p(H|DI) = p(H|I) p(D|HI) p(D|I) posterior prior likelihood evidence p(D|I) = X i p(Hi|I)p(D|HiI) marginalisation
  19. 19. Walter Del Pozzo Waves on the Lake, Como, 2018 Example • To the (bat) board! 19
  20. 20. Walter Del Pozzo Waves on the Lake, Como, 2018 Model selection • Consider 2 competing hypotheses (models) • Given an observation , which model is to be preferred? 20 H1, H2<latexit sha1_base64="W5c69Jujx1u+4d23LRpRjHEBC5U=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBQym7RdBj0UuPFeyHtMuSTbNtaJJdkqxQlv4KLx4U8erP8ea/MW33oK0PBh7vzTAzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6eg4VYS2Scxj1QuxppxJ2jbMcNpLFMUi5LQbTu7mfveJKs1i+WCmCfUFHkkWMYKNlR6bgVcdVJtBPShX3Jq7AFonXk4qkKMVlL8Gw5ikgkpDONa677mJ8TOsDCOczkqDVNMEkwke0b6lEguq/Wxx8AxdWGWIoljZkgYt1N8TGRZaT0VoOwU2Y73qzcX/vH5qohs/YzJJDZVkuShKOTIxmn+PhkxRYvjUEkwUs7ciMsYKE2MzKtkQvNWX10mnXvPcmnd/VWnc5nEU4QzO4RI8uIYGNKEFbSAg4Ble4c1Rzovz7nwsWwtOPnMKf+B8/gACzo85</latexit><latexit 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sha1_base64="NsLos3laW3WlkybC0/eUHSYSJS0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNRDx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS465frrhVdw6ySrycVCBHvV/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ66LquVWvcVmp3eRxFOEETuEcPLiCGtxDHZrAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHl2mMyA==</latexit> p(H1|DI) = p(H1|I) p(D|H1I) p(D|I)<latexit 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sha1_base64="MQhyoXzHLLF/5epKjoTLsmhboLs=">AAACFnicbZDLSgMxFIYzXmu9jbp0EyxCu7DMiKAboWgX7a6CvUA7DJk004ZmZkKSEcq0T+HGV3HjQhG34s63MZ3OQlsPhHz5zzmcnN/jjEplWd/Gyura+sZmbiu/vbO7t28eHLZkFAtMmjhikeh4SBJGQ9JUVDHS4YKgwGOk7Y1uZ/n2AxGSRuG9GnPiBGgQUp9ipLTkmme8WHPtSbVegtdwzvVSzxcIJ7xYneg3rJemKevbNQtW2UoDLoOdQQFk0XDNr14/wnFAQoUZkrJrW1w5CRKKYkam+V4sCUd4hAakqzFEAZFOkq41hada6UM/EvqECqbq744EBVKOA09XBkgN5WJuJv6X68bKv3ISGvJYkRDPB/kxgyqCM49gnwqCFRtrQFhQ/VeIh0h7orSTeW2CvbjyMrTOy7ZVtu8uCpWbzI4cOAYnoAhscAkqoAYaoAkweATP4BW8GU/Gi/FufMxLV4ys5wj8CePzB1TJnF4=</latexit><latexit sha1_base64="MQhyoXzHLLF/5epKjoTLsmhboLs=">AAACFnicbZDLSgMxFIYzXmu9jbp0EyxCu7DMiKAboWgX7a6CvUA7DJk004ZmZkKSEcq0T+HGV3HjQhG34s63MZ3OQlsPhHz5zzmcnN/jjEplWd/Gyura+sZmbiu/vbO7t28eHLZkFAtMmjhikeh4SBJGQ9JUVDHS4YKgwGOk7Y1uZ/n2AxGSRuG9GnPiBGgQUp9ipLTkmme8WHPtSbVegtdwzvVSzxcIJ7xYneg3rJemKevbNQtW2UoDLoOdQQFk0XDNr14/wnFAQoUZkrJrW1w5CRKKYkam+V4sCUd4hAakqzFEAZFOkq41hada6UM/EvqECqbq744EBVKOA09XBkgN5WJuJv6X68bKv3ISGvJYkRDPB/kxgyqCM49gnwqCFRtrQFhQ/VeIh0h7orSTeW2CvbjyMrTOy7ZVtu8uCpWbzI4cOAYnoAhscAkqoAYaoAkweATP4BW8GU/Gi/FufMxLV4ys5wj8CePzB1TJnF4=</latexit><latexit sha1_base64="MQhyoXzHLLF/5epKjoTLsmhboLs=">AAACFnicbZDLSgMxFIYzXmu9jbp0EyxCu7DMiKAboWgX7a6CvUA7DJk004ZmZkKSEcq0T+HGV3HjQhG34s63MZ3OQlsPhHz5zzmcnN/jjEplWd/Gyura+sZmbiu/vbO7t28eHLZkFAtMmjhikeh4SBJGQ9JUVDHS4YKgwGOk7Y1uZ/n2AxGSRuG9GnPiBGgQUp9ipLTkmme8WHPtSbVegtdwzvVSzxcIJ7xYneg3rJemKevbNQtW2UoDLoOdQQFk0XDNr14/wnFAQoUZkrJrW1w5CRKKYkam+V4sCUd4hAakqzFEAZFOkq41hada6UM/EvqECqbq744EBVKOA09XBkgN5WJuJv6X68bKv3ISGvJYkRDPB/kxgyqCM49gnwqCFRtrQFhQ/VeIh0h7orSTeW2CvbjyMrTOy7ZVtu8uCpWbzI4cOAYnoAhscAkqoAYaoAkweATP4BW8GU/Gi/FufMxLV4ys5wj8CePzB1TJnF4=</latexit> p(H2|DI) = p(H2|I) p(D|H2I) p(D|I)<latexit 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  21. 21. Walter Del Pozzo Waves on the Lake, Como, 2018 Model selection • The troublesome term • Simplifies taking the ratio 21 p(D|I) = X i p(Hi|I)p(D|HiI) <latexit sha1_base64="yi9bql7SYzVbfFhNKvTYG5LajtU=">AAACC3icbZDLSgMxFIYzXmu9jbp0E1qEdlNmRNCNUNRFu6tgL9AOQybNtKFJZkgyQhm7d+OruHGhiFtfwJ1vY9rOQlt/CPx85xxOzh/EjCrtON/Wyura+sZmbiu/vbO7t28fHLZUlEhMmjhikewESBFGBWlqqhnpxJIgHjDSDkbX03r7nkhFI3GnxzHxOBoIGlKMtEG+XYhLNw/1MryEPZVwn8K4VPOpIVNuHKyXfbvoVJyZ4LJxM1MEmRq+/dXrRzjhRGjMkFJd14m1lyKpKWZkku8lisQIj9CAdI0ViBPlpbNbJvDEkD4MI2me0HBGf0+kiCs15oHp5EgP1WJtCv+rdRMdXngpFXGiicDzRWHCoI7gNBjYp5JgzcbGICyp+SvEQyQR1ia+vAnBXTx52bROK65TcW/PitWrLI4cOAYFUAIuOAdVUAMN0AQYPIJn8ArerCfrxXq3PuatK1Y2cwT+yPr8Ae/EmG8=</latexit><latexit sha1_base64="yi9bql7SYzVbfFhNKvTYG5LajtU=">AAACC3icbZDLSgMxFIYzXmu9jbp0E1qEdlNmRNCNUNRFu6tgL9AOQybNtKFJZkgyQhm7d+OruHGhiFtfwJ1vY9rOQlt/CPx85xxOzh/EjCrtON/Wyura+sZmbiu/vbO7t28fHLZUlEhMmjhikewESBFGBWlqqhnpxJIgHjDSDkbX03r7nkhFI3GnxzHxOBoIGlKMtEG+XYhLNw/1MryEPZVwn8K4VPOpIVNuHKyXfbvoVJyZ4LJxM1MEmRq+/dXrRzjhRGjMkFJd14m1lyKpKWZkku8lisQIj9CAdI0ViBPlpbNbJvDEkD4MI2me0HBGf0+kiCs15oHp5EgP1WJtCv+rdRMdXngpFXGiicDzRWHCoI7gNBjYp5JgzcbGICyp+SvEQyQR1ia+vAnBXTx52bROK65TcW/PitWrLI4cOAYFUAIuOAdVUAMN0AQYPIJn8ArerCfrxXq3PuatK1Y2cwT+yPr8Ae/EmG8=</latexit><latexit sha1_base64="yi9bql7SYzVbfFhNKvTYG5LajtU=">AAACC3icbZDLSgMxFIYzXmu9jbp0E1qEdlNmRNCNUNRFu6tgL9AOQybNtKFJZkgyQhm7d+OruHGhiFtfwJ1vY9rOQlt/CPx85xxOzh/EjCrtON/Wyura+sZmbiu/vbO7t28fHLZUlEhMmjhikewESBFGBWlqqhnpxJIgHjDSDkbX03r7nkhFI3GnxzHxOBoIGlKMtEG+XYhLNw/1MryEPZVwn8K4VPOpIVNuHKyXfbvoVJyZ4LJxM1MEmRq+/dXrRzjhRGjMkFJd14m1lyKpKWZkku8lisQIj9CAdI0ViBPlpbNbJvDEkD4MI2me0HBGf0+kiCs15oHp5EgP1WJtCv+rdRMdXngpFXGiicDzRWHCoI7gNBjYp5JgzcbGICyp+SvEQyQR1ia+vAnBXTx52bROK65TcW/PitWrLI4cOAYFUAIuOAdVUAMN0AQYPIJn8ArerCfrxXq3PuatK1Y2cwT+yPr8Ae/EmG8=</latexit><latexit sha1_base64="yi9bql7SYzVbfFhNKvTYG5LajtU=">AAACC3icbZDLSgMxFIYzXmu9jbp0E1qEdlNmRNCNUNRFu6tgL9AOQybNtKFJZkgyQhm7d+OruHGhiFtfwJ1vY9rOQlt/CPx85xxOzh/EjCrtON/Wyura+sZmbiu/vbO7t28fHLZUlEhMmjhikewESBFGBWlqqhnpxJIgHjDSDkbX03r7nkhFI3GnxzHxOBoIGlKMtEG+XYhLNw/1MryEPZVwn8K4VPOpIVNuHKyXfbvoVJyZ4LJxM1MEmRq+/dXrRzjhRGjMkFJd14m1lyKpKWZkku8lisQIj9CAdI0ViBPlpbNbJvDEkD4MI2me0HBGf0+kiCs15oHp5EgP1WJtCv+rdRMdXngpFXGiicDzRWHCoI7gNBjYp5JgzcbGICyp+SvEQyQR1ia+vAnBXTx52bROK65TcW/PitWrLI4cOAYFUAIuOAdVUAMN0AQYPIJn8ArerCfrxXq3PuatK1Y2cwT+yPr8Ae/EmG8=</latexit> O12 = p(H1|I)p(D|H1I) p(H2|I)p(D|H2I) = p(H1|I) p(H2|I) B12 <latexit sha1_base64="bngykZNNDe8Ch0TiY0WkUMkp7CI=">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</latexit><latexit sha1_base64="bngykZNNDe8Ch0TiY0WkUMkp7CI=">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</latexit><latexit sha1_base64="bngykZNNDe8Ch0TiY0WkUMkp7CI=">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</latexit><latexit sha1_base64="bngykZNNDe8Ch0TiY0WkUMkp7CI=">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</latexit> Bayes’ factor prior odds
