Helper function for PPI++ logistic regression
Usage
ppi_plusplus_logistic(
X_l,
Y_l,
f_l,
X_u,
f_u,
lhat = NULL,
coord = NULL,
opts = NULL,
w_l = NULL,
w_u = NULL
)Arguments
- X_l
(matrix): n x p matrix of covariates in the labeled data.
- Y_l
(vector): n-vector of labeled outcomes.
- f_l
(vector): n-vector of predictions in the labeled data.
- X_u
(matrix): N x p matrix of covariates in the unlabeled data.
- f_u
(vector): N-vector of predictions in the unlabeled data.
- lhat
(float, optional): Power-tuning parameter (see https://arxiv.org/abs/2311.01453). The default value,
NULL, will estimate the optimal value from the data. Settinglhat = 1recovers PPI with no power tuning, and settinglhat = 0recovers the classical point estimate.- coord
(int, optional): Coordinate for which to optimize
lhat = 1. IfNULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.- opts
(list, optional): Options to pass to the optimizer. See ?optim for details.
- w_l
(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
- w_u
(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.
Value
(list): A list containing the following:
- est
(vector): vector of PPI++ logistic regression coefficient estimates.
- se
(vector): vector of standard errors of the coefficients.
- lambda
(float): estimated power-tuning parameter.
- rectifier_est
(vector): vector of the rectifier logistic regression coefficient estimates.
- var_u
(matrix): covariance matrix for the gradients in the unlabeled data.
- var_l
(matrix): covariance matrix for the gradients in the labeled data.
- grads
(matrix): matrix of gradients for the labeled data.
- grads_hat_unlabeled
(matrix): matrix of predicted gradients for the unlabeled data.
- grads_hat
(matrix): matrix of predicted gradients for the labeled data.
- inv_hessian
(matrix): inverse Hessian matrix.
Details
PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) https://arxiv.org/abs/2311.01453
Examples
dat <- simdat(model = "logistic")
form <- Y - f ~ X1
X_l <- model.matrix(form, data = dat[dat$set_label == "labeled", ])
Y_l <- dat[dat$set_label == "labeled", all.vars(form)[1]] |>
matrix(ncol = 1)
f_l <- dat[dat$set_label == "labeled", all.vars(form)[2]] |>
matrix(ncol = 1)
X_u <- model.matrix(form, data = dat[dat$set_label == "unlabeled", ])
f_u <- dat[dat$set_label == "unlabeled", all.vars(form)[2]] |>
matrix(ncol = 1)
ppi_plusplus_logistic(X_l, Y_l, f_l, X_u, f_u)
#> $est
#> [,1]
#> [1,] 0.4865938
#> [2,] 0.7040716
#>
#> $se
#> [1] 0.1268517 0.1347547
#>
#> $lambda
#> [1] 0.081643
#>
#> $rectifier_est
#> [,1]
#> [1,] -0.006157121
#> [2,] -0.006461035
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.0013050030 -0.0005640268
#> [2,] -0.0005640268 0.0011172474
#>
#> $var_l
#> [,1] [,2]
#> [1,] 0.2178305 -0.0413854
#> [2,] -0.0413854 0.1667002
#>
#> $grads
#> [,1] [,2]
#> [1,] 0.64981379 0.121483408
#> [2,] -0.34865053 -0.068525846
#> [3,] 0.55159384 -0.218955189
#> [4,] -0.67080193 1.141779389
#> [5,] -0.29953318 -0.154400444
#> [6,] 0.56397541 -0.183659285
#> [7,] 0.54753800 -0.230087001
#> [8,] -0.28019681 -0.181827539
#> [9,] -0.36687570 -0.030769545
#> [10,] -0.53069983 0.459451715
#> [11,] -0.28516830 -0.175123236
