The document discusses Bayesian data analysis and gravitational wave detection. It provides an outline of topics to be covered including fundamentals of probability theory, stochastic processes, detection of gravitational wave signals, and inference of gravitational wave physics. It explains that Bayesian analysis is useful for gravitational wave detection because events are rare and detectors are noise dominated, requiring accurate models to detect signals in the noise.

1. Walter Del Pozzo Waves on the Lake, Como, 2018
Gravitational waves
data analysis
Walter Del Pozzo
1

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. Walter Del Pozzo Waves on the Lake, Como, 2018
What is “data analysis”?
3

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. 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. 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. 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. Walter Del Pozzo Waves on the Lake, Como, 2018
Part 1
Fundaments of probability theory
8
“Probability is a property of the observer”

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. 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. 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. 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. 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. 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. 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
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I’: The die is biased
p(6|I0
) >
1
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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. 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. 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. Walter Del Pozzo Waves on the Lake, Como, 2018
Example
• To the (bat) board!
19

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 sha1_base64="W5c69Jujx1u+4d23LRpRjHEBC5U=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBQym7RdBj0UuPFeyHtMuSTbNtaJJdkqxQlv4KLx4U8erP8ea/MW33oK0PBh7vzTAzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6eg4VYS2Scxj1QuxppxJ2jbMcNpLFMUi5LQbTu7mfveJKs1i+WCmCfUFHkkWMYKNlR6bgVcdVJtBPShX3Jq7AFonXk4qkKMVlL8Gw5ikgkpDONa677mJ8TOsDCOczkqDVNMEkwke0b6lEguq/Wxx8AxdWGWIoljZkgYt1N8TGRZaT0VoOwU2Y73qzcX/vH5qohs/YzJJDZVkuShKOTIxmn+PhkxRYvjUEkwUs7ciMsYKE2MzKtkQvNWX10mnXvPcmnd/VWnc5nEU4QzO4RI8uIYGNKEFbSAg4Ble4c1Rzovz7nwsWwtOPnMKf+B8/gACzo85</latexit><latexit sha1_base64="W5c69Jujx1u+4d23LRpRjHEBC5U=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBQym7RdBj0UuPFeyHtMuSTbNtaJJdkqxQlv4KLx4U8erP8ea/MW33oK0PBh7vzTAzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6eg4VYS2Scxj1QuxppxJ2jbMcNpLFMUi5LQbTu7mfveJKs1i+WCmCfUFHkkWMYKNlR6bgVcdVJtBPShX3Jq7AFonXk4qkKMVlL8Gw5ikgkpDONa677mJ8TOsDCOczkqDVNMEkwke0b6lEguq/Wxx8AxdWGWIoljZkgYt1N8TGRZaT0VoOwU2Y73qzcX/vH5qohs/YzJJDZVkuShKOTIxmn+PhkxRYvjUEkwUs7ciMsYKE2MzKtkQvNWX10mnXvPcmnd/VWnc5nEU4QzO4RI8uIYGNKEFbSAg4Ble4c1Rzovz7nwsWwtOPnMKf+B8/gACzo85</latexit><latexit sha1_base64="W5c69Jujx1u+4d23LRpRjHEBC5U=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBQym7RdBj0UuPFeyHtMuSTbNtaJJdkqxQlv4KLx4U8erP8ea/MW33oK0PBh7vzTAzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6eg4VYS2Scxj1QuxppxJ2jbMcNpLFMUi5LQbTu7mfveJKs1i+WCmCfUFHkkWMYKNlR6bgVcdVJtBPShX3Jq7AFonXk4qkKMVlL8Gw5ikgkpDONa677mJ8TOsDCOczkqDVNMEkwke0b6lEguq/Wxx8AxdWGWIoljZkgYt1N8TGRZaT0VoOwU2Y73qzcX/vH5qohs/YzJJDZVkuShKOTIxmn+PhkxRYvjUEkwUs7ciMsYKE2MzKtkQvNWX10mnXvPcmnd/VWnc5nEU4QzO4RI8uIYGNKEFbSAg4Ble4c1Rzovz7nwsWwtOPnMKf+B8/gACzo85</latexit>