  22. 22. Walter Del Pozzo Waves on the Lake, Como, 2018 Exhaustivity • Note, the models might be exhaustive • But in general they are not • Notable exception, “null test hypothesis” 22 p(H1|I) + p(H2|I) = 1 =) H1 = H2<latexit sha1_base64="XDfKmQQEfVE2W99qo1OthGwT8MA=">AAACHHicbZBNS8MwGMfT+TbnW9Wjl+AQJsJop6AXYehl3ia4F1hLSbN0C0ubkqTCqPsgXvwqXjwo4sWD4Lcx3XrQzQcCP/7/J3ny/P2YUaks69soLC2vrK4V10sbm1vbO+buXlvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rO/M49EZLy6E6NY+KGaBDRgGKktOSZp3Gl4dkPN8cnGdQ0wEtoQ4eGejqRUJtacLh+IxuRNiZezTPLVtWaFlwEO4cyyKvpmZ9On+MkJJHCDEnZs61YuSkSimJGJiUnkSRGeIQGpKcxQiGRbjpdbgKPtNKHARf6RApO1d83UhRKOQ593RkiNZTzXib+5/USFVy4KY3iRJEIzwYFCYOKwywp2KeCYMXGGhAWVP8V4iESCCudZ0mHYM+vvAjtWtW2qvbtWbl+lcdRBAfgEFSADc5BHTRAE7QABo/gGbyCN+PJeDHejY9Za8HI7+yDP2V8/QC0wJ60</latexit><latexit sha1_base64="XDfKmQQEfVE2W99qo1OthGwT8MA=">AAACHHicbZBNS8MwGMfT+TbnW9Wjl+AQJsJop6AXYehl3ia4F1hLSbN0C0ubkqTCqPsgXvwqXjwo4sWD4Lcx3XrQzQcCP/7/J3ny/P2YUaks69soLC2vrK4V10sbm1vbO+buXlvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rO/M49EZLy6E6NY+KGaBDRgGKktOSZp3Gl4dkPN8cnGdQ0wEtoQ4eGejqRUJtacLh+IxuRNiZezTPLVtWaFlwEO4cyyKvpmZ9On+MkJJHCDEnZs61YuSkSimJGJiUnkSRGeIQGpKcxQiGRbjpdbgKPtNKHARf6RApO1d83UhRKOQ593RkiNZTzXib+5/USFVy4KY3iRJEIzwYFCYOKwywp2KeCYMXGGhAWVP8V4iESCCudZ0mHYM+vvAjtWtW2qvbtWbl+lcdRBAfgEFSADc5BHTRAE7QABo/gGbyCN+PJeDHejY9Za8HI7+yDP2V8/QC0wJ60</latexit><latexit sha1_base64="XDfKmQQEfVE2W99qo1OthGwT8MA=">AAACHHicbZBNS8MwGMfT+TbnW9Wjl+AQJsJop6AXYehl3ia4F1hLSbN0C0ubkqTCqPsgXvwqXjwo4sWD4Lcx3XrQzQcCP/7/J3ny/P2YUaks69soLC2vrK4V10sbm1vbO+buXlvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rO/M49EZLy6E6NY+KGaBDRgGKktOSZp3Gl4dkPN8cnGdQ0wEtoQ4eGejqRUJtacLh+IxuRNiZezTPLVtWaFlwEO4cyyKvpmZ9On+MkJJHCDEnZs61YuSkSimJGJiUnkSRGeIQGpKcxQiGRbjpdbgKPtNKHARf6RApO1d83UhRKOQ593RkiNZTzXib+5/USFVy4KY3iRJEIzwYFCYOKwywp2KeCYMXGGhAWVP8V4iESCCudZ0mHYM+vvAjtWtW2qvbtWbl+lcdRBAfgEFSADc5BHTRAE7QABo/gGbyCN+PJeDHejY9Za8HI7+yDP2V8/QC0wJ60</latexit><latexit sha1_base64="XDfKmQQEfVE2W99qo1OthGwT8MA=">AAACHHicbZBNS8MwGMfT+TbnW9Wjl+AQJsJop6AXYehl3ia4F1hLSbN0C0ubkqTCqPsgXvwqXjwo4sWD4Lcx3XrQzQcCP/7/J3ny/P2YUaks69soLC2vrK4V10sbm1vbO+buXlvyRGDSwpxx0fWRJIxGpKWoYqQbC4JCn5GOP7rO/M49EZLy6E6NY+KGaBDRgGKktOSZp3Gl4dkPN8cnGdQ0wEtoQ4eGejqRUJtacLh+IxuRNiZezTPLVtWaFlwEO4cyyKvpmZ9On+MkJJHCDEnZs61YuSkSimJGJiUnkSRGeIQGpKcxQiGRbjpdbgKPtNKHARf6RApO1d83UhRKOQ593RkiNZTzXib+5/USFVy4KY3iRJEIzwYFCYOKwywp2KeCYMXGGhAWVP8V4iESCCudZ0mHYM+vvAjtWtW2qvbtWbl+lcdRBAfgEFSADc5BHTRAE7QABo/gGbyCN+PJeDHejY9Za8HI7+yDP2V8/QC0wJ60</latexit> X j p(Hj|I) = 1 <latexit sha1_base64="/wzzppSREfZAgm2HRRxCnALB7k4=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBHqpiQi6EYouqm7CvYBbQiT6aSddmYSZiZCiPVX3LhQxK0f4s6/cdpmoa0HLhzOuZd77wliRpV2nG9rZXVtfWOzsFXc3tnd27cPDlsqSiQmTRyxSHYCpAijgjQ11Yx0YkkQDxhpB+Obqd9+IFLRSNzrNCYeRwNBQ4qRNpJvl3oq4f4IxpW6P3q8PYVX0PXtslN1ZoDLxM1JGeRo+PZXrx/hhBOhMUNKdV0n1l6GpKaYkUmxlygSIzxGA9I1VCBOlJfNjp/AE6P0YRhJU0LDmfp7IkNcqZQHppMjPVSL3lT8z+smOrz0MiriRBOB54vChEEdwWkSsE8lwZqlhiAsqbkV4iGSCGuTV9GE4C6+vExaZ1XXqbp35+XadR5HARyBY1ABLrgANVAHDdAEGKTgGbyCN+vJerHerY9564qVz5TAH1ifP2UPk0g=</latexit><latexit sha1_base64="/wzzppSREfZAgm2HRRxCnALB7k4=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBHqpiQi6EYouqm7CvYBbQiT6aSddmYSZiZCiPVX3LhQxK0f4s6/cdpmoa0HLhzOuZd77wliRpV2nG9rZXVtfWOzsFXc3tnd27cPDlsqSiQmTRyxSHYCpAijgjQ11Yx0YkkQDxhpB+Obqd9+IFLRSNzrNCYeRwNBQ4qRNpJvl3oq4f4IxpW6P3q8PYVX0PXtslN1ZoDLxM1JGeRo+PZXrx/hhBOhMUNKdV0n1l6GpKaYkUmxlygSIzxGA9I1VCBOlJfNjp/AE6P0YRhJU0LDmfp7IkNcqZQHppMjPVSL3lT8z+smOrz0MiriRBOB54vChEEdwWkSsE8lwZqlhiAsqbkV4iGSCGuTV9GE4C6+vExaZ1XXqbp35+XadR5HARyBY1ABLrgANVAHDdAEGKTgGbyCN+vJerHerY9564qVz5TAH1ifP2UPk0g=</latexit><latexit sha1_base64="/wzzppSREfZAgm2HRRxCnALB7k4=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBHqpiQi6EYouqm7CvYBbQiT6aSddmYSZiZCiPVX3LhQxK0f4s6/cdpmoa0HLhzOuZd77wliRpV2nG9rZXVtfWOzsFXc3tnd27cPDlsqSiQmTRyxSHYCpAijgjQ11Yx0YkkQDxhpB+Obqd9+IFLRSNzrNCYeRwNBQ4qRNpJvl3oq4f4IxpW6P3q8PYVX0PXtslN1ZoDLxM1JGeRo+PZXrx/hhBOhMUNKdV0n1l6GpKaYkUmxlygSIzxGA9I1VCBOlJfNjp/AE6P0YRhJU0LDmfp7IkNcqZQHppMjPVSL3lT8z+smOrz0MiriRBOB54vChEEdwWkSsE8lwZqlhiAsqbkV4iGSCGuTV9GE4C6+vExaZ1XXqbp35+XadR5HARyBY1ABLrgANVAHDdAEGKTgGbyCN+vJerHerY9564qVz5TAH1ifP2UPk0g=</latexit><latexit sha1_base64="/wzzppSREfZAgm2HRRxCnALB7k4=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBHqpiQi6EYouqm7CvYBbQiT6aSddmYSZiZCiPVX3LhQxK0f4s6/cdpmoa0HLhzOuZd77wliRpV2nG9rZXVtfWOzsFXc3tnd27cPDlsqSiQmTRyxSHYCpAijgjQ11Yx0YkkQDxhpB+Obqd9+IFLRSNzrNCYeRwNBQ4qRNpJvl3oq4f4IxpW6P3q8PYVX0PXtslN1ZoDLxM1JGeRo+PZXrx/hhBOhMUNKdV0n1l6GpKaYkUmxlygSIzxGA9I1VCBOlJfNjp/AE6P0YRhJU0LDmfp7IkNcqZQHppMjPVSL3lT8z+smOrz0MiriRBOB54vChEEdwWkSsE8lwZqlhiAsqbkV4iGSCGuTV9GE4C6+vExaZ1XXqbp35+XadR5HARyBY1ABLrgANVAHDdAEGKTgGbyCN+vJerHerY9564qVz5TAH1ifP2UPk0g=</latexit>
  23. 23. Walter Del Pozzo Waves on the Lake, Como, 2018 Model selection • If the model depends on a set of parameters need to marginalise over them • Odds 23 Hi<latexit sha1_base64="RK6tMOZxx6ELHaUJ//auteCmvvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTHgg3LFrboLkHXi5aQCOZqD8ld/GLM0QmmYoFr3PDcxfkaV4UzgrNRPNSaUTegIe5ZKGqH2s8WpM3JhlSEJY2VLGrJQf09kNNJ6GgW2M6JmrFe9ufif10tNeONnXCapQcmWi8JUEBOT+d9kyBUyI6aWUKa4vZWwMVWUGZtOyYbgrb68TtpXVc+tevfXlfptHkcRzuAcLsGDGtShAU1oAYMRPMMrvDnCeXHenY9la8HJZ07hD5zPHxdwjag=</latexit><latexit 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sha1_base64="CyRu65jRuhI7PpAgSv8Jv2PVWuc=">AAACN3icbVDLSgMxFM34rPVVdekmWIQKUmZE0I0g6qJupEJrhU4ZMpnUBjOZIbkjlGn/yo2/4U43LhRx6x+Yaafg60LIybnnJLnHjwXXYNtP1tT0zOzcfGGhuLi0vLJaWlu/0lGiKGvSSETq2ieaCS5ZEzgIdh0rRkJfsJZ/e5r1W3dMaR7JBvRj1gnJjeRdTgkYyitdxJWzAa55HJ/v4CPscgme2+gxIDhwIds97u5iHFcmp0EuzowTbnIB9kplu2qPCv8FTg7KKK+6V3p0g4gmIZNABdG67dgxdFKigFPBhkU30Swm9JbcsLaBkoRMd9LR3EO8bZgAdyNllgQ8Yr87UhJq3Q99owwJ9PTvXkb+12sn0D3spFzGCTBJxw91E4EhwlmIOOCKURB9AwhV3PwV0x5RhIKJumhCcH6P/Bdc7VUdu+pc7pePT/I4CmgTbaEKctABOkY1VEdNRNE9ekav6M16sF6sd+tjLJ2ycs8G+lHW5xe7A6jQ</latexit><latexit sha1_base64="CyRu65jRuhI7PpAgSv8Jv2PVWuc=">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</latexit> Oij = p(Hi|I) p(Hj|I) R ⇥i d✓i p(✓i|HiI)p(D|✓iHiI) R ⇥j d✓j p(✓j|HjI)p(D|✓jHjI) <latexit 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sha1_base64="IxsmlT/ZLy79KqURxkCtYIMVx1U=">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</latexit>
  24. 24. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning probabilities • Indifference principle: “if among the possible outcomes, there is no reason to prefer any of them over any other, then all outcomes should be equally probable.” 24
  25. 25. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning probabilities • Indifference principle: “if among the possible outcomes, there is no reason to prefer any of them over any other, then all outcomes should be equally probable.” • Invariance arguments: 25 p( |I)d = p( 0 |I)d 0 0 = k () p( |I) / 1