#> [12,] 0.41170112 -0.493249199
#> [13,] 0.37593880 -0.530434067
#> [14,] 0.50406934 -0.336715636
#> [15,] 0.47235425 -0.400715374
#> [16,] 0.62060624 0.004873018
#> [17,] -0.21294580 -0.248209545
#> [18,] -0.17319888 -0.264822150
#> [19,] -0.47280342 0.253635917
#> [20,] 0.58752019 -0.110875203
#> [21,] 0.27273457 -0.568417965
#> [22,] -0.55328453 0.550512664
#> [23,] 0.44853982 -0.441592210
#> [24,] -0.31170860 -0.135273833
#> [25,] -0.34191820 -0.081665409
#> [26,] -0.33408434 -0.096407560
#> [27,] 0.73882457 0.580583978
#> [28,] -0.38665732 0.013844019
#> [29,] -0.14390616 -0.265018904
#> [30,] -0.65956354 1.075376028
#> [31,] 0.53629900 -0.259851493
#> [32,] 0.36566398 -0.538810162
#> [33,] -0.23376174 -0.232608103
#> [34,] -0.38185370 0.002660886
#> [35,] 0.70804096 0.401545717
#> [36,] -0.40782411 0.065816400
#> [37,] 0.38773758 -0.519551856
#> [38,] -0.19255699 -0.258964580
#> [39,] 0.49004503 -0.366395951
#> [40,] -0.37440244 -0.014241907
#> [41,] -0.37323324 -0.016845275
#> [42,] -0.37737464 -0.007564277
#> [43,] 0.47317933 -0.399190709
#> [44,] -0.27284229 -0.191301755
#> [45,] -0.27444274 -0.189285242
#> [46,] -0.34057269 -0.084239323
#> [47,] -0.45818541 0.207557627
#> [48,] 0.15968778 -0.486986115
#> [49,] -0.23638102 -0.230323878
#> [50,] -0.37260416 -0.018240529
#> [51,] -0.41869881 0.094243217
#> [52,] 0.70125021 0.365195221
#> [53,] 0.43045030 -0.468683339
#> [54,] -0.44430724 0.165900673
#> [55,] -0.18877292 -0.260450470
#> [56,] 0.57769557 -0.142171872
#> [57,] -0.50369016 0.358667240
#> [58,] -0.14569671 -0.265324239
#> [59,] -0.39765028 0.040285390
#> [60,] 0.38480410 -0.522384106
#> [61,] -0.36688923 -0.030740320
#> [62,] -0.30050342 -0.152928717
#> [63,] 0.38868949 -0.518614730
#> [64,] 0.49317236 -0.359969437
#> [65,] -0.11389706 -0.253159685
#> [66,] -0.51348100 0.394210375
#> [67,] 0.41565022 -0.488368515
#> [68,] -0.26336085 -0.202729518
#> [69,] -0.49157438 0.316201073
#> [70,] -0.24103058 -0.226095128
#> [71,] -0.20686027 -0.251897877
#> [72,] -0.25371565 -0.213437745
#> [73,] 0.50809014 -0.327793344
#> [74,] -0.18238363 -0.262586825
#> [75,] -0.39308637 0.029163980
#> [76,] 0.58836060 -0.108134954
#> [77,] 0.42479247 -0.476469698
#> [78,] 0.71688530 0.450522602
#> [79,] -0.14631699 -0.265420010
#> [80,] 0.61443605 -0.017973930
#> [81,] -0.21262401 -0.248414606
#> [82,] -0.39866588 0.042788108
#> [83,] -0.29954920 -0.154376221
#> [84,] -0.36067404 -0.043976254
#> [85,] 0.44982345 -0.439541324
#> [86,] 0.42796296 -0.472146417
#> [87,] -0.08646345 -0.229768931
#> [88,] -0.39857539 0.042564691
#> [89,] 0.71846942 0.459496369
#> [90,] 0.68034657 0.259714369
#> [91,] -0.25121153 -0.216065383
#> [92,] -0.26292888 -0.203228951
#> [93,] -0.33851627 -0.088139620
#> [94,] 0.51939225 -0.301708065
#> [95,] -0.58056018 0.669280865
#> [96,] -0.44273106 0.161296392
#> [97,] 0.65693971 0.152174527
#> [98,] 0.66275415 0.177907030
#> [99,] -0.41407472 0.082011373
#> [100,] -0.55265216 0.547875426
#> [101,] -0.12503347 -0.259100241
#> [102,] -0.56034047 0.580289531
#> [103,] -0.27401322 -0.189828891
#> [104,] -0.38694108 0.014511677
#> [105,] -0.59808372 0.750995038
#> [106,] -0.17385918 -0.264694851