D<latexit sha1_base64="NsLos3laW3WlkybC0/eUHSYSJS0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNRDx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS465frrhVdw6ySrycVCBHvV/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ66LquVWvcVmp3eRxFOEETuEcPLiCGtxDHZrAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHl2mMyA==</latexit><latexit sha1_base64="NsLos3laW3WlkybC0/eUHSYSJS0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNRDx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS465frrhVdw6ySrycVCBHvV/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ66LquVWvcVmp3eRxFOEETuEcPLiCGtxDHZrAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHl2mMyA==</latexit><latexit sha1_base64="NsLos3laW3WlkybC0/eUHSYSJS0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNRDx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFS465frrhVdw6ySrycVCBHvV/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJ66LquVWvcVmp3eRxFOEETuEcPLiCGtxDHZrAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHl2mMyA==</latexit><latexit 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 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><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 sha1_base64="5rX0BkBVuAqhHgLowFjlbSXJMlg=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EYp20e4q2Au0Q8mkmTY0MxOSjFCmfQo3voobF4q4FXe+jel0Ftp6IOTLf87h5PwuZ1Qqy/o2MmvrG5tb2e3czu7e/oF5eNSSYSQwaeKQhaLjIkkYDUhTUcVIhwuCfJeRtju+nefbD0RIGgb3asKJ46NhQD2KkdJS3zznhVq/PK3Wi/AaLrhe7HkC4ZgXqlP9hvXiLGF99828VbKSgKtgp5AHaTT65ldvEOLIJ4HCDEnZtS2unBgJRTEjs1wvkoQjPEZD0tUYIJ9IJ07WmsEzrQygFwp9AgUT9XdHjHwpJ76rK32kRnI5Nxf/y3Uj5V05MQ14pEiAF4O8iEEVwrlHcEAFwYpNNCAsqP4rxCOkPVHayZw2wV5eeRVa5ZJtley7i3zlJrUjC07AKSgAG1yCCqiBBmgCDB7BM3gFb8aT8WK8Gx+L0oyR9hyDP2F8/gBZpZxh</latexit><latexit sha1_base64="5rX0BkBVuAqhHgLowFjlbSXJMlg=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EYp20e4q2Au0Q8mkmTY0MxOSjFCmfQo3voobF4q4FXe+jel0Ftp6IOTLf87h5PwuZ1Qqy/o2MmvrG5tb2e3czu7e/oF5eNSSYSQwaeKQhaLjIkkYDUhTUcVIhwuCfJeRtju+nefbD0RIGgb3asKJ46NhQD2KkdJS3zznhVq/PK3Wi/AaLrhe7HkC4ZgXqlP9hvXiLGF99828VbKSgKtgp5AHaTT65ldvEOLIJ4HCDEnZtS2unBgJRTEjs1wvkoQjPEZD0tUYIJ9IJ07WmsEzrQygFwp9AgUT9XdHjHwpJ76rK32kRnI5Nxf/y3Uj5V05MQ14pEiAF4O8iEEVwrlHcEAFwYpNNCAsqP4rxCOkPVHayZw2wV5eeRVa5ZJtley7i3zlJrUjC07AKSgAG1yCCqiBBmgCDB7BM3gFb8aT8WK8Gx+L0oyR9hyDP2F8/gBZpZxh</latexit><latexit sha1_base64="5rX0BkBVuAqhHgLowFjlbSXJMlg=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EYp20e4q2Au0Q8mkmTY0MxOSjFCmfQo3voobF4q4FXe+jel0Ftp6IOTLf87h5PwuZ1Qqy/o2MmvrG5tb2e3czu7e/oF5eNSSYSQwaeKQhaLjIkkYDUhTUcVIhwuCfJeRtju+nefbD0RIGgb3asKJ46NhQD2KkdJS3zznhVq/PK3Wi/AaLrhe7HkC4ZgXqlP9hvXiLGF99828VbKSgKtgp5AHaTT65ldvEOLIJ4HCDEnZtS2unBgJRTEjs1wvkoQjPEZD0tUYIJ9IJ07WmsEzrQygFwp9AgUT9XdHjHwpJ76rK32kRnI5Nxf/y3Uj5V05MQ14pEiAF4O8iEEVwrlHcEAFwYpNNCAsqP4rxCOkPVHayZw2wV5eeRVa5ZJtley7i3zlJrUjC07AKSgAG1yCCqiBBmgCDB7BM3gFb8aT8WK8Gx+L0oyR9hyDP2F8/gBZpZxh</latexit><latexit sha1_base64="5rX0BkBVuAqhHgLowFjlbSXJMlg=">AAACFnicbZDLSgMxFIYz9VbrbdSlm2AR2oVlpgi6EYp20e4q2Au0Q8mkmTY0MxOSjFCmfQo3voobF4q4FXe+jel0Ftp6IOTLf87h5PwuZ1Qqy/o2MmvrG5tb2e3czu7e/oF5eNSSYSQwaeKQhaLjIkkYDUhTUcVIhwuCfJeRtju+nefbD0RIGgb3asKJ46NhQD2KkdJS3zznhVq/PK3Wi/AaLrhe7HkC4ZgXqlP9hvXiLGF99828VbKSgKtgp5AHaTT65ldvEOLIJ4HCDEnZtS2unBgJRTEjs1wvkoQjPEZD0tUYIJ9IJ07WmsEzrQygFwp9AgUT9XdHjHwpJ76rK32kRnI5Nxf/y3Uj5V05MQ14pEiAF4O8iEEVwrlHcEAFwYpNNCAsqP4rxCOkPVHayZw2wV5eeRVa5ZJtley7i3zlJrUjC07AKSgAG1yCCqiBBmgCDB7BM3gFb8aT8WK8Gx+L0oyR9hyDP2F8/gBZpZxh</latexit>