  26. 26. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning probabilities • Indifference principle: “if among the possible outcomes, there is no reason to prefer any of them over any other, then all outcomes should be equally probable.” • Invariance arguments: • Maximum entropy: 26 p( |I)d = p( 0 |I)d 0 0 = k () p( |I) / 1 H(p) = X j pj log ✓ pj mj ◆ <latexit sha1_base64="ubDlyYEuSaCPnAAx82UATQhIwF8=">AAACHHicbVDLSsNAFJ3UV62vqks3g0VoF5ZEBd0IRTddVrAPaEqYTCft2JkkzNwIJeRD3Pgrblwo4saF4N84fSy0euDC4Zx7ufcePxZcg21/Wbml5ZXVtfx6YWNza3unuLvX0lGiKGvSSESq4xPNBA9ZEzgI1okVI9IXrO2Prid++54pzaPwFsYx60kyCHnAKQEjecXTejmu4Et87OpEenc4NuWKaOAKFkDZDRShqdGy1JiZq/hgCBWvWLKr9hT4L3HmpITmaHjFD7cf0USyEKggWncdO4ZeShRwKlhWcBPNYkJHZMC6hoZEMt1Lp89l+MgofRxEylQIeKr+nEiJ1HosfdMpCQz1ojcR//O6CQQXvZSHcQIspLNFQSIwRHiSFO5zxSiIsSGEKm5uxXRITCBg8iyYEJzFl/+S1knVsavOzVmpdjWPI48O0CEqIwedoxqqowZqIooe0BN6Qa/Wo/VsvVnvs9acNZ/ZR79gfX4Dg1ehFg==</latexit><latexit sha1_base64="ubDlyYEuSaCPnAAx82UATQhIwF8=">AAACHHicbVDLSsNAFJ3UV62vqks3g0VoF5ZEBd0IRTddVrAPaEqYTCft2JkkzNwIJeRD3Pgrblwo4saF4N84fSy0euDC4Zx7ufcePxZcg21/Wbml5ZXVtfx6YWNza3unuLvX0lGiKGvSSESq4xPNBA9ZEzgI1okVI9IXrO2Prid++54pzaPwFsYx60kyCHnAKQEjecXTejmu4Et87OpEenc4NuWKaOAKFkDZDRShqdGy1JiZq/hgCBWvWLKr9hT4L3HmpITmaHjFD7cf0USyEKggWncdO4ZeShRwKlhWcBPNYkJHZMC6hoZEMt1Lp89l+MgofRxEylQIeKr+nEiJ1HosfdMpCQz1ojcR//O6CQQXvZSHcQIspLNFQSIwRHiSFO5zxSiIsSGEKm5uxXRITCBg8iyYEJzFl/+S1knVsavOzVmpdjWPI48O0CEqIwedoxqqowZqIooe0BN6Qa/Wo/VsvVnvs9acNZ/ZR79gfX4Dg1ehFg==</latexit><latexit sha1_base64="ubDlyYEuSaCPnAAx82UATQhIwF8=">AAACHHicbVDLSsNAFJ3UV62vqks3g0VoF5ZEBd0IRTddVrAPaEqYTCft2JkkzNwIJeRD3Pgrblwo4saF4N84fSy0euDC4Zx7ufcePxZcg21/Wbml5ZXVtfx6YWNza3unuLvX0lGiKGvSSESq4xPNBA9ZEzgI1okVI9IXrO2Prid++54pzaPwFsYx60kyCHnAKQEjecXTejmu4Et87OpEenc4NuWKaOAKFkDZDRShqdGy1JiZq/hgCBWvWLKr9hT4L3HmpITmaHjFD7cf0USyEKggWncdO4ZeShRwKlhWcBPNYkJHZMC6hoZEMt1Lp89l+MgofRxEylQIeKr+nEiJ1HosfdMpCQz1ojcR//O6CQQXvZSHcQIspLNFQSIwRHiSFO5zxSiIsSGEKm5uxXRITCBg8iyYEJzFl/+S1knVsavOzVmpdjWPI48O0CEqIwedoxqqowZqIooe0BN6Qa/Wo/VsvVnvs9acNZ/ZR79gfX4Dg1ehFg==</latexit><latexit sha1_base64="ubDlyYEuSaCPnAAx82UATQhIwF8=">AAACHHicbVDLSsNAFJ3UV62vqks3g0VoF5ZEBd0IRTddVrAPaEqYTCft2JkkzNwIJeRD3Pgrblwo4saF4N84fSy0euDC4Zx7ufcePxZcg21/Wbml5ZXVtfx6YWNza3unuLvX0lGiKGvSSESq4xPNBA9ZEzgI1okVI9IXrO2Prid++54pzaPwFsYx60kyCHnAKQEjecXTejmu4Et87OpEenc4NuWKaOAKFkDZDRShqdGy1JiZq/hgCBWvWLKr9hT4L3HmpITmaHjFD7cf0USyEKggWncdO4ZeShRwKlhWcBPNYkJHZMC6hoZEMt1Lp89l+MgofRxEylQIeKr+nEiJ1HosfdMpCQz1ojcR//O6CQQXvZSHcQIspLNFQSIwRHiSFO5zxSiIsSGEKm5uxXRITCBg8iyYEJzFl/+S1knVsavOzVmpdjWPI48O0CEqIwedoxqqowZqIooe0BN6Qa/Wo/VsvVnvs9acNZ/ZR79gfX4Dg1ehFg==</latexit>
  27. 27. Walter Del Pozzo Waves on the Lake, Como, 2018 Indifference principle • Take a 6-face die, if you have no reason to suspect the die is biased • In general, for N equally probable cases 27 6X i=1 pi = 1 <latexit sha1_base64="ZKgUoPu4u8+vQH11U1C4EaFzkew=">AAAB/XicbVDLSgMxFM3UV62v8bFzEyyCqzIRUTeFohuXFewD2nHIpJk2NMkMSUaoQ/FX3LhQxK3/4c6/MW1noa0HLhzOuZd77wkTzrTxvG+nsLS8srpWXC9tbG5t77i7e00dp4rQBol5rNoh1pQzSRuGGU7biaJYhJy2wuH1xG89UKVZLO/MKKG+wH3JIkawsVLgHnR1KoKMVdH4/hwmAYNViAK37FW8KeAiQTkpgxz1wP3q9mKSCioN4VjrDvIS42dYGUY4HZe6qaYJJkPcpx1LJRZU+9n0+jE8tkoPRrGyJQ2cqr8nMiy0HonQdgpsBnrem4j/eZ3URJd+xmSSGirJbFGUcmhiOIkC9piixPCRJZgoZm+FZIAVJsYGVrIhoPmXF0nztIK8Cro9K9eu8jiK4BAcgROAwAWogRtQBw1AwCN4Bq/gzXlyXpx352PWWnDymX3wB87nD30Bk+w=</latexit><latexit sha1_base64="ZKgUoPu4u8+vQH11U1C4EaFzkew=">AAAB/XicbVDLSgMxFM3UV62v8bFzEyyCqzIRUTeFohuXFewD2nHIpJk2NMkMSUaoQ/FX3LhQxK3/4c6/MW1noa0HLhzOuZd77wkTzrTxvG+nsLS8srpWXC9tbG5t77i7e00dp4rQBol5rNoh1pQzSRuGGU7biaJYhJy2wuH1xG89UKVZLO/MKKG+wH3JIkawsVLgHnR1KoKMVdH4/hwmAYNViAK37FW8KeAiQTkpgxz1wP3q9mKSCioN4VjrDvIS42dYGUY4HZe6qaYJJkPcpx1LJRZU+9n0+jE8tkoPRrGyJQ2cqr8nMiy0HonQdgpsBnrem4j/eZ3URJd+xmSSGirJbFGUcmhiOIkC9piixPCRJZgoZm+FZIAVJsYGVrIhoPmXF0nztIK8Cro9K9eu8jiK4BAcgROAwAWogRtQBw1AwCN4Bq/gzXlyXpx352PWWnDymX3wB87nD30Bk+w=</latexit><latexit sha1_base64="ZKgUoPu4u8+vQH11U1C4EaFzkew=">AAAB/XicbVDLSgMxFM3UV62v8bFzEyyCqzIRUTeFohuXFewD2nHIpJk2NMkMSUaoQ/FX3LhQxK3/4c6/MW1noa0HLhzOuZd77wkTzrTxvG+nsLS8srpWXC9tbG5t77i7e00dp4rQBol5rNoh1pQzSRuGGU7biaJYhJy2wuH1xG89UKVZLO/MKKG+wH3JIkawsVLgHnR1KoKMVdH4/hwmAYNViAK37FW8KeAiQTkpgxz1wP3q9mKSCioN4VjrDvIS42dYGUY4HZe6qaYJJkPcpx1LJRZU+9n0+jE8tkoPRrGyJQ2cqr8nMiy0HonQdgpsBnrem4j/eZ3URJd+xmSSGirJbFGUcmhiOIkC9piixPCRJZgoZm+FZIAVJsYGVrIhoPmXF0nztIK8Cro9K9eu8jiK4BAcgROAwAWogRtQBw1AwCN4Bq/gzXlyXpx352PWWnDymX3wB87nD30Bk+w=</latexit><latexit sha1_base64="ZKgUoPu4u8+vQH11U1C4EaFzkew=">AAAB/XicbVDLSgMxFM3UV62v8bFzEyyCqzIRUTeFohuXFewD2nHIpJk2NMkMSUaoQ/FX3LhQxK3/4c6/MW1noa0HLhzOuZd77wkTzrTxvG+nsLS8srpWXC9tbG5t77i7e00dp4rQBol5rNoh1pQzSRuGGU7biaJYhJy2wuH1xG89UKVZLO/MKKG+wH3JIkawsVLgHnR1KoKMVdH4/hwmAYNViAK37FW8KeAiQTkpgxz1wP3q9mKSCioN4VjrDvIS42dYGUY4HZe6qaYJJkPcpx1LJRZU+9n0+jE8tkoPRrGyJQ2cqr8nMiy0HonQdgpsBnrem4j/eZ3URJd+xmSSGirJbFGUcmhiOIkC9piixPCRJZgoZm+FZIAVJsYGVrIhoPmXF0nztIK8Cro9K9eu8jiK4BAcgROAwAWogRtQBw1AwCN4Bq/gzXlyXpx352PWWnDymX3wB87nD30Bk+w=</latexit> pi = pj 8i, j<latexit sha1_base64="b3w9f+SGXnhaPtwHFgkM8N1obkQ=">AAACAnicbVBNS8NAEJ34WetX1JN4WSyCh1ISEfQiFL14rGA/oClhs920226SZXcjlFC8+Fe8eFDEq7/Cm//GbZuDtj4YeLw3w8y8QHCmtON8W0vLK6tr64WN4ubW9s6uvbffUEkqCa2ThCeyFWBFOYtpXTPNaUtIiqOA02YwvJn4zQcqFUviez0StBPhXsxCRrA2km8fCp+hKyT8gVf2ysgLE4k5R6w88O2SU3GmQIvEzUkJctR8+8vrJiSNaKwJx0q1XUfoToalZoTTcdFLFRWYDHGPtg2NcURVJ5u+MEYnRukis91UrNFU/T2R4UipURSYzgjrvpr3JuJ/XjvV4WUnY7FINY3JbFGYcqQTNMkDdZmkRPORIZhIZm5FpI8lJtqkVjQhuPMvL5LGWcV1Ku7deal6ncdRgCM4hlNw4QKqcAs1qAOBR3iGV3iznqwX6936mLUuWfnMAfyB9fkDx16Vvg==</latexit><latexit sha1_base64="b3w9f+SGXnhaPtwHFgkM8N1obkQ=">AAACAnicbVBNS8NAEJ34WetX1JN4WSyCh1ISEfQiFL14rGA/oClhs920226SZXcjlFC8+Fe8eFDEq7/Cm//GbZuDtj4YeLw3w8y8QHCmtON8W0vLK6tr64WN4ubW9s6uvbffUEkqCa2ThCeyFWBFOYtpXTPNaUtIiqOA02YwvJn4zQcqFUviez0StBPhXsxCRrA2km8fCp+hKyT8gVf2ysgLE4k5R6w88O2SU3GmQIvEzUkJctR8+8vrJiSNaKwJx0q1XUfoToalZoTTcdFLFRWYDHGPtg2NcURVJ5u+MEYnRukis91UrNFU/T2R4UipURSYzgjrvpr3JuJ/XjvV4WUnY7FINY3JbFGYcqQTNMkDdZmkRPORIZhIZm5FpI8lJtqkVjQhuPMvL5LGWcV1Ku7deal6ncdRgCM4hlNw4QKqcAs1qAOBR3iGV3iznqwX6936mLUuWfnMAfyB9fkDx16Vvg==</latexit><latexit sha1_base64="b3w9f+SGXnhaPtwHFgkM8N1obkQ=">AAACAnicbVBNS8NAEJ34WetX1JN4WSyCh1ISEfQiFL14rGA/oClhs920226SZXcjlFC8+Fe8eFDEq7/Cm//GbZuDtj4YeLw3w8y8QHCmtON8W0vLK6tr64WN4ubW9s6uvbffUEkqCa2ThCeyFWBFOYtpXTPNaUtIiqOA02YwvJn4zQcqFUviez0StBPhXsxCRrA2km8fCp+hKyT8gVf2ysgLE4k5R6w88O2SU3GmQIvEzUkJctR8+8vrJiSNaKwJx0q1XUfoToalZoTTcdFLFRWYDHGPtg2NcURVJ5u+MEYnRukis91UrNFU/T2R4UipURSYzgjrvpr3JuJ/XjvV4WUnY7FINY3JbFGYcqQTNMkDdZmkRPORIZhIZm5FpI8lJtqkVjQhuPMvL5LGWcV1Ku7deal6ncdRgCM4hlNw4QKqcAs1qAOBR3iGV3iznqwX6936mLUuWfnMAfyB9fkDx16Vvg==</latexit><latexit sha1_base64="b3w9f+SGXnhaPtwHFgkM8N1obkQ=">AAACAnicbVBNS8NAEJ34WetX1JN4WSyCh1ISEfQiFL14rGA/oClhs920226SZXcjlFC8+Fe8eFDEq7/Cm//GbZuDtj4YeLw3w8y8QHCmtON8W0vLK6tr64WN4ubW9s6uvbffUEkqCa2ThCeyFWBFOYtpXTPNaUtIiqOA02YwvJn4zQcqFUviez0StBPhXsxCRrA2km8fCp+hKyT8gVf2ysgLE4k5R6w88O2SU3GmQIvEzUkJctR8+8vrJiSNaKwJx0q1XUfoToalZoTTcdFLFRWYDHGPtg2NcURVJ5u+MEYnRukis91UrNFU/T2R4UipURSYzgjrvpr3JuJ/XjvV4WUnY7FINY3JbFGYcqQTNMkDdZmkRPORIZhIZm5FpI8lJtqkVjQhuPMvL5LGWcV1Ku7deal6ncdRgCM4hlNw4QKqcAs1qAOBR3iGV3iznqwX6936mLUuWfnMAfyB9fkDx16Vvg==</latexit> pi = 1 6<latexit sha1_base64="690e9kxg+RvpTk/0gH3/LUKdoj0=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIqBeh6MVjBfsBbQib7aZdutmE3Y1SYn6KFw+KePWXePPfuG1z0NYHA4/3ZpiZFyScKe0431ZpZXVtfaO8Wdna3tnds6v7bRWnktAWiXksuwFWlDNBW5ppTruJpDgKOO0E45up33mgUrFY3OtJQr0IDwULGcHaSL5dTXyGrlA/lJhkbp6d575dc+rODGiZuAWpQYGmb3/1BzFJIyo04Vipnusk2suw1Ixwmlf6qaIJJmM8pD1DBY6o8rLZ6Tk6NsoAhbE0JTSaqb8nMhwpNYkC0xlhPVKL3lT8z+ulOrz0MiaSVFNB5ovClCMdo2kOaMAkJZpPDMFEMnMrIiNsUtAmrYoJwV18eZm0T+uuU3fvzmqN6yKOMhzCEZyACxfQgFtoQgsIPMIzvMKb9WS9WO/Wx7y1ZBUzB/AH1ucPQ/CTWQ==</latexit><latexit