#> [107,] 0.74647724 0.629052748
#> [108,] 0.61408899 -0.019241304
#> [109,] 0.47986005 -0.386573298
#> [110,] 0.72259351 0.483153554
#> [111,] 0.74094662 0.593855268
#> [112,] 0.52849703 -0.279596054
#> [113,] -0.11032917 -0.250846303
#> [114,] 0.69875228 0.352089213
#> [115,] -0.32233253 -0.117419275
#> [116,] 0.47353305 -0.398534817
#> [117,] 0.54100271 -0.247586392
#> [118,] -0.33841607 -0.088328625
#> [119,] -0.32344829 -0.115481251
#> [120,] -0.53239410 0.466063600
#> [121,] -0.43629188 0.142752507
#> [122,] -0.65468810 1.047305767
#> [123,] 0.59608577 -0.082471427
#> [124,] -0.21559247 -0.246480706
#> [125,] 0.59897741 -0.072642164
#> [126,] 0.64683064 0.108905436
#> [127,] 0.66842766 0.203629801
#> [128,] -0.20369820 -0.253655114
#> [129,] 0.24268172 -0.559981640
#> [130,] 0.50257979 -0.339973891
#> [131,] 0.86212376 1.648703378
#> [132,] -0.11388179 -0.253150218
#> [133,] 0.35985977 -0.543091029
#> [134,] 0.80482131 1.063205913
#> [135,] -0.58463148 0.687876332
#> [136,] -0.60770828 0.797780248
#> [137,] -0.25103346 -0.216249836
#> [138,] 0.51091890 -0.321404414
#> [139,] -0.36251580 -0.040092858
#> [140,] -0.42213669 0.103476302
#> [141,] -0.27057984 -0.194109336
#> [142,] 0.72790129 0.514241716
#> [143,] -0.24983776 -0.217480078
#> [144,] -0.33919524 -0.086856331
#> [145,] -0.51551722 0.401742273
#> [146,] 0.32901680 -0.560406785
#> [147,] -0.38961974 0.020852987
#> [148,] -0.15203039 -0.266062943
#> [149,] 0.54470045 -0.237750969
#> [150,] -0.50778685 0.373404410
#> [151,] -0.37879876 -0.004334323
#> [152,] -0.18648518 -0.261269500
#> [153,] -0.17875979 -0.263587726
#> [154,] -0.41962409 0.096716528
#> [155,] 0.48234777 -0.381750449
#> [156,] -0.36991803 -0.024155018
#> [157,] -0.14367857 -0.264977007
#> [158,] -0.35308189 -0.059639648
#> [159,] -0.33434752 -0.095921850
#> [160,] -0.43918904 0.151043120
#> [161,] -0.41833563 0.093274759
#> [162,] 0.80485517 1.063497083
#> [163,] -0.22866073 -0.236852544
#> [164,] -0.29594209 -0.159768508
#> [165,] -0.41778245 0.091802201
#> [166,] -0.27362095 -0.190323819
#> [167,] 0.48224806 -0.381945043
#> [168,] -0.31011931 -0.137851532
#> [169,] 0.31979636 -0.563811579
#> [170,] 0.65440178 0.141138265
#> [171,] -0.60995549 0.808904898
#> [172,] 0.68610723 0.287851823
#> [173,] 0.75741641 0.701370499
#> [174,] -0.29598632 -0.159703164
#> [175,] 0.82147547 1.213165834
#> [176,] 0.56106903 -0.192124283
#> [177,] 0.68121545 0.263914459
#> [178,] -0.23053175 -0.235327127
#> [179,] 0.71561434 0.443367731
#> [180,] 0.80844196 1.094643315
#> [181,] -0.59137209 0.719178308
#> [182,] -0.39792899 0.040971211
#> [183,] 0.38585442 -0.521379686
#> [184,] -0.31714229 -0.126277423
#> [185,] -0.24464094 -0.222658922
#> [186,] 0.58984631 -0.103266010
#> [187,] 0.53023581 -0.275259894
#> [188,] -0.13917820 -0.264004182
#> [189,] -0.44141153 0.157461569
#> [190,] 0.45128365 -0.437187439
#> [191,] 0.41728440 -0.486303226
#> [192,] -0.68369222 1.220990787
#> [193,] -0.57458088 0.642387764
#> [194,] -0.38928001 0.020044848
#> [195,] -0.10214638 -0.244750087
#> [196,] -0.25895505 -0.207736216
#> [197,] 0.49090436 -0.364640956
#> [198,] -0.52899202 0.452822300
#> [199,] 0.74025318 0.589504409