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)
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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=">AAACQXicbZBLTwIxFIU7PhFfqEs3jUQDGzIzMdENCUEXsBITeSRAJp3SgYbOI23HhAzz19z4D9y5d+NCY9y6sQxEBbxJk5PvnnvbHjtgVEhdf9ZWVtfWNzZTW+ntnd29/czBYUP4Icekjn3m85aNBGHUI3VJJSOtgBPk2ow07eHVpN+8J1xQ37uTo4B0XdT3qEMxkgpZmdaNFRlmDM+KsONwhKMgV7GMcTUf5K7HSsFqPk6Y+cPMCYPFefuvKS4nG61MVi/oScFlYcxEFsyqZmWeOj0fhy7xJGZIiLahB7IbIS4pZiROd0JBAoSHqE/aSnrIJaIbJQnE8FSRHnR8ro4nYUL/TkTIFWLk2srpIjkQi70J/K/XDqVz2Y2oF4SSeHh6kRMyKH04iRP2KCdYspESCHOq3grxAKlkpAo9rUIwFr+8LBpmwdALxu15tlSexZECx+AE5IABLkAJVEAN1AEGD+AFvIF37VF71T60z6l1RZvNHIG50r6+AS+qq+8=</latexit>
Bayes’ factor
prior odds

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
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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 sha1_base64="RK6tMOZxx6ELHaUJ//auteCmvvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTHgg3LFrboLkHXi5aQCOZqD8ld/GLM0QmmYoFr3PDcxfkaV4UzgrNRPNSaUTegIe5ZKGqH2s8WpM3JhlSEJY2VLGrJQf09kNNJ6GgW2M6JmrFe9ufif10tNeONnXCapQcmWi8JUEBOT+d9kyBUyI6aWUKa4vZWwMVWUGZtOyYbgrb68TtpXVc+tevfXlfptHkcRzuAcLsGDGtShAU1oAYMRPMMrvDnCeXHenY9la8HJZ07hD5zPHxdwjag=</latexit><latexit sha1_base64="RK6tMOZxx6ELHaUJ//auteCmvvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTHgg3LFrboLkHXi5aQCOZqD8ld/GLM0QmmYoFr3PDcxfkaV4UzgrNRPNSaUTegIe5ZKGqH2s8WpM3JhlSEJY2VLGrJQf09kNNJ6GgW2M6JmrFe9ufif10tNeONnXCapQcmWi8JUEBOT+d9kyBUyI6aWUKa4vZWwMVWUGZtOyYbgrb68TtpXVc+tevfXlfptHkcRzuAcLsGDGtShAU1oAYMRPMMrvDnCeXHenY9la8HJZ07hD5zPHxdwjag=</latexit><latexit sha1_base64="RK6tMOZxx6ELHaUJ//auteCmvvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeilx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJaPZpqgH9GR5CFn1FjpoTHgg3LFrboLkHXi5aQCOZqD8ld/GLM0QmmYoFr3PDcxfkaV4UzgrNRPNSaUTegIe5ZKGqH2s8WpM3JhlSEJY2VLGrJQf09kNNJ6GgW2M6JmrFe9ufif10tNeONnXCapQcmWi8JUEBOT+d9kyBUyI6aWUKa4vZWwMVWUGZtOyYbgrb68TtpXVc+tevfXlfptHkcRzuAcLsGDGtShAU1oAYMRPMMrvDnCeXHenY9la8HJZ07hD5zPHxdwjag=</latexit>
✓i<latexit sha1_base64="69Cwcaham9M/tUrpfobI6ehtXNA=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhsw9neoUQ8hNePCji1d/x5t84SfagiQUNRVU33V1BqqQh1/12CmvrG5tbxe3Szu7e/kH58KhpkkwLbIhEJbodcINKxtggSQrbqUYeBQpbweh25reeUBuZxA80TtGP+CCWoRScrNTu0hCJ92SvXHGr7hxslXg5qUCOeq/81e0nIoswJqG4MR3PTcmfcE1SKJyWupnBlIsRH2DH0phHaPzJ/N4pO7NKn4WJthUTm6u/JyY8MmYcBbYz4jQ0y95M/M/rZBRe+xMZpxlhLBaLwkwxStjsedaXGgWpsSVcaGlvZWLINRdkIyrZELzll1dJ86LquVXv/rJSu8njKMIJnMI5eHAFNbiDOjRAgIJneIU359F5cd6dj0VrwclnjuEPnM8fIlaQBA==</latexit><latexit