sha1_base64="690e9kxg+RvpTk/0gH3/LUKdoj0=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIqBeh6MVjBfsBbQib7aZdutmE3Y1SYn6KFw+KePWXePPfuG1z0NYHA4/3ZpiZFyScKe0431ZpZXVtfaO8Wdna3tnds6v7bRWnktAWiXksuwFWlDNBW5ppTruJpDgKOO0E45up33mgUrFY3OtJQr0IDwULGcHaSL5dTXyGrlA/lJhkbp6d575dc+rODGiZuAWpQYGmb3/1BzFJIyo04Vipnusk2suw1Ixwmlf6qaIJJmM8pD1DBY6o8rLZ6Tk6NsoAhbE0JTSaqb8nMhwpNYkC0xlhPVKL3lT8z+ulOrz0MiaSVFNB5ovClCMdo2kOaMAkJZpPDMFEMnMrIiNsUtAmrYoJwV18eZm0T+uuU3fvzmqN6yKOMhzCEZyACxfQgFtoQgsIPMIzvMKb9WS9WO/Wx7y1ZBUzB/AH1ucPQ/CTWQ==</latexit><latexit sha1_base64="690e9kxg+RvpTk/0gH3/LUKdoj0=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIqBeh6MVjBfsBbQib7aZdutmE3Y1SYn6KFw+KePWXePPfuG1z0NYHA4/3ZpiZFyScKe0431ZpZXVtfaO8Wdna3tnds6v7bRWnktAWiXksuwFWlDNBW5ppTruJpDgKOO0E45up33mgUrFY3OtJQr0IDwULGcHaSL5dTXyGrlA/lJhkbp6d575dc+rODGiZuAWpQYGmb3/1BzFJIyo04Vipnusk2suw1Ixwmlf6qaIJJmM8pD1DBY6o8rLZ6Tk6NsoAhbE0JTSaqb8nMhwpNYkC0xlhPVKL3lT8z+ulOrz0MiaSVFNB5ovClCMdo2kOaMAkJZpPDMFEMnMrIiNsUtAmrYoJwV18eZm0T+uuU3fvzmqN6yKOMhzCEZyACxfQgFtoQgsIPMIzvMKb9WS9WO/Wx7y1ZBUzB/AH1ucPQ/CTWQ==</latexit><latexit sha1_base64="690e9kxg+RvpTk/0gH3/LUKdoj0=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIqBeh6MVjBfsBbQib7aZdutmE3Y1SYn6KFw+KePWXePPfuG1z0NYHA4/3ZpiZFyScKe0431ZpZXVtfaO8Wdna3tnds6v7bRWnktAWiXksuwFWlDNBW5ppTruJpDgKOO0E45up33mgUrFY3OtJQr0IDwULGcHaSL5dTXyGrlA/lJhkbp6d575dc+rODGiZuAWpQYGmb3/1BzFJIyo04Vipnusk2suw1Ixwmlf6qaIJJmM8pD1DBY6o8rLZ6Tk6NsoAhbE0JTSaqb8nMhwpNYkC0xlhPVKL3lT8z+ulOrz0MiaSVFNB5ovClCMdo2kOaMAkJZpPDMFEMnMrIiNsUtAmrYoJwV18eZm0T+uuU3fvzmqN6yKOMhzCEZyACxfQgFtoQgsIPMIzvMKb9WS9WO/Wx7y1ZBUzB/AH1ucPQ/CTWQ==</latexit> pi = 1 N<latexit sha1_base64="HFCq+DAUetU0AmfJjdg7wb531Mk=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIoBeh6MWTVLAf0Iaw2W7apZtN2N0oJeanePGgiFd/iTf/jds2B219MPB4b4aZeUHCmdKO822VVlbX1jfKm5Wt7Z3dPbu631ZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6e+p0HKhWLxb2eJNSL8FCwkBGsjeTb1cRn6BL1Q4lJ5ubZbe7bNafuzICWiVuQGhRo+vZXfxCTNKJCE46V6rlOor0MS80Ip3mlnyqaYDLGQ9ozVOCIKi+bnZ6jY6MMUBhLU0Kjmfp7IsORUpMoMJ0R1iO16E3F/7xeqsMLL2MiSTUVZL4oTDnSMZrmgAZMUqL5xBBMJDO3IjLCJgVt0qqYENzFl5dJ+7TuOnX37qzWuCriKMMhHMEJuHAODbiBJrSAwCM8wyu8WU/Wi/VufcxbS1YxcwB/YH3+AGhok3E=</latexit><latexit sha1_base64="HFCq+DAUetU0AmfJjdg7wb531Mk=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIoBeh6MWTVLAf0Iaw2W7apZtN2N0oJeanePGgiFd/iTf/jds2B219MPB4b4aZeUHCmdKO822VVlbX1jfKm5Wt7Z3dPbu631ZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6e+p0HKhWLxb2eJNSL8FCwkBGsjeTb1cRn6BL1Q4lJ5ubZbe7bNafuzICWiVuQGhRo+vZXfxCTNKJCE46V6rlOor0MS80Ip3mlnyqaYDLGQ9ozVOCIKi+bnZ6jY6MMUBhLU0Kjmfp7IsORUpMoMJ0R1iO16E3F/7xeqsMLL2MiSTUVZL4oTDnSMZrmgAZMUqL5xBBMJDO3IjLCJgVt0qqYENzFl5dJ+7TuOnX37qzWuCriKMMhHMEJuHAODbiBJrSAwCM8wyu8WU/Wi/VufcxbS1YxcwB/YH3+AGhok3E=</latexit><latexit sha1_base64="HFCq+DAUetU0AmfJjdg7wb531Mk=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIoBeh6MWTVLAf0Iaw2W7apZtN2N0oJeanePGgiFd/iTf/jds2B219MPB4b4aZeUHCmdKO822VVlbX1jfKm5Wt7Z3dPbu631ZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6e+p0HKhWLxb2eJNSL8FCwkBGsjeTb1cRn6BL1Q4lJ5ubZbe7bNafuzICWiVuQGhRo+vZXfxCTNKJCE46V6rlOor0MS80Ip3mlnyqaYDLGQ9ozVOCIKi+bnZ6jY6MMUBhLU0Kjmfp7IsORUpMoMJ0R1iO16E3F/7xeqsMLL2MiSTUVZL4oTDnSMZrmgAZMUqL5xBBMJDO3IjLCJgVt0qqYENzFl5dJ+7TuOnX37qzWuCriKMMhHMEJuHAODbiBJrSAwCM8wyu8WU/Wi/VufcxbS1YxcwB/YH3+AGhok3E=</latexit><latexit sha1_base64="HFCq+DAUetU0AmfJjdg7wb531Mk=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5KIoBeh6MWTVLAf0Iaw2W7apZtN2N0oJeanePGgiFd/iTf/jds2B219MPB4b4aZeUHCmdKO822VVlbX1jfKm5Wt7Z3dPbu631ZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6e+p0HKhWLxb2eJNSL8FCwkBGsjeTb1cRn6BL1Q4lJ5ubZbe7bNafuzICWiVuQGhRo+vZXfxCTNKJCE46V6rlOor0MS80Ip3mlnyqaYDLGQ9ozVOCIKi+bnZ6jY6MMUBhLU0Kjmfp7IsORUpMoMJ0R1iO16E3F/7xeqsMLL2MiSTUVZL4oTDnSMZrmgAZMUqL5xBBMJDO3IjLCJgVt0qqYENzFl5dJ+7TuOnX37qzWuCriKMMhHMEJuHAODbiBJrSAwCM8wyu8WU/Wi/VufcxbS1YxcwB/YH3+AGhok3E=</latexit>
  28. 28. Walter Del Pozzo Waves on the Lake, Como, 2018 Invariance • I: we live in Friedmann-Robertson-Walker-LeMaitre Universe, how should I expect GW sources to be distributed (ignoring evolutionary effects)? 28 N = Z dV dN dV dN dV = n0 dV = D2 L cos(✓)dDLd✓d ; z << 1 p(DL, ✓, |I) / D2 L cos(✓)<latexit sha1_base64="P2fRX6culq0F4WLJK6Wez5uwHG8=">AAAChnicbVFda9swFJXdbk2zj2br415Ew0oCIdjZSgdrIWx9aGGUDha3EGVGluRGVJY0SR5kXn5K/9Te+m8qJ3lom10QOvfccyXdo0wLbl0U3QXhxuaz51uN7eaLl69e77TevE2sKg1hI6KEMlcZtkxwyUaOO8GutGG4yAS7zG6+1vXL38xYruQPN9NsUuBryXNOsPNU2ro9h/vHEHHpIE1QD+UGk4qezyuazBFqPsprpUwjT9Okxifpt58DRJTtIDdlDnepZyBdJn7XU/4Z/SoxhX/g0RGMfaPueElvqejVgr9nXbiPtFHaqfUD01Y76keLgOsgXoE2WMVF2vqHqCJlwaQjAls7jiPtJhU2jhPB5k1UWqYxucHXbOyhxAWzk2ph4xy+9wyFuTJ+eT8W7MOOChfWzorMKwvspvZprSb/VxuXLv80qbjUpWOSLC/KSwH9yPWfQMoNI07MPMDEcP9WSKbYO+/8zzW9CfHTkddBMujHUT/+/rE9/LKyowHegT3QATE4BENwCi7ACJBgI+gGg+BD2Aj74UF4uJSGwapnFzyKcHgPIZq98g==</latexit><latexit sha1_base64="P2fRX6culq0F4WLJK6Wez5uwHG8=">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</latexit><latexit sha1_base64="P2fRX6culq0F4WLJK6Wez5uwHG8=">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</latexit><latexit sha1_base64="P2fRX6culq0F4WLJK6Wez5uwHG8=">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</latexit>
  29. 29. Walter Del Pozzo Waves on the Lake, Como, 2018 Uniform distribution 29 H(p) = NX j=1 pj log ✓ pj mj ◆ X j pj = 1 2 4 NX j=1 pj log ✓ pj mj ◆ ( NX j=1 pj 1) 3 5 = 0 · · · = 1 pj = mj mj = 1 N<latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">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</latexit><latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">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</latexit><latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">AAAC3XiclVJLj9MwEHbCaymvLhy5WFSs2kOreIXEXiqt4LKn1SLR3RV1iBzHad21k8ieIFVRJC4cQIgr/4sb/4MfgPM4wJYLI9n6/M18M+Ox40JJC0Hw0/Nv3Lx1+87e3cG9+w8ePhruPz63eWm4WPBc5eYyZlYomYkFSFDisjCC6ViJi/jqdeO/+CCMlXn2FraFCDVbZTKVnIGjouGvk3ExwQdzPKW21FG1mZP6/Skuog2mKl9RJVIY09QwXjmurrTbqJGrNUwoHbSaTRvtUhDHHNBEKGCtbvn/OfHUhbjuEzbe0U4xmXRh4TxoS/EkB9u00UnaazRN9P24vO6gu0NXj9TVaT2IhqNgFrSGdwHpwQj1dhYNf9Ak56UWGXDFrF2SoICwYgYkV6Ie0NKKgvErthJLBzOmhQ2r9nVq/NwxCU5z41YGuGX/VFRMW7vVsYvUDNb2uq8h/+VblpAehZXMihJExrtCaakw5Lh5apxIIziorQOMG+l6xXzN3BjAfYhmCOT6lXfB+eGMBDPy5sXo+FU/jj30FD1DY0TQS3SMTtAZWiDuvfM+ep+9L37kf/K/+t+6UN/rNU/QX+Z//w3zY9/2</latexit><latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">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</latexit> We want to maximise the entropy with some constraint Only constraint Lagrange multiplier
  30. 30. Walter Del Pozzo Waves on the Lake, Como, 2018 Gaussian distribution • Similar calculation shows that if the constraint is • The maximum entropy distribution is 30
  31. 31. Walter Del Pozzo Waves on the Lake, Como, 2018 Gaussian distribution • Similar calculation shows that if the constraint is • The maximum entropy distribution is 31 If the only known constraint is the variance, the Gaussian distribution is the least informative probability distribution.