#> [200,] 0.70331693 0.376145698
#> [201,] -0.39554593 0.035132056
#> [202,] -0.67618080 1.174426673
#> [203,] -0.54321127 0.509109327
#> [204,] 0.55609332 -0.206359164
#> [205,] -0.22095903 -0.242745411
#> [206,] 0.74677383 0.630965518
#> [207,] -0.22268499 -0.241479106
#> [208,] -0.24428575 -0.223002865
#> [209,] 0.49154819 -0.363320678
#> [210,] 0.65562502 0.146442876
#> [211,] -0.34723896 -0.071316877
#> [212,] -0.47622592 0.264755847
#> [213,] -0.54508522 0.516714410
#> [214,] -0.34018333 -0.084980904
#> [215,] -0.44525458 0.168680375
#> [216,] -0.68736814 1.244202561
#> [217,] 0.37170464 -0.534009535
#> [218,] 0.25347506 -0.564053298
#> [219,] -0.72551104 1.502973046
#> [220,] 0.51967954 -0.301025454
#> [221,] 0.51476583 -0.312566552
#> [222,] -0.69638565 1.302364973
#> [223,] -0.24794139 -0.219401600
#> [224,] -0.15160669 -0.266029952
#> [225,] 0.60752569 -0.042859366
#> [226,] 0.73923852 0.583162782
#> [227,] -0.28792610 -0.171300248
#> [228,] 0.73639832 0.565565573
#> [229,] -0.19928784 -0.255922252
#> [230,] -0.40721240 0.064252372
#> [231,] -0.39031313 0.022505928
#> [232,] 0.58426208 -0.121403963
#> [233,] -0.36733815 -0.029769869
#> [234,] 0.43240209 -0.465921835
#> [235,] 0.25394992 -0.564205413
#> [236,] 0.31962163 -0.563868226
#> [237,] -0.47075024 0.247025812
#> [238,] -0.31391648 -0.131652536
#> [239,] 0.52782987 -0.281250090
#> [240,] 0.47776262 -0.390586886
#> [241,] -0.36548763 -0.033757718
#> [242,] -0.31066969 -0.136961609
#> [243,] 0.50968035 -0.324213179
#> [244,] 0.52658856 -0.284313290
#> [245,] 0.72355183 0.488712630
#> [246,] 0.69363076 0.325653024
#> [247,] -0.30596844 -0.144469194
#> [248,] 0.55923755 -0.197401520
#> [249,] -0.27324921 -0.190791434
#> [250,] 0.62855486 0.035199335
#> [251,] 0.71570287 0.443864855
#> [252,] -0.35042070 -0.064998759
#> [253,] 0.56657504 -0.175992939
#> [254,] -0.26510535 -0.200693718
#> [255,] 0.33138567 -0.559400397
#> [256,] 0.40757479 -0.498182855
#> [257,] -0.43466018 0.138121047
#> [258,] 0.51394173 -0.314474376
#> [259,] -0.12498879 -0.259080167
#> [260,] -0.50399080 0.359742197
#> [261,] -0.42263219 0.104816874
#> [262,] -0.54295223 0.508061553
#> [263,] -0.52616695 0.441933272
#> [264,] -0.22534902 -0.239462867
#> [265,] 0.53042433 -0.274787562
#> [266,] 0.57791630 -0.141483497
#> [267,] -0.41713013 0.090069666
#> [268,] -0.35535691 -0.055004221
#> [269,] -0.39034161 0.022573915
#> [270,] -0.53610961 0.480686160
#> [271,] -0.40568818 0.060371377
#> [272,] 0.51506095 -0.311881418
#> [273,] -0.36928737 -0.025533521
#> [274,] -0.36262860 -0.039853953
#> [275,] -0.27285122 -0.191290564
#> [276,] 0.48524828 -0.376041440
#> [277,] 0.62867610 0.035669867
#> [278,] -0.42543499 0.112446438
#> [279,] -0.13453565 -0.262708237
#> [280,] -0.42575751 0.113329488
#> [281,] 0.39795080 -0.509033893
#> [282,] -0.57319830 0.636239157
#> [283,] 0.49055524 -0.365354955
#> [284,] -0.45658854 0.202661665
#> [285,] 0.40676747 -0.499128342
#> [286,] -0.41088569 0.073700151
#> [287,] 0.38936244 -0.517946886
#> [288,] -0.49028842 0.311790725
#> [289,] -0.45337911 0.192902835
#> [290,] -0.29528596 -0.160735829
#> [291,] -0.50200962 0.352677462
#> [292,] -0.26626540 -0.199323247
