WEBVTT 00:00:00.250 --> 00:00:04.135 X means dot and Y means stripe on the original four cards: 00:00:04.416 --> 00:00:08.041 A has only a dot, B both, C only a stripe, and 00:00:08.052 --> 00:00:08.803 D neither. 00:00:09.340 --> 00:00:14.456 A selected signature records the chosen formula answers in their displayed order. 00:00:14.957 --> 00:00:18.587 For a pair of bits, XOR is one when they differ and 00:00:18.609 --> 00:00:20.224 zero when they agree. 00:00:21.022 --> 00:00:25.515 Keep the four valuations and select X and Y in that order. 00:00:26.010 --> 00:00:30.373 Give the X coordinate weight one and the Y coordinate weight three. 00:00:30.809 --> 00:00:34.822 All selected weights in our construction are strictly positive. 00:00:35.163 --> 00:00:40.795 These weights describe a chosen measurement; they are not probabilities, truth values 00:00:40.816 --> 00:00:43.182 or costs of acquiring tests. 00:00:44.316 --> 00:00:47.257 A and C disagree on X and on Y. 00:00:47.780 --> 00:00:52.404 Their coordinatewise XOR has a one at each disagreement. 00:00:52.864 --> 00:00:57.043 XOR tells us where the profiles differ; the weights tell us how 00:00:57.063 --> 00:00:59.993 much each differing coordinate contributes. 00:01:01.142 --> 00:01:05.348 At the first coordinate the bits differ, so include the first weight, 00:01:05.537 --> 00:01:06.249 one. 00:01:06.570 --> 00:01:10.132 If they agreed, this coordinate would contribute zero. 00:01:10.689 --> 00:01:14.726 The indicator on screen is one when the bits differ and zero 00:01:14.824 --> 00:01:15.874 when they agree. 00:01:16.403 --> 00:01:21.231 Multiplying that indicator by the weight includes the weight exactly when the 00:01:21.231 --> 00:01:22.403 condition holds. 00:01:23.537 --> 00:01:28.244 At the second coordinate the bits also differ, so include weight three. 00:01:28.792 --> 00:01:32.675 We have kept the two contributions visible separately. 00:01:33.740 --> 00:01:39.568 Adding the two contributions gives weighted disagreement four between A and C. 00:01:40.416 --> 00:01:45.940 Retain both labelled contributions beside the result so that the total can 00:01:45.950 --> 00:01:46.719 be checked. 00:01:47.804 --> 00:01:52.817 The general formula sums the weight times the absolute difference of the 00:01:52.839 --> 00:01:55.485 bits at each selected coordinate. 00:01:55.934 --> 00:02:00.659 For zero and one, that absolute difference is one when they differ 00:02:00.810 --> 00:02:02.631 and zero when they agree. 00:02:03.125 --> 00:02:06.901 Reversing the endpoints changes no contribution. 00:02:08.037 --> 00:02:11.770 Nonnegative weights give nonnegative contributions. 00:02:12.379 --> 00:02:16.519 Comparing a profile with itself gives no disagreement. 00:02:16.962 --> 00:02:21.316 Reversing a comparison leaves the disagreement set unchanged. 00:02:21.812 --> 00:02:27.748 These observations explain self-distance and symmetry without relying on the picture' 00:02:27.748 --> 00:02:28.738 s geometry. 00:02:29.835 --> 00:02:34.752 If a coordinate differs between v and u, the intermediate bit at 00:02:34.762 --> 00:02:38.479 w must disagree with at least one endpoint. 00:02:38.878 --> 00:02:42.111 Its weight is therefore counted on at least one leg. 00:02:42.670 --> 00:02:49.213 Sum over coordinates: the direct weighted disagreement cannot exceed the total disagreement 00:02:49.213 --> 00:02:50.912 along the two-leg detour. 00:02:51.429 --> 00:02:53.695 That is the triangle inequality. 00:02:54.728 --> 00:02:59.371 A metric allows zero distance only between identical points. 00:03:00.041 --> 00:03:04.472 A pseudometric may give distinct points zero distance. 00:03:05.383 --> 00:03:11.448 Now explicitly replace the selected tuple by X alone while retaining all 00:03:11.481 --> 00:03:12.804 four valuations. 00:03:13.504 --> 00:03:17.555 A and B both have the one-entry true signature. 00:03:18.010 --> 00:03:22.893 Their measured disagreement is zero even though they differ on the unobserved 00:03:22.893 --> 00:03:24.168 variable Y. 00:03:25.108 --> 00:03:30.240 Strictly positive weights ensure that a sum can vanish only when every 00:03:30.273 --> 00:03:31.901 selected bit agrees. 00:03:32.505 --> 00:03:39.218 Thus zero distance means equal signatures, not necessarily equal valuations. 00:03:39.239 --> 00:03:42.277 This is a pseudometric on the original carrier. 00:03:43.288 --> 00:03:46.917 Replace each equal-signature class by one point. 00:03:47.465 --> 00:03:53.358 The distance is well-defined between classes because changing representatives changes no 00:03:53.389 --> 00:03:54.490 selected bit. 00:03:55.012 --> 00:04:01.555 Distinct classes have distinct signatures and therefore positive distance, giving a metric 00:04:01.587 --> 00:04:02.927 on that class set. 00:04:03.922 --> 00:04:08.581 This statement concerns a metric on an underlying quotient set. 00:04:09.073 --> 00:04:14.357 It does not yet claim that its metric topology equals the topological 00:04:14.411 --> 00:04:17.916 quotient of the original observation space. 00:04:18.352 --> 00:04:21.901 The next lecture isolates that important distinction. 00:04:23.055 --> 00:04:28.568 With the original two-observation menu and weights one and three, compute 00:04:28.568 --> 00:04:31.400 the symmetric distance between A and B. 00:04:32.263 --> 00:04:38.019 Which coordinate contributes, and would reversing the endpoints change the answer? 00:04:54.290 --> 00:04:57.766 A and B agree on X and disagree on Y. 00:04:58.262 --> 00:05:02.928 Only the second weight contributes, so the distance is three in either 00:05:02.928 --> 00:05:03.671 direction. 00:05:04.168 --> 00:05:09.517 A different positive weight choice changes this number without changing the selected 00:05:09.517 --> 00:05:10.560 truth regions. 00:05:11.699 --> 00:05:17.262 Weighted Hamming distance measures disagreement between complete selected profiles. 00:05:17.845 --> 00:05:24.027 Its zero classes are qualitative signature classes, but its positive values depend 00:05:24.037 --> 00:05:26.816 on the chosen indexed coordinates and weights. 00:05:27.440 --> 00:05:30.500 They cannot be read from the topology alone.