  32. 32. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning likelihoods • Assume the hypothesis H depends on parameters • The likelihood: • Data D = D1,…,Dn = d1,…,dn • H = X1,…,Xn: datum di in [xi,xi+dxi] • E = E1,…,En: the error on di in [ei,ei+dei] 32 ✓ p(D|✓HI)
  33. 33. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning likelihoods • One can write • Consider joint distribution for Di, Ei, Xi • Marginalise over Ei, Xi 33 di = xi + ei p(Xi|✓HI) ⌘ f(xi) p(Ei|✓HI) ⌘ g(ei) p(DiEiXi|✓HI) p(Di|✓HI) = Z dEidXi p(DiEiXi|✓HI) = Z dEidXi p(Di|EiXi✓HI)p(Ei|✓HI)p(Xi|✓HI)
  34. 34. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning likelihoods • Deterministic models • If the errors are independent 34 di = xi + ei =) p(Di|EiXi✓HI) = (di xi ei) p(Di|✓HI) = Z dxif(xi) Z deig(ei) (di xi ei) = Z dxif(xi)g(di xi) f(xi) = (xi m(xi; ✓)) p(Di|✓HI) = g(di m(xi; ✓)) p(D|✓HI) = NY i=1 g(di m(xi; ✓))
  35. 35. Walter Del Pozzo Waves on the Lake, Como, 2018 Assigning likelihoods • Deterministic models • If the errors are independent 35 di = xi + ei =) p(Di|EiXi✓HI) = (di xi ei) p(Di|✓HI) = Z dxif(xi) Z deig(ei) (di xi ei) = Z dxif(xi)g(di xi) f(xi) = (xi m(xi; ✓)) p(Di|✓HI) = g(di m(xi; ✓)) p(D|✓HI) = NY i=1 g(di m(xi; ✓)) The noise probability distribution defines the likelihood function
  36. 36. Walter Del Pozzo Waves on the Lake, Como, 2018 Part 2 Stochastic processes 36
  37. 37. Walter Del Pozzo Waves on the Lake, Como, 2018 Stochastic processes • Stochastic process is a family of random variables • We will deal with discrete time processes where the set of time samples is potentially infinite, but countable • The state space (the set of possible realisations of the process) is infinite 37 {X(t) t 2 T}<latexit sha1_base64="PHFlBqJ9cA2Xoz+uSNfT6+90rz8=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRKkhJRNBj0YvHCq0tNKFstpt26WYTdidCCQX/ihcPinj1d3jz37hpc9DWNww83pthZ1+QCK7Bcb6t0srq2vpGebOytb2zu2fvHzzoOFWUtWksYtUNiGaCS9YGDoJ1E8VIFAjWCca3ud95ZErzWLZgkjA/IkPJQ04JGKlvH3lZtwZn3nleGDwuccub9u2qU3dmwMvELUgVFWj27S9vENM0YhKoIFr3XCcBPyMKOBVsWvFSzRJCx2TIeoZKEjHtZ7Pzp/jUKAMcxsq0BDxTf29kJNJ6EgVmMiIw0oteLv7n9VIIr/2MyyQFJun8oTAVGGKcZ4EHXDEKYmIIoYqbWzEdEUUomMQqJgR38cvL5OGi7jp19/6y2rgp4iijY3SCashFV6iB7lATtRFFGXpGr+jNerJerHfrYz5asoqdQ/QH1ucP2ECUHQ==</latexit><latexit sha1_base64="PHFlBqJ9cA2Xoz+uSNfT6+90rz8=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRKkhJRNBj0YvHCq0tNKFstpt26WYTdidCCQX/ihcPinj1d3jz37hpc9DWNww83pthZ1+QCK7Bcb6t0srq2vpGebOytb2zu2fvHzzoOFWUtWksYtUNiGaCS9YGDoJ1E8VIFAjWCca3ud95ZErzWLZgkjA/IkPJQ04JGKlvH3lZtwZn3nleGDwuccub9u2qU3dmwMvELUgVFWj27S9vENM0YhKoIFr3XCcBPyMKOBVsWvFSzRJCx2TIeoZKEjHtZ7Pzp/jUKAMcxsq0BDxTf29kJNJ6EgVmMiIw0oteLv7n9VIIr/2MyyQFJun8oTAVGGKcZ4EHXDEKYmIIoYqbWzEdEUUomMQqJgR38cvL5OGi7jp19/6y2rgp4iijY3SCashFV6iB7lATtRFFGXpGr+jNerJerHfrYz5asoqdQ/QH1ucP2ECUHQ==</latexit><latexit sha1_base64="PHFlBqJ9cA2Xoz+uSNfT6+90rz8=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRKkhJRNBj0YvHCq0tNKFstpt26WYTdidCCQX/ihcPinj1d3jz37hpc9DWNww83pthZ1+QCK7Bcb6t0srq2vpGebOytb2zu2fvHzzoOFWUtWksYtUNiGaCS9YGDoJ1E8VIFAjWCca3ud95ZErzWLZgkjA/IkPJQ04JGKlvH3lZtwZn3nleGDwuccub9u2qU3dmwMvELUgVFWj27S9vENM0YhKoIFr3XCcBPyMKOBVsWvFSzRJCx2TIeoZKEjHtZ7Pzp/jUKAMcxsq0BDxTf29kJNJ6EgVmMiIw0oteLv7n9VIIr/2MyyQFJun8oTAVGGKcZ4EHXDEKYmIIoYqbWzEdEUUomMQqJgR38cvL5OGi7jp19/6y2rgp4iijY3SCashFV6iB7lATtRFFGXpGr+jNerJerHfrYz5asoqdQ/QH1ucP2ECUHQ==</latexit><latexit sha1_base64="PHFlBqJ9cA2Xoz+uSNfT6+90rz8=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRKkhJRNBj0YvHCq0tNKFstpt26WYTdidCCQX/ihcPinj1d3jz37hpc9DWNww83pthZ1+QCK7Bcb6t0srq2vpGebOytb2zu2fvHzzoOFWUtWksYtUNiGaCS9YGDoJ1E8VIFAjWCca3ud95ZErzWLZgkjA/IkPJQ04JGKlvH3lZtwZn3nleGDwuccub9u2qU3dmwMvELUgVFWj27S9vENM0YhKoIFr3XCcBPyMKOBVsWvFSzRJCx2TIeoZKEjHtZ7Pzp/jUKAMcxsq0BDxTf29kJNJ6EgVmMiIw0oteLv7n9VIIr/2MyyQFJun8oTAVGGKcZ4EHXDEKYmIIoYqbWzEdEUUomMQqJgR38cvL5OGi7jp19/6y2rgp4iijY3SCashFV6iB7lATtRFFGXpGr+jNerJerHfrYz5asoqdQ/QH1ucP2ECUHQ==</latexit> {x(t1), . . . , x(tn)} ⌘ {x1, . . . , xn}<latexit sha1_base64="VMG3b9xePxQoLMRWR3P9U4pwNYs=">AAACG3icbVDLSgMxFM34rPVVdekmWIQWSpkpgi6LblxWsA/olCGTpm1oJjMmd4pl6H+48VfcuFDEleDCvzHTFtHWAxdOzrmX3Hv8SHANtv1lrayurW9sZray2zu7e/u5g8OGDmNFWZ2GIlQtn2gmuGR14CBYK1KMBL5gTX94lfrNEVOah/IWxhHrBKQveY9TAkbychU3uS+A5xRLruiGoEvpSxbdicvuYj7CxvacH8+T7gR7ubxdtqfAy8SZkzyao+blPtxuSOOASaCCaN127Ag6CVHAqWCTrBtrFhE6JH3WNlSSgOlOMr1tgk+N0sW9UJmSgKfq74mEBFqPA990BgQGetFLxf+8dgy9i07CZRQDk3T2US8WGEKcBoW7XDEKYmwIoYqbXTEdEEUomDizJgRn8eRl0qiUHbvs3Jzlq5fzODLoGJ2gAnLQOaqia1RDdUTRA3pCL+jVerSerTfrfda6Ys1njtAfWJ/fv8WgnQ==</latexit><latexit sha1_base64="VMG3b9xePxQoLMRWR3P9U4pwNYs=">AAACG3icbVDLSgMxFM34rPVVdekmWIQWSpkpgi6LblxWsA/olCGTpm1oJjMmd4pl6H+48VfcuFDEleDCvzHTFtHWAxdOzrmX3Hv8SHANtv1lrayurW9sZray2zu7e/u5g8OGDmNFWZ2GIlQtn2gmuGR14CBYK1KMBL5gTX94lfrNEVOah/IWxhHrBKQveY9TAkbychU3uS+A5xRLruiGoEvpSxbdicvuYj7CxvacH8+T7gR7ubxdtqfAy8SZkzyao+blPtxuSOOASaCCaN127Ag6CVHAqWCTrBtrFhE6JH3WNlSSgOlOMr1tgk+N0sW9UJmSgKfq74mEBFqPA990BgQGetFLxf+8dgy9i07CZRQDk3T2US8WGEKcBoW7XDEKYmwIoYqbXTEdEEUomDizJgRn8eRl0qiUHbvs3Jzlq5fzODLoGJ2gAnLQOaqia1RDdUTRA3pCL+jVerSerTfrfda6Ys1njtAfWJ/fv8WgnQ==</latexit><latexit sha1_base64="VMG3b9xePxQoLMRWR3P9U4pwNYs=">AAACG3icbVDLSgMxFM34rPVVdekmWIQWSpkpgi6LblxWsA/olCGTpm1oJjMmd4pl6H+48VfcuFDEleDCvzHTFtHWAxdOzrmX3Hv8SHANtv1lrayurW9sZray2zu7e/u5g8OGDmNFWZ2GIlQtn2gmuGR14CBYK1KMBL5gTX94lfrNEVOah/IWxhHrBKQveY9TAkbychU3uS+A5xRLruiGoEvpSxbdicvuYj7CxvacH8+T7gR7ubxdtqfAy8SZkzyao+blPtxuSOOASaCCaN127Ag6CVHAqWCTrBtrFhE6JH3WNlSSgOlOMr1tgk+N0sW9UJmSgKfq74mEBFqPA990BgQGetFLxf+8dgy9i07CZRQDk3T2US8WGEKcBoW7XDEKYmwIoYqbXTEdEEUomDizJgRn8eRl0qiUHbvs3Jzlq5fzODLoGJ2gAnLQOaqia1RDdUTRA3pCL+jVerSerTfrfda6Ys1njtAfWJ/fv8WgnQ==</latexit><latexit sha1_base64="VMG3b9xePxQoLMRWR3P9U4pwNYs=">AAACG3icbVDLSgMxFM34rPVVdekmWIQWSpkpgi6LblxWsA/olCGTpm1oJjMmd4pl6H+48VfcuFDEleDCvzHTFtHWAxdOzrmX3Hv8SHANtv1lrayurW9sZray2zu7e/u5g8OGDmNFWZ2GIlQtn2gmuGR14CBYK1KMBL5gTX94lfrNEVOah/IWxhHrBKQveY9TAkbychU3uS+A5xRLruiGoEvpSxbdicvuYj7CxvacH8+T7gR7ubxdtqfAy8SZkzyao+blPtxuSOOASaCCaN127Ag6CVHAqWCTrBtrFhE6JH3WNlSSgOlOMr1tgk+N0sW9UJmSgKfq74mEBFqPA990BgQGetFLxf+8dgy9i07CZRQDk3T2US8WGEKcBoW7XDEKYmwIoYqbXTEdEEUomDizJgRn8eRl0qiUHbvs3Jzlq5fzODLoGJ2gAnLQOaqia1RDdUTRA3pCL+jVerSerTfrfda6Ys1njtAfWJ/fv8WgnQ==</latexit>