#> [293,] -0.48481313 0.293218511
#> [294,] -0.13982851 -0.264161855
#> [295,] -0.55319116 0.550122950
#> [296,] 0.39246739 -0.514808090
#> [297,] -0.09351397 -0.237064475
#> [298,] -0.16823942 -0.265610292
#> [299,] -0.61397328 0.828987703
#> [300,] 0.36558493 -0.538870648
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] 0.78114738 0.8717910571
#> [2,] 0.36319112 -0.5406737221
#> [3,] -0.38258628 0.0043518486
#> [4,] 0.68812850 0.2978887822
#> [5,] -0.14724436 -0.2655536387
#> [6,] 0.44135589 -0.4527542750
#> [7,] -0.33089892 -0.1022337770
#> [8,] -0.25424883 -0.2128701149
#> [9,] 0.33903353 -0.5557840624
#> [10,] -0.29065159 -0.1674493726
#> [11,] -0.65680298 1.0594284916
#> [12,] -0.63219340 0.9232603850
#> [13,] -0.31667260 -0.1270662840
#> [14,] -0.28986663 -0.1685658504
#> [15,] -0.48650959 0.2989373646
#> [16,] -0.14501763 -0.2652134986
#> [17,] 0.49846424 -0.3488447661
#> [18,] -0.22188969 -0.2420665159
#> [19,] 0.67481590 0.2333385516
#> [20,] 0.61015837 -0.0334650368
#> [21,] 0.36740728 -0.5374609276
#> [22,] -0.33173011 -0.1007229015
#> [23,] -0.34896557 -0.0679003232
#> [24,] -0.44345201 0.1633992109
#> [25,] 0.55198746 -0.2178636740
#> [26,] -0.28115053 -0.1805601185
#> [27,] -0.57068214 0.6251159409
#> [28,] 0.62139175 0.0078247413
#> [29,] 0.49006108 -0.3663632396
#> [30,] 0.57654514 -0.1457488054
#> [31,] -0.43383766 0.1357966931
#> [32,] -0.10443327 -0.2465671743
#> [33,] 0.37265445 -0.5332225874
#> [34,] -0.49046908 0.3124092046
#> [35,] 0.50537553 -0.3338375423
#> [36,] 0.37660016 -0.5298598913
#> [37,] -0.29400192 -0.1626167944
#> [38,] 0.40652201 -0.4994145167
#> [39,] -0.12680495 -0.2598723853
#> [40,] -0.36278426 -0.0395240758
#> [41,] -0.25515988 -0.2118935919
#> [42,] -0.35664888 -0.0523496432
#> [43,] -0.53402740 0.4724706918
#> [44,] -0.20310542 -0.2539723273
#> [45,] -0.47399884 0.2575054934
#> [46,] -0.30371346 -0.1479946725
#> [47,] -0.41685345 0.0893361139
#> [48,] -0.66274322 1.0939203594
#> [49,] -0.69348242 1.2834468202
#> [50,] -0.30543822 -0.1453025588
#> [51,] -0.30882461 -0.1399334521
#> [52,] 0.30502612 -0.5675640842
#> [53,] -0.50771923 0.3731595983
#> [54,] -0.20862279 -0.2508709804
#> [55,] -0.30774262 -0.1416609418
#> [56,] 0.65305561 0.1353319638
#> [57,] 0.61043096 -0.0324862701
#> [58,] -0.29673195 -0.1585985064
#> [59,] -0.59944728 0.7575395361
#> [60,] -0.40994145 0.0712587405
#> [61,] -0.08242722 -0.2251558889
#> [62,] 0.15351001 -0.4783457339
#> [63,] 0.41866911 -0.4845322647
#> [64,] -0.46675016 0.2342780304
#> [65,] -0.34798904 -0.0698361571
#> [66,] -0.32073259 -0.1201774645
#> [67,] -0.28188470 -0.1795784027
#> [68,] 0.63867242 0.0753020589
#> [69,] -0.24447191 -0.2228227624
#> [70,] -0.28912873 -0.1696099402
#> [71,] -0.30692905 -0.1429524627
#> [72,] -0.29664424 -0.1587287359
#> [73,] -0.50564155 0.3656629962
#> [74,] -0.59552805 0.7388024976
#> [75,] -0.48907527 0.3076470073
#> [76,] 0.63816229 0.0732389294
#> [77,] -0.57550854 0.6465278516
#> [78,] -0.31828601 -0.1243476565
#> [79,] -0.32256348 -0.1170191058
#> [80,] -0.19586746 -0.2575317419
#> [81,] -0.33176587 -0.1006577407
#> [82,] -0.17551510 -0.2643526792
#> [83,] 0.57938077 -0.1368992487