sha1_base64="69Cwcaham9M/tUrpfobI6ehtXNA=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhsw9neoUQ8hNePCji1d/x5t84SfagiQUNRVU33V1BqqQh1/12CmvrG5tbxe3Szu7e/kH58KhpkkwLbIhEJbodcINKxtggSQrbqUYeBQpbweh25reeUBuZxA80TtGP+CCWoRScrNTu0hCJ92SvXHGr7hxslXg5qUCOeq/81e0nIoswJqG4MR3PTcmfcE1SKJyWupnBlIsRH2DH0phHaPzJ/N4pO7NKn4WJthUTm6u/JyY8MmYcBbYz4jQ0y95M/M/rZBRe+xMZpxlhLBaLwkwxStjsedaXGgWpsSVcaGlvZWLINRdkIyrZELzll1dJ86LquVXv/rJSu8njKMIJnMI5eHAFNbiDOjRAgIJneIU359F5cd6dj0VrwclnjuEPnM8fIlaQBA==</latexit><latexit sha1_base64="69Cwcaham9M/tUrpfobI6ehtXNA=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhsw9neoUQ8hNePCji1d/x5t84SfagiQUNRVU33V1BqqQh1/12CmvrG5tbxe3Szu7e/kH58KhpkkwLbIhEJbodcINKxtggSQrbqUYeBQpbweh25reeUBuZxA80TtGP+CCWoRScrNTu0hCJ92SvXHGr7hxslXg5qUCOeq/81e0nIoswJqG4MR3PTcmfcE1SKJyWupnBlIsRH2DH0phHaPzJ/N4pO7NKn4WJthUTm6u/JyY8MmYcBbYz4jQ0y95M/M/rZBRe+xMZpxlhLBaLwkwxStjsedaXGgWpsSVcaGlvZWLINRdkIyrZELzll1dJ86LquVXv/rJSu8njKMIJnMI5eHAFNbiDOjRAgIJneIU359F5cd6dj0VrwclnjuEPnM8fIlaQBA==</latexit><latexit sha1_base64="69Cwcaham9M/tUrpfobI6ehtXNA=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhsw9neoUQ8hNePCji1d/x5t84SfagiQUNRVU33V1BqqQh1/12CmvrG5tbxe3Szu7e/kH58KhpkkwLbIhEJbodcINKxtggSQrbqUYeBQpbweh25reeUBuZxA80TtGP+CCWoRScrNTu0hCJ92SvXHGr7hxslXg5qUCOeq/81e0nIoswJqG4MR3PTcmfcE1SKJyWupnBlIsRH2DH0phHaPzJ/N4pO7NKn4WJthUTm6u/JyY8MmYcBbYz4jQ0y95M/M/rZBRe+xMZpxlhLBaLwkwxStjsedaXGgWpsSVcaGlvZWLINRdkIyrZELzll1dJ86LquVXv/rJSu8njKMIJnMI5eHAFNbiDOjRAgIJneIU359F5cd6dj0VrwclnjuEPnM8fIlaQBA==</latexit>
p(D|HiI) =
Z
⇥
d✓i p(✓i|HiI)p(D|✓iHiI)
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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)
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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. 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. 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
◆
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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. 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=">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</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. 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=">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><latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">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</latexit><latexit sha1_base64="PQfHc6ytV7TwbhKVs0akDO49U1k=">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</latexit>
We want to maximise
the entropy with some constraint
Only constraint
Lagrange multiplier