  38. 38. Walter Del Pozzo Waves on the Lake, Como, 2018 Process properties • Process is characterised by its statistics • Mean: • Autocovariance: • If the statistics do not depend on time we have a stationary process • e.g. white noise Gaussian process 38 < n(t) >= µ(t)<latexit sha1_base64="yoYj9zSKCRKVZcxI9kMR9rZQvpw=">AAAB+HicbZDLSgMxFIbP1Futl466dBMsQt2UGRF0oVJ047KCvUA7lEyaaUOTzJBkhDr0Sdy4UMStj+LOtzG9LLT1h8DHf87hnPxhwpk2nvft5FZW19Y38puFre2d3aK7t9/QcaoIrZOYx6oVYk05k7RumOG0lSiKRchpMxzeTurNR6o0i+WDGSU0ELgvWcQINtbqusVLWTYn1+gKdURqqeuWvIo3FVoGfw4lmKvWdb86vZikgkpDONa67XuJCTKsDCOcjgudVNMEkyHu07ZFiQXVQTY9fIyOrdNDUazskwZN3d8TGRZaj0RoOwU2A71Ym5j/1dqpiS6CjMkkNVSS2aIo5cjEaJIC6jFFieEjC5goZm9FZIAVJsZmVbAh+ItfXobGacX3Kv79Wal6M48jD4dwBGXw4RyqcAc1qAOBFJ7hFd6cJ+fFeXc+Zq05Zz5zAH/kfP4AHoeRbg==</latexit><latexit sha1_base64="yoYj9zSKCRKVZcxI9kMR9rZQvpw=">AAAB+HicbZDLSgMxFIbP1Futl466dBMsQt2UGRF0oVJ047KCvUA7lEyaaUOTzJBkhDr0Sdy4UMStj+LOtzG9LLT1h8DHf87hnPxhwpk2nvft5FZW19Y38puFre2d3aK7t9/QcaoIrZOYx6oVYk05k7RumOG0lSiKRchpMxzeTurNR6o0i+WDGSU0ELgvWcQINtbqusVLWTYn1+gKdURqqeuWvIo3FVoGfw4lmKvWdb86vZikgkpDONa67XuJCTKsDCOcjgudVNMEkyHu07ZFiQXVQTY9fIyOrdNDUazskwZN3d8TGRZaj0RoOwU2A71Ym5j/1dqpiS6CjMkkNVSS2aIo5cjEaJIC6jFFieEjC5goZm9FZIAVJsZmVbAh+ItfXobGacX3Kv79Wal6M48jD4dwBGXw4RyqcAc1qAOBFJ7hFd6cJ+fFeXc+Zq05Zz5zAH/kfP4AHoeRbg==</latexit><latexit sha1_base64="yoYj9zSKCRKVZcxI9kMR9rZQvpw=">AAAB+HicbZDLSgMxFIbP1Futl466dBMsQt2UGRF0oVJ047KCvUA7lEyaaUOTzJBkhDr0Sdy4UMStj+LOtzG9LLT1h8DHf87hnPxhwpk2nvft5FZW19Y38puFre2d3aK7t9/QcaoIrZOYx6oVYk05k7RumOG0lSiKRchpMxzeTurNR6o0i+WDGSU0ELgvWcQINtbqusVLWTYn1+gKdURqqeuWvIo3FVoGfw4lmKvWdb86vZikgkpDONa67XuJCTKsDCOcjgudVNMEkyHu07ZFiQXVQTY9fIyOrdNDUazskwZN3d8TGRZaj0RoOwU2A71Ym5j/1dqpiS6CjMkkNVSS2aIo5cjEaJIC6jFFieEjC5goZm9FZIAVJsZmVbAh+ItfXobGacX3Kv79Wal6M48jD4dwBGXw4RyqcAc1qAOBFJ7hFd6cJ+fFeXc+Zq05Zz5zAH/kfP4AHoeRbg==</latexit><latexit sha1_base64="yoYj9zSKCRKVZcxI9kMR9rZQvpw=">AAAB+HicbZDLSgMxFIbP1Futl466dBMsQt2UGRF0oVJ047KCvUA7lEyaaUOTzJBkhDr0Sdy4UMStj+LOtzG9LLT1h8DHf87hnPxhwpk2nvft5FZW19Y38puFre2d3aK7t9/QcaoIrZOYx6oVYk05k7RumOG0lSiKRchpMxzeTurNR6o0i+WDGSU0ELgvWcQINtbqusVLWTYn1+gKdURqqeuWvIo3FVoGfw4lmKvWdb86vZikgkpDONa67XuJCTKsDCOcjgudVNMEkyHu07ZFiQXVQTY9fIyOrdNDUazskwZN3d8TGRZaj0RoOwU2A71Ym5j/1dqpiS6CjMkkNVSS2aIo5cjEaJIC6jFFieEjC5goZm9FZIAVJsZmVbAh+ItfXobGacX3Kv79Wal6M48jD4dwBGXw4RyqcAc1qAOBFJ7hFd6cJ+fFeXc+Zq05Zz5zAH/kfP4AHoeRbg==</latexit> < n(t)n(t0 ) >= C(t, t0 )<latexit sha1_base64="Phk2avkHdJrf5t9bewXAi4a39WY=">AAACCnicbVDLSgNBEJyNrxhfqx69jAYhAQm7IuhBJZiLxwjmAUkMs5PZZMjs7DLTK4SQsxd/xYsHRbz6Bd78GyfJIppY0FBUddPd5UWCa3CcLyu1sLi0vJJezaytb2xu2ds7VR3GirIKDUWo6h7RTHDJKsBBsHqkGAk8wWpevzT2a/dMaR7KWxhErBWQruQ+pwSM1Lb3z2UO8qbumpHiActf4gtcysHRj9C2s07BmQDPEzchWZSg3LY/m52QxgGTQAXRuuE6EbSGRAGngo0yzViziNA+6bKGoZIETLeGk1dG+NAoHeyHypQEPFF/TwxJoPUg8ExnQKCnZ72x+J/XiME/aw25jGJgkk4X+bHAEOJxLrjDFaMgBoYQqri5FdMeUYSCSS9jQnBnX54n1eOC6xTcm5Ns8SqJI4320AHKIRedoiK6RmVUQRQ9oCf0gl6tR+vZerPep60pK5nZRX9gfXwDK/CYqA==</latexit><latexit sha1_base64="Phk2avkHdJrf5t9bewXAi4a39WY=">AAACCnicbVDLSgNBEJyNrxhfqx69jAYhAQm7IuhBJZiLxwjmAUkMs5PZZMjs7DLTK4SQsxd/xYsHRbz6Bd78GyfJIppY0FBUddPd5UWCa3CcLyu1sLi0vJJezaytb2xu2ds7VR3GirIKDUWo6h7RTHDJKsBBsHqkGAk8wWpevzT2a/dMaR7KWxhErBWQruQ+pwSM1Lb3z2UO8qbumpHiActf4gtcysHRj9C2s07BmQDPEzchWZSg3LY/m52QxgGTQAXRuuE6EbSGRAGngo0yzViziNA+6bKGoZIETLeGk1dG+NAoHeyHypQEPFF/TwxJoPUg8ExnQKCnZ72x+J/XiME/aw25jGJgkk4X+bHAEOJxLrjDFaMgBoYQqri5FdMeUYSCSS9jQnBnX54n1eOC6xTcm5Ns8SqJI4320AHKIRedoiK6RmVUQRQ9oCf0gl6tR+vZerPep60pK5nZRX9gfXwDK/CYqA==</latexit><latexit sha1_base64="Phk2avkHdJrf5t9bewXAi4a39WY=">AAACCnicbVDLSgNBEJyNrxhfqx69jAYhAQm7IuhBJZiLxwjmAUkMs5PZZMjs7DLTK4SQsxd/xYsHRbz6Bd78GyfJIppY0FBUddPd5UWCa3CcLyu1sLi0vJJezaytb2xu2ds7VR3GirIKDUWo6h7RTHDJKsBBsHqkGAk8wWpevzT2a/dMaR7KWxhErBWQruQ+pwSM1Lb3z2UO8qbumpHiActf4gtcysHRj9C2s07BmQDPEzchWZSg3LY/m52QxgGTQAXRuuE6EbSGRAGngo0yzViziNA+6bKGoZIETLeGk1dG+NAoHeyHypQEPFF/TwxJoPUg8ExnQKCnZ72x+J/XiME/aw25jGJgkk4X+bHAEOJxLrjDFaMgBoYQqri5FdMeUYSCSS9jQnBnX54n1eOC6xTcm5Ns8SqJI4320AHKIRedoiK6RmVUQRQ9oCf0gl6tR+vZerPep60pK5nZRX9gfXwDK/CYqA==</latexit><latexit sha1_base64="Phk2avkHdJrf5t9bewXAi4a39WY=">AAACCnicbVDLSgNBEJyNrxhfqx69jAYhAQm7IuhBJZiLxwjmAUkMs5PZZMjs7DLTK4SQsxd/xYsHRbz6Bd78GyfJIppY0FBUddPd5UWCa3CcLyu1sLi0vJJezaytb2xu2ds7VR3GirIKDUWo6h7RTHDJKsBBsHqkGAk8wWpevzT2a/dMaR7KWxhErBWQruQ+pwSM1Lb3z2UO8qbumpHiActf4gtcysHRj9C2s07BmQDPEzchWZSg3LY/m52QxgGTQAXRuuE6EbSGRAGngo0yzViziNA+6bKGoZIETLeGk1dG+NAoHeyHypQEPFF/TwxJoPUg8ExnQKCnZ72x+J/XiME/aw25jGJgkk4X+bHAEOJxLrjDFaMgBoYQqri5FdMeUYSCSS9jQnBnX54n1eOC6xTcm5Ns8SqJI4320AHKIRedoiK6RmVUQRQ9oCf0gl6tR+vZerPep60pK5nZRX9gfXwDK/CYqA==</latexit>
  39. 39. Walter Del Pozzo Waves on the Lake, Como, 2018 Wide-sense stationarity • For a wide sense stationary process • Constant mean • Covariance function depends only on the time difference 39 < n(t) > = µ(t) = µ(t + ⌧) < n(t)n(t0 ) > = C(t t0 , 0)<latexit sha1_base64="EWzoafYddbx2ebNU0AUOOKD2vKE=">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</latexit><latexit sha1_base64="EWzoafYddbx2ebNU0AUOOKD2vKE=">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</latexit><latexit sha1_base64="EWzoafYddbx2ebNU0AUOOKD2vKE=">AAACLnicbZDLSgMxFIYzXmu9VV26CRalRS0zIuhCRRTBZQVbhc5YMmlag0lmSM4IZegTufFVdCGoiFsfw7QdwduBAx//+Q/J+cNYcAOu++yMjI6NT0zmpvLTM7Nz84WFxbqJEk1ZjUYi0pchMUxwxWrAQbDLWDMiQ8Euwpvj/vzilmnDI3UO3ZgFknQUb3NKwErNwsmeKkH5AK/tY18mFnEG6z6QpOz7+YHB9pUfay7Z0Htcgs0vZcMtNwtFt+IOCv8FL4MiyqraLDz6rYgmkimgghjT8NwYgpRo4FSwXt5PDIsJvSEd1rCoiGQmSAfn9vCqVVq4HWnbCvBA/b6REmlMV4bWKQlcm9+zvvjfrJFAezdIuYoTYIoOH2onAkOE+9nhFteMguhaIFRz+1dMr4kmFGzCeRuC9/vkv1DfqnhuxTvbLh4eZXHk0DJaQSXkoR10iE5RFdUQRXfoAb2gV+feeXLenPehdcTJdpbQj3I+PgF1xKP5</latexit><latexit sha1_base64="EWzoafYddbx2ebNU0AUOOKD2vKE=">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</latexit>
  40. 40. Walter Del Pozzo Waves on the Lake, Como, 2018 Detector output process • GW detector output is modelled as a wide-sense stationary stochastic process • Given a set of (regularly spaced) times • The detector registers • In the absence of a GW signal • When (if) a signal is present 40 {t1, . . . , tn}<latexit sha1_base64="iF6xtB6M1Qs2PhqrixuOhc8Bn4o=">AAAB/HicbVBNS8NAEN3Ur1q/oj16CRbFQymJCHosevFYwX5AE8Jms2mXbjZhdyKEUP+KFw+KePWHePPfuG1z0NYHA4/3ZpiZF6ScKbDtb6Oytr6xuVXdru3s7u0fmIdHPZVkktAuSXgiBwFWlDNBu8CA00EqKY4DTvvB5Hbm9x+pVCwRD5Cn1IvxSLCIEQxa8s36mVuA7zRdHiagmuALd+qbDbtlz2GtEqckDVSi45tfbpiQLKYCCMdKDR07Ba/AEhjhdFpzM0VTTCZ4RIeaChxT5RXz46fWqVZCK0qkLgHWXP09UeBYqTwOdGeMYayWvZn4nzfMILr2CibSDKggi0VRxi1IrFkSVsgkJcBzTTCRTN9qkTGWmIDOq6ZDcJZfXiW9i5Zjt5z7y0b7poyjio7RCTpHDrpCbXSHOqiLCMrRM3pFb8aT8WK8Gx+L1opRztTRHxifPwj0lF4=</latexit><latexit 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sha1_base64="iF6xtB6M1Qs2PhqrixuOhc8Bn4o=">AAAB/HicbVBNS8NAEN3Ur1q/oj16CRbFQymJCHosevFYwX5AE8Jms2mXbjZhdyKEUP+KFw+KePWHePPfuG1z0NYHA4/3ZpiZF6ScKbDtb6Oytr6xuVXdru3s7u0fmIdHPZVkktAuSXgiBwFWlDNBu8CA00EqKY4DTvvB5Hbm9x+pVCwRD5Cn1IvxSLCIEQxa8s36mVuA7zRdHiagmuALd+qbDbtlz2GtEqckDVSi45tfbpiQLKYCCMdKDR07Ba/AEhjhdFpzM0VTTCZ4RIeaChxT5RXz46fWqVZCK0qkLgHWXP09UeBYqTwOdGeMYayWvZn4nzfMILr2CibSDKggi0VRxi1IrFkSVsgkJcBzTTCRTN9qkTGWmIDOq6ZDcJZfXiW9i5Zjt5z7y0b7poyjio7RCTpHDrpCbXSHOqiLCMrRM3pFb8aT8WK8Gx+L1opRztTRHxifPwj0lF4=</latexit> {d1, . . . , dn}<latexit sha1_base64="99vNcHRje/inpcaSF0PZw/8+uDE=">AAAB/HicbVDLSsNAFJ3UV62vaJduBoviopREBF0W3bisYB/QhDCZTNqhk5kwMxFCqL/ixoUibv0Qd/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuVtfWNza3qdm1nd2//wD486imRSUy6WDAhByFShFFOuppqRgapJCgJGemHk9uZ338kUlHBH3SeEj9BI05jipE2UmDXz7wiCtymxyKhVTMKuDcN7IbTcuaAq8QtSQOU6AT2lxcJnCWEa8yQUkPXSbVfIKkpZmRa8zJFUoQnaESGhnKUEOUX8+On8NQoEYyFNMU1nKu/JwqUKJUnoelMkB6rZW8m/ucNMx1f+wXlaaYJx4tFccagFnCWBIyoJFiz3BCEJTW3QjxGEmFt8qqZENzll1dJ76LlOi33/rLRvinjqIJjcALOgQuuQBvcgQ7oAgxy8AxewZv1ZL1Y79bHorVilTN18AfW5w/XNZQ+</latexit><latexit sha1_base64="99vNcHRje/inpcaSF0PZw/8+uDE=">AAAB/HicbVDLSsNAFJ3UV62vaJduBoviopREBF0W3bisYB/QhDCZTNqhk5kwMxFCqL/ixoUibv0Qd/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuVtfWNza3qdm1nd2//wD486imRSUy6WDAhByFShFFOuppqRgapJCgJGemHk9uZ338kUlHBH3SeEj9BI05jipE2UmDXz7wiCtymxyKhVTMKuDcN7IbTcuaAq8QtSQOU6AT2lxcJnCWEa8yQUkPXSbVfIKkpZmRa8zJFUoQnaESGhnKUEOUX8+On8NQoEYyFNMU1nKu/JwqUKJUnoelMkB6rZW8m/ucNMx1f+wXlaaYJx4tFccagFnCWBIyoJFiz3BCEJTW3QjxGEmFt8qqZENzll1dJ76LlOi33/rLRvinjqIJjcALOgQuuQBvcgQ7oAgxy8AxewZv1ZL1Y79bHorVilTN18AfW5w/XNZQ+</latexit><latexit