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. 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. 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. 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. 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. 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. Walter Del Pozzo Waves on the Lake, Como, 2018
Part 2
Stochastic processes
36

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
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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
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< 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. 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=">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</latexit><latexit sha1_base64="EWzoafYddbx2ebNU0AUOOKD2vKE=">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</latexit>

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
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{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. 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. 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. 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. 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. 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. 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. 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. 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. 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
)
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50. Walter Del Pozzo Waves on the Lake, Como, 2018
Power spectral density
50

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 sha1_base64="0C284gaAY4umyf7z8kXGuYgyDRA=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBEEoigl6EohePFawttKFsNpt26WYTdidCCf0RXjwo4tXf481/47bNQVsfDDzem2FmXpBKYdB1v53Syura+kZ5s7K1vbO7V90/eDRJphlvsUQmuhNQw6VQvIUCJe+kmtM4kLwdjG6nfvuJayMS9YDjlPsxHSgRCUbRSu2QXJPhmepXa27dnYEsE68gNSjQ7Fe/emHCspgrZJIa0/XcFP2cahRM8kmllxmeUjaiA961VNGYGz+fnTshJ1YJSZRoWwrJTP09kdPYmHEc2M6Y4tAselPxP6+bYXTl50KlGXLF5ouiTBJMyPR3EgrNGcqxJZRpYW8lbEg1ZWgTqtgQvMWXl8njed1z6979Ra1xU8RRhiM4hlPw4BIacAdNaAGDETzDK7w5qfPivDsf89aSU8wcwh84nz/ty46i</latexit><latexit sha1_base64="0C284gaAY4umyf7z8kXGuYgyDRA=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBEEoigl6EohePFawttKFsNpt26WYTdidCCf0RXjwo4tXf481/47bNQVsfDDzem2FmXpBKYdB1v53Syura+kZ5s7K1vbO7V90/eDRJphlvsUQmuhNQw6VQvIUCJe+kmtM4kLwdjG6nfvuJayMS9YDjlPsxHSgRCUbRSu2QXJPhmepXa27dnYEsE68gNSjQ7Fe/emHCspgrZJIa0/XcFP2cahRM8kmllxmeUjaiA961VNGYGz+fnTshJ1YJSZRoWwrJTP09kdPYmHEc2M6Y4tAselPxP6+bYXTl50KlGXLF5ouiTBJMyPR3EgrNGcqxJZRpYW8lbEg1ZWgTqtgQvMWXl8njed1z6979Ra1xU8RRhiM4hlPw4BIacAdNaAGDETzDK7w5qfPivDsf89aSU8wcwh84nz/ty46i</latexit><latexit 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 sha1_base64="VWGDnaEmiAsaRaxQ2VxtxhihqME=">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</latexit><latexit sha1_base64="VWGDnaEmiAsaRaxQ2VxtxhihqME=">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</latexit><latexit sha1_base64="VWGDnaEmiAsaRaxQ2VxtxhihqME=">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</latexit><latexit sha1_base64="VWGDnaEmiAsaRaxQ2VxtxhihqME=">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</latexit>
Z
df ˜F⇤
(f) ˜d(f) =
Z
df ˜F⇤
(f)˜h(f) +
Z
df ˜F⇤
(f)˜n(f)
D = H + N<latexit sha1_base64="RgCY+hxsJtswysnwVIiho3FgaSI=">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</latexit><latexit sha1_base64="RgCY+hxsJtswysnwVIiho3FgaSI=">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</latexit><latexit sha1_base64="RgCY+hxsJtswysnwVIiho3FgaSI=">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</latexit><latexit 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>