sha1_base64="99vNcHRje/inpcaSF0PZw/8+uDE=">AAAB/HicbVDLSsNAFJ3UV62vaJduBoviopREBF0W3bisYB/QhDCZTNqhk5kwMxFCqL/ixoUibv0Qd/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuVtfWNza3qdm1nd2//wD486imRSUy6WDAhByFShFFOuppqRgapJCgJGemHk9uZ338kUlHBH3SeEj9BI05jipE2UmDXz7wiCtymxyKhVTMKuDcN7IbTcuaAq8QtSQOU6AT2lxcJnCWEa8yQUkPXSbVfIKkpZmRa8zJFUoQnaESGhnKUEOUX8+On8NQoEYyFNMU1nKu/JwqUKJUnoelMkB6rZW8m/ucNMx1f+wXlaaYJx4tFccagFnCWBIyoJFiz3BCEJTW3QjxGEmFt8qqZENzll1dJ76LlOi33/rLRvinjqIJjcALOgQuuQBvcgQ7oAgxy8AxewZv1ZL1Y79bHorVilTN18AfW5w/XNZQ+</latexit><latexit sha1_base64="99vNcHRje/inpcaSF0PZw/8+uDE=">AAAB/HicbVDLSsNAFJ3UV62vaJduBoviopREBF0W3bisYB/QhDCZTNqhk5kwMxFCqL/ixoUibv0Qd/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuVtfWNza3qdm1nd2//wD486imRSUy6WDAhByFShFFOuppqRgapJCgJGemHk9uZ338kUlHBH3SeEj9BI05jipE2UmDXz7wiCtymxyKhVTMKuDcN7IbTcuaAq8QtSQOU6AT2lxcJnCWEa8yQUkPXSbVfIKkpZmRa8zJFUoQnaESGhnKUEOUX8+On8NQoEYyFNMU1nKu/JwqUKJUnoelMkB6rZW8m/ucNMx1f+wXlaaYJx4tFccagFnCWBIyoJFiz3BCEJTW3QjxGEmFt8qqZENzll1dJ76LlOi33/rLRvinjqIJjcALOgQuuQBvcgQ7oAgxy8AxewZv1ZL1Y79bHorVilTN18AfW5w/XNZQ+</latexit> di = ni 8i<latexit sha1_base64="qf0hNmRLUNJHxAM7p3l6co9kym0=">AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuJCSiKAboejGZQX7gDaEyWTSDp1MwsxEKKVu/BU3LhRx61+482+cpllo6xkGDufcy733BClnSjvOt1VaWl5ZXSuvVzY2t7Z37N29lkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4c3Ubz9QqVgi7vUopV6M+4JFjGBtJN8+CH2GrpDwWe80f1EiMeeI+XbVqTk50CJxC1KFAg3f/uqFCcliKjThWKmu66TaG2OpGeF0UulliqaYDHGfdg0VOKbKG+cXTNCxUUJkZpsvNMrV3x1jHCs1igNTGWM9UPPeVPzP62Y6uvTGTKSZpoLMBkUZRzpB0zhQyCQlmo8MwUQysysiAywx0Sa0ignBnT95kbTOaq5Tc+/Oq/XrIo4yHMIRnIALF1CHW2hAEwg8wjO8wpv1ZL1Y79bHrLRkFT378AfW5w89RJV3</latexit><latexit sha1_base64="qf0hNmRLUNJHxAM7p3l6co9kym0=">AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuJCSiKAboejGZQX7gDaEyWTSDp1MwsxEKKVu/BU3LhRx61+482+cpllo6xkGDufcy733BClnSjvOt1VaWl5ZXSuvVzY2t7Z37N29lkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4c3Ubz9QqVgi7vUopV6M+4JFjGBtJN8+CH2GrpDwWe80f1EiMeeI+XbVqTk50CJxC1KFAg3f/uqFCcliKjThWKmu66TaG2OpGeF0UulliqaYDHGfdg0VOKbKG+cXTNCxUUJkZpsvNMrV3x1jHCs1igNTGWM9UPPeVPzP62Y6uvTGTKSZpoLMBkUZRzpB0zhQyCQlmo8MwUQysysiAywx0Sa0ignBnT95kbTOaq5Tc+/Oq/XrIo4yHMIRnIALF1CHW2hAEwg8wjO8wpv1ZL1Y79bHrLRkFT378AfW5w89RJV3</latexit><latexit sha1_base64="qf0hNmRLUNJHxAM7p3l6co9kym0=">AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuJCSiKAboejGZQX7gDaEyWTSDp1MwsxEKKVu/BU3LhRx61+482+cpllo6xkGDufcy733BClnSjvOt1VaWl5ZXSuvVzY2t7Z37N29lkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4c3Ubz9QqVgi7vUopV6M+4JFjGBtJN8+CH2GrpDwWe80f1EiMeeI+XbVqTk50CJxC1KFAg3f/uqFCcliKjThWKmu66TaG2OpGeF0UulliqaYDHGfdg0VOKbKG+cXTNCxUUJkZpsvNMrV3x1jHCs1igNTGWM9UPPeVPzP62Y6uvTGTKSZpoLMBkUZRzpB0zhQyCQlmo8MwUQysysiAywx0Sa0ignBnT95kbTOaq5Tc+/Oq/XrIo4yHMIRnIALF1CHW2hAEwg8wjO8wpv1ZL1Y79bHrLRkFT378AfW5w89RJV3</latexit><latexit sha1_base64="qf0hNmRLUNJHxAM7p3l6co9kym0=">AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuJCSiKAboejGZQX7gDaEyWTSDp1MwsxEKKVu/BU3LhRx61+482+cpllo6xkGDufcy733BClnSjvOt1VaWl5ZXSuvVzY2t7Z37N29lkoySWiTJDyRnQArypmgTc00p51UUhwHnLaD4c3Ubz9QqVgi7vUopV6M+4JFjGBtJN8+CH2GrpDwWe80f1EiMeeI+XbVqTk50CJxC1KFAg3f/uqFCcliKjThWKmu66TaG2OpGeF0UulliqaYDHGfdg0VOKbKG+cXTNCxUUJkZpsvNMrV3x1jHCs1igNTGWM9UPPeVPzP62Y6uvTGTKSZpoLMBkUZRzpB0zhQyCQlmo8MwUQysysiAywx0Sa0ignBnT95kbTOaq5Tc+/Oq/XrIo4yHMIRnIALF1CHW2hAEwg8wjO8wpv1ZL1Y79bHrLRkFT378AfW5w89RJV3</latexit> di = ni + hi(✓) hi(✓) = h(ti, ✓)<latexit sha1_base64="WRkZu2J421XfxjdCuvP0GhW+fEs=">AAACIXicbVDLSgMxFM3UV62vUZdugkWpKGVGBLsRim5cVrAP6JQhk6adYCYzJHeEUvorbvwVNy4U6U78GdN2FrX1QMjJuedyc0+QCK7Bcb6t3Mrq2vpGfrOwtb2zu2fvHzR0nCrK6jQWsWoFRDPBJasDB8FaiWIkCgRrBk93k3rzmSnNY/kIg4R1ItKXvMcpASP5dqXrc3x6g6W5znHo85IHIQNyhj2vMP80nrAEPr/IBN8uOmVnCrxM3IwUUYaab4+9bkzTiEmggmjddp0EOkOigFPBRgUv1Swh9In0WdtQSSKmO8PphiN8YpQu7sXKHAl4qs53DEmk9SAKjDMiEOrF2kT8r9ZOoVfpDLlMUmCSzgb1UoEhxpO4cJcrRkEMDCFUcfNXTEOiCAUTasGE4C6uvEwal2XXKbsPV8XqbRZHHh2hY1RCLrpGVXSPaqiOKHpBb+gDfVqv1rv1ZY1n1pyV9RyiP7B+fgHmDaBJ</latexit><latexit sha1_base64="WRkZu2J421XfxjdCuvP0GhW+fEs=">AAACIXicbVDLSgMxFM3UV62vUZdugkWpKGVGBLsRim5cVrAP6JQhk6adYCYzJHeEUvorbvwVNy4U6U78GdN2FrX1QMjJuedyc0+QCK7Bcb6t3Mrq2vpGfrOwtb2zu2fvHzR0nCrK6jQWsWoFRDPBJasDB8FaiWIkCgRrBk93k3rzmSnNY/kIg4R1ItKXvMcpASP5dqXrc3x6g6W5znHo85IHIQNyhj2vMP80nrAEPr/IBN8uOmVnCrxM3IwUUYaab4+9bkzTiEmggmjddp0EOkOigFPBRgUv1Swh9In0WdtQSSKmO8PphiN8YpQu7sXKHAl4qs53DEmk9SAKjDMiEOrF2kT8r9ZOoVfpDLlMUmCSzgb1UoEhxpO4cJcrRkEMDCFUcfNXTEOiCAUTasGE4C6uvEwal2XXKbsPV8XqbRZHHh2hY1RCLrpGVXSPaqiOKHpBb+gDfVqv1rv1ZY1n1pyV9RyiP7B+fgHmDaBJ</latexit><latexit sha1_base64="WRkZu2J421XfxjdCuvP0GhW+fEs=">AAACIXicbVDLSgMxFM3UV62vUZdugkWpKGVGBLsRim5cVrAP6JQhk6adYCYzJHeEUvorbvwVNy4U6U78GdN2FrX1QMjJuedyc0+QCK7Bcb6t3Mrq2vpGfrOwtb2zu2fvHzR0nCrK6jQWsWoFRDPBJasDB8FaiWIkCgRrBk93k3rzmSnNY/kIg4R1ItKXvMcpASP5dqXrc3x6g6W5znHo85IHIQNyhj2vMP80nrAEPr/IBN8uOmVnCrxM3IwUUYaab4+9bkzTiEmggmjddp0EOkOigFPBRgUv1Swh9In0WdtQSSKmO8PphiN8YpQu7sXKHAl4qs53DEmk9SAKjDMiEOrF2kT8r9ZOoVfpDLlMUmCSzgb1UoEhxpO4cJcrRkEMDCFUcfNXTEOiCAUTasGE4C6uvEwal2XXKbsPV8XqbRZHHh2hY1RCLrpGVXSPaqiOKHpBb+gDfVqv1rv1ZY1n1pyV9RyiP7B+fgHmDaBJ</latexit><latexit sha1_base64="WRkZu2J421XfxjdCuvP0GhW+fEs=">AAACIXicbVDLSgMxFM3UV62vUZdugkWpKGVGBLsRim5cVrAP6JQhk6adYCYzJHeEUvorbvwVNy4U6U78GdN2FrX1QMjJuedyc0+QCK7Bcb6t3Mrq2vpGfrOwtb2zu2fvHzR0nCrK6jQWsWoFRDPBJasDB8FaiWIkCgRrBk93k3rzmSnNY/kIg4R1ItKXvMcpASP5dqXrc3x6g6W5znHo85IHIQNyhj2vMP80nrAEPr/IBN8uOmVnCrxM3IwUUYaab4+9bkzTiEmggmjddp0EOkOigFPBRgUv1Swh9In0WdtQSSKmO8PphiN8YpQu7sXKHAl4qs53DEmk9SAKjDMiEOrF2kT8r9ZOoVfpDLlMUmCSzgb1UoEhxpO4cJcrRkEMDCFUcfNXTEOiCAUTasGE4C6uvEwal2XXKbsPV8XqbRZHHh2hY1RCLrpGVXSPaqiOKHpBb+gDfVqv1rv1ZY1n1pyV9RyiP7B+fgHmDaBJ</latexit> see also Finn, arXiv:9209010
  41. 41. Walter Del Pozzo Waves on the Lake, Como, 2018 Noise distribution • Assume that the only non-null moment of the probability distribution for the process realisation is the second (the autocovariance) • Using the maximum entropy principle 41 < ~n >= 0 p(~n|I) / e 1 2 ~nt C 1 ~n , ~n = n(t1), . . . , n(tK)
  42. 42. Walter Del Pozzo Waves on the Lake, Como, 2018 The noise model • The matrix C is the covariance matrix C(ti, tj) =< n(ti)n(tj) > C = 0 B B B B @ C(t1, t1) C(t1, t2) · · · C(t1, tK) C(t2, t1) C(t2, t2) ... ... ... ... ... ... C(tK, t1) · · · · · · C(tK, tK) 1 C C C C A 42
  43. 43. Walter Del Pozzo Waves on the Lake, Como, 2018 The noise model • In general, the auto-covariance function depends on 2 time values (non-stationary process) • However, we assumed that is a wide-sense stationary process • Assume now that we are sampling the process at regular intervals n(t) C(ti, tj) ⌘ C(ti tj, 0) ⌘ C(⌧) ⌘< n(t)n(t + ⌧) > n(t) tj = t0 + j t 43
  44. 44. Walter Del Pozzo Waves on the Lake, Como, 2018 The noise model • The covariance matrix becomes a Toeplitz matrix • Toeplitz matrices are asymptotically equivalent to circulant matrices • Diagonalised by same base in the limit K ! 1 0 B B B B @ C(0) C( t) · · · C(K t) C( t) C(0) ... ... ... ... ... ... C( K t) · · · · · · C(0) 1 C C C C A 44
  45. 45. Walter Del Pozzo Waves on the Lake, Como, 2018 Circulant matrices • Circulant matrices have the general form • All circulant matrices have the same eigenvectors and eigenvalues 0 B B B B @ a0 an 1 · · · a1 a1 a0 ... ... ... ... ... ... an 1 · · · · · · a0 1 C C C C A 45
  46. 46. Walter Del Pozzo Waves on the Lake, Como, 2018 Eigenvalues & eigenvectors • They are: • Which are nothing more than the Discrete Fourier Transform series vj = 1 p n (1, !j, !2 j , . . . , !n 1 j )t !j = e 2ij⇡ n j = n 1X k=0 ake 2ij⇡ k n 46
  47. 47. Walter Del Pozzo Waves on the Lake, Como, 2018 Fourier Matrix • The DFT matrix • diagonalises any circulant matrix and thus any wide-sense stationary process covariance matrix F = 0 B B B B B B @ 1 1 · · · · · · 1 1 ! !2 · · · !K 1 ... !2 !4 · · · !2(K 1) ... ... ... ... ... 1 !K 1 !2(K 1) · · · !(K 1)(K 1) 1 C C C C C C A F · F 1 = F 1 · F = I det(F) = 1 47 F 1 = F†
  48. 48. Walter Del Pozzo Waves on the Lake, Como, 2018 Diagonalised circulant matrix • It thus follows that any circulant matrix is diagonalised as • It follows that the same is true for a wide-sense stationary covariance matrix • The elements of a diagonalised covariance matrix are the DFT of the autocorrelation function F 1 AF = Diag( K 1X j=0 aje 2⇡in j K ) 48
  49. 49. Walter Del Pozzo Waves on the Lake, Como, 2018 The noise distribution • We have then: • And • Parseval’s Theorem Sn = K 1X j=0 Cnje 2⇡in j K 49 ~nt C 1 ~n = ~nt (F† F)C 1 (F† F)~n = = (F~n)† (FC 1 F† )(F~n) = = ˜~nt S 1˜~n < n(t)n(t + ⌧) > = C(⌧) < ˜n(f)˜n(f0 ) > = 1 2 Sn(f) (f f0 ) <latexit sha1_base64="R/UGXv+TX8VJr6XEcMDXKJ+VcEw=">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</latexit><latexit sha1_base64="R/UGXv+TX8VJr6XEcMDXKJ+VcEw=">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</latexit><latexit sha1_base64="R/UGXv+TX8VJr6XEcMDXKJ+VcEw=">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</latexit><latexit sha1_base64="R/UGXv+TX8VJr6XEcMDXKJ+VcEw=">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</latexit>
  50. 50. Walter Del Pozzo Waves on the Lake, Como, 2018 Power spectral density 50
  51. 51. Walter Del Pozzo Waves on the Lake, Como, 2018 Matched filtering • Detector output • Filter • Filtered output • signal-to-noise ratio 51 d = h + n<latexit sha1_base64="0C284gaAY4umyf7z8kXGuYgyDRA=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBEEoigl6EohePFawttKFsNpt26WYTdidCCf0RXjwo4tXf481/47bNQVsfDDzem2FmXpBKYdB1v53Syura+kZ5s7K1vbO7V90/eDRJphlvsUQmuhNQw6VQvIUCJe+kmtM4kLwdjG6nfvuJayMS9YDjlPsxHSgRCUbRSu2QXJPhmepXa27dnYEsE68gNSjQ7Fe/emHCspgrZJIa0/XcFP2cahRM8kmllxmeUjaiA961VNGYGz+fnTshJ1YJSZRoWwrJTP09kdPYmHEc2M6Y4tAselPxP6+bYXTl50KlGXLF5ouiTBJMyPR3EgrNGcqxJZRpYW8lbEg1ZWgTqtgQvMWXl8njed1z6979Ra1xU8RRhiM4hlPw4BIacAdNaAGDETzDK7w5qfPivDsf89aSU8wcwh84nz/ty46i</latexit><latexit 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sha1_base64="0C284gaAY4umyf7z8kXGuYgyDRA=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBEEoigl6EohePFawttKFsNpt26WYTdidCCf0RXjwo4tXf481/47bNQVsfDDzem2FmXpBKYdB1v53Syura+kZ5s7K1vbO7V90/eDRJphlvsUQmuhNQw6VQvIUCJe+kmtM4kLwdjG6nfvuJayMS9YDjlPsxHSgRCUbRSu2QXJPhmepXa27dnYEsE68gNSjQ7Fe/emHCspgrZJIa0/XcFP2cahRM8kmllxmeUjaiA961VNGYGz+fnTshJ1YJSZRoWwrJTP09kdPYmHEc2M6Y4tAselPxP6+bYXTl50KlGXLF5ouiTBJMyPR3EgrNGcqxJZRpYW8lbEg1ZWgTqtgQvMWXl8njed1z6979Ra1xU8RRhiM4hlPw4BIacAdNaAGDETzDK7w5qfPivDsf89aSU8wcwh84nz/ty46i</latexit> A(t) = Z d⌧F(t + ⌧)a(⌧) = Z df ˜F⇤ (f)˜a(f) <latexit 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sha1_base64="VWGDnaEmiAsaRaxQ2VxtxhihqME=">AAACMXicbVDLSgMxFM34tr6qLt0EizBVKDMi6EbwAeKygrWFTi13MpkazDxI7ghl6C+58U/EjQtF3PoTpu0Ivg6EnJxzL7n3+KkUGh3n2ZqYnJqemZ2bLy0sLi2vlFfXrnSSKcYbLJGJavmguRQxb6BAyVup4hD5kjf929Oh37zjSoskvsR+yjsR9GIRCgZopG75/NjGKj2knoiRBh5CRs9s3BmSKgV7fH/ZIfVQyIDnZ4PrbTusFi8YGN4tV5yaMwL9S9yCVEiBerf86AUJyyIeI5Ogddt1UuzkoFAwyQclL9M8BXYLPd42NIaI604+2nhAt4wS0DBR5pjJRur3jhwirfuRbyojwBv92xuK/3ntDMODTi7iNEMes/FHYSYpJnQYHw2E4gxl3xBgSphZKbsBBQxNyCUTgvt75b/karfmOjX3Yq9ydFLEMUc2yCaxiUv2yRE5J3XSIIzckyfyQl6tB+vZerPex6UTVtGzTn7A+vgEhwOm2g==</latexit> Z df ˜F⇤ (f) ˜d(f) = Z df ˜F⇤ (f)˜h(f) + Z df ˜F⇤ (f)˜n(f) D = H + N<latexit 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sha1_base64="RgCY+hxsJtswysnwVIiho3FgaSI=">AAACYnicfVFbS8MwGE3rvV626aM+BIfiBUYrgr4IoiJ7kgnODdY50vTrFpamJUmFUfYnffPJF3+I6eyDN/wgcHLO+ZIvJ0HKmdKu+2rZc/MLi0vLK87q2vpGpVrbfFRJJim0acIT2Q2IAs4EtDXTHLqpBBIHHDrB+LrQO88gFUvEg56k0I/JULCIUaINNahO9n0mNA4j7GvGQ8hvp09HB9FhuQunBuML/K9pVJiO/7WIwuL7zv6NOayJj/GdM6jW3YY7K/wbeCWoo7Jag+qLHyY0i0FoyolSPc9NdT8nUjPKYer4mYKU0DEZQs9AQWJQ/XwW0RTvGSbEUSLNMnPO2K8dOYmVmsSBccZEj9RPrSD/0nqZjs77ORNppkHQz4uijGOd4CJvHDIJVPOJAYRKZmbFdEQkodr8ShGC9/PJv8HjScNzG979af3yqoxjGW2jXXSAPHSGLlETtVAbUfRmLVgbVsV6tx27Zm99Wm2r7NlC38re+QCPZLJT</latexit><latexit 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sha1_base64="RgCY+hxsJtswysnwVIiho3FgaSI=">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</latexit> SNR ⌘ ⇢ = H2 < N2 ><latexit sha1_base64="2QnKvrGOfZ0172gykMqKaTcDmzQ=">AAACCnicbVDLSsNAFJ34rPUVdelmtAiuSlIEXagU3XRV6qMPaNIymU7aoZOHM5NCCVm78VfcuFDErV/gzr9x0mahrQcuHM65l3vvcUJGhTSMb21hcWl5ZTW3ll/f2Nza1nd2GyKIOCZ1HLCAtxwkCKM+qUsqGWmFnCDPYaTpDK9TvzkiXNDAv5fjkNge6vvUpRhJJXX1g7vqrUUeIjqy+CCAF5bLEY4rnVISn1c7pcsk39ULRtGYAM4TMyMFkKHW1b+sXoAjj/gSMyRE2zRCaceIS4oZSfJWJEiI8BD1SVtRH3lE2PHklQQeKaUH3YCr8iWcqL8nYuQJMfYc1ekhORCzXir+57Uj6Z7ZMfXDSBIfTxe5EYMygGkusEc5wZKNFUGYU3UrxAOkwpAqvTQEc/bledIoFU2jaN6cFMpXWRw5sA8OwTEwwSkogwqogTrA4BE8g1fwpj1pL9q79jFtXdCymT3wB9rnD6jemZI=</latexit><latexit sha1_base64="2QnKvrGOfZ0172gykMqKaTcDmzQ=">AAACCnicbVDLSsNAFJ34rPUVdelmtAiuSlIEXagU3XRV6qMPaNIymU7aoZOHM5NCCVm78VfcuFDErV/gzr9x0mahrQcuHM65l3vvcUJGhTSMb21hcWl5ZTW3ll/f2Nza1nd2GyKIOCZ1HLCAtxwkCKM+qUsqGWmFnCDPYaTpDK9TvzkiXNDAv5fjkNge6vvUpRhJJXX1g7vqrUUeIjqy+CCAF5bLEY4rnVISn1c7pcsk39ULRtGYAM4TMyMFkKHW1b+sXoAjj/gSMyRE2zRCaceIS4oZSfJWJEiI8BD1SVtRH3lE2PHklQQeKaUH3YCr8iWcqL8nYuQJMfYc1ekhORCzXir+57Uj6Z7ZMfXDSBIfTxe5EYMygGkusEc5wZKNFUGYU3UrxAOkwpAqvTQEc/bledIoFU2jaN6cFMpXWRw5sA8OwTEwwSkogwqogTrA4BE8g1fwpj1pL9q79jFtXdCymT3wB9rnD6jemZI=</latexit><latexit sha1_base64="2QnKvrGOfZ0172gykMqKaTcDmzQ=">AAACCnicbVDLSsNAFJ34rPUVdelmtAiuSlIEXagU3XRV6qMPaNIymU7aoZOHM5NCCVm78VfcuFDErV/gzr9x0mahrQcuHM65l3vvcUJGhTSMb21hcWl5ZTW3ll/f2Nza1nd2GyKIOCZ1HLCAtxwkCKM+qUsqGWmFnCDPYaTpDK9TvzkiXNDAv5fjkNge6vvUpRhJJXX1g7vqrUUeIjqy+CCAF5bLEY4rnVISn1c7pcsk39ULRtGYAM4TMyMFkKHW1b+sXoAjj/gSMyRE2zRCaceIS4oZSfJWJEiI8BD1SVtRH3lE2PHklQQeKaUH3YCr8iWcqL8nYuQJMfYc1ekhORCzXir+57Uj6Z7ZMfXDSBIfTxe5EYMygGkusEc5wZKNFUGYU3UrxAOkwpAqvTQEc/bledIoFU2jaN6cFMpXWRw5sA8OwTEwwSkogwqogTrA4BE8g1fwpj1pL9q79jFtXdCymT3wB9rnD6jemZI=</latexit><latexit sha1_base64="2QnKvrGOfZ0172gykMqKaTcDmzQ=">AAACCnicbVDLSsNAFJ34rPUVdelmtAiuSlIEXagU3XRV6qMPaNIymU7aoZOHM5NCCVm78VfcuFDErV/gzr9x0mahrQcuHM65l3vvcUJGhTSMb21hcWl5ZTW3ll/f2Nza1nd2GyKIOCZ1HLCAtxwkCKM+qUsqGWmFnCDPYaTpDK9TvzkiXNDAv5fjkNge6vvUpRhJJXX1g7vqrUUeIjqy+CCAF5bLEY4rnVISn1c7pcsk39ULRtGYAM4TMyMFkKHW1b+sXoAjj/gSMyRE2zRCaceIS4oZSfJWJEiI8BD1SVtRH3lE2PHklQQeKaUH3YCr8iWcqL8nYuQJMfYc1ekhORCzXir+57Uj6Z7ZMfXDSBIfTxe5EYMygGkusEc5wZKNFUGYU3UrxAOkwpAqvTQEc/bledIoFU2jaN6cFMpXWRw5sA8OwTEwwSkogwqogTrA4BE8g1fwpj1pL9q79jFtXdCymT3wB9rnD6jemZI=</latexit>

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