#> [84,] 0.61331081 -0.0220762510
#> [85,] 0.28833435 -0.5692718793
#> [86,] 0.54347714 -0.2410237569
#> [87,] -0.13320048 -0.2622795970
#> [88,] -0.18565955 -0.2615503129
#> [89,] -0.35566344 -0.0543758618
#> [90,] 0.59157672 -0.0975552933
#> [91,] -0.22974691 -0.2359714365
#> [92,] 0.39635761 -0.5107420253
#> [93,] -0.43243477 0.1318482550
#> [94,] -0.28091783 -0.1808701758
#> [95,] -0.39952138 0.0449041668
#> [96,] -0.68865110 1.2523708916
#> [97,] 0.66456997 0.1860729150
#> [98,] -0.37732048 -0.0076867297
#> [99,] -0.56029533 0.5800969833
#> [100,] -0.39275756 0.0283706267
#> [101,] 0.55316466 -0.2145873974
#> [102,] -0.64075796 0.9694566627
#> [103,] -0.24563191 -0.2216925187
#> [104,] -0.46367424 0.2245919407
#> [105,] -0.25408549 -0.2130443127
#> [106,] -0.41943759 0.0962173127
#> [107,] 0.60746364 -0.0430795155
#> [108,] -0.34451414 -0.0766504104
#> [109,] -0.55013001 0.5374081406
#> [110,] -0.39507968 0.0339961398
#> [111,] 0.51334126 -0.3158594758
#> [112,] 0.48703882 -0.3724707935
#> [113,] -0.19946487 -0.2558353997
#> [114,] -0.52415202 0.4342255440
#> [115,] -0.19000968 -0.2599827601
#> [116,] 0.57069555 -0.1636564453
#> [117,] 0.29460351 -0.5689467920
#> [118,] 0.68234919 0.2694181400
#> [119,] -0.22777343 -0.2375631762
#> [120,] 0.48103926 -0.3842956675
#> [121,] -0.45700461 0.2039347172
#> [122,] 0.66286210 0.1783907430
#> [123,] -0.34718260 -0.0714279119
#> [124,] -0.20667188 -0.2520056302
#> [125,] 0.70048902 0.3611865432
#> [126,] 0.54979043 -0.2239307657
#> [127,] -0.21467333 -0.2470896270
#> [128,] -0.45190184 0.1884472474
#> [129,] 0.56249587 -0.1879824508
#> [130,] 0.74566685 0.6238394309
#> [131,] -0.17233183 -0.2649813635
#> [132,] -0.54231174 0.5054744188
#> [133,] 0.36966299 -0.5356715322
#> [134,] -0.53536511 0.4777425943
#> [135,] 0.77259786 0.8081219173
#> [136,] 0.41657863 -0.4871984634
#> [137,] 0.47515388 -0.3955120159
#> [138,] -0.49099157 0.3141999629
#> [139,] -0.49613894 0.3320052651
#> [140,] 0.29073956 -0.5691922700
#> [141,] -0.29720040 -0.1579017532
#> [142,] -0.10940435 -0.2502130693
#> [143,] -0.32094076 -0.1198199777
#> [144,] -0.57845638 0.6597623608
#> [145,] -0.58883295 0.7073110710
#> [146,] -0.18746867 -0.2609247624
#> [147,] -0.46694702 0.2349013833
#> [148,] -0.34630979 -0.0731436236
#> [149,] -0.47789452 0.2702233582
#> [150,] -0.55909699 0.5749948732
#> [151,] -0.33686301 -0.0912458628
#> [152,] -0.32105038 -0.1196315666
#> [153,] -0.43942509 0.1517223993
#> [154,] -0.18180283 -0.2627575793
#> [155,] -0.62452376 0.8829264071
#> [156,] 0.61445150 -0.0179174668
#> [157,] -0.36972788 -0.0245710603
#> [158,] 0.40565232 -0.5004236950
#> [159,] 0.51221299 -0.3184506781
#> [160,] -0.13504522 -0.2628651946
#> [161,] 0.51733448 -0.3065689084
#> [162,] -0.54384135 0.5116614875
#> [163,] -0.33233435 -0.0996203792
#> [164,] 0.64209876 0.0892747655
#> [165,] 0.23408846 -0.5558906982
#> [166,] 0.29240526 -0.5691042072
#> [167,] 0.39659669 -0.5104872927
#> [168,] 0.56228744 -0.1885891580
#> [169,] -0.34394419 -0.0777570063
#> [170,] 0.41490491 -0.4893015747
#> [171,] 0.46889335 -0.4070308133
#> [172,] -0.27432440 -0.1894352048
#> [173,] -0.42323683 0.1064560643
#> [174,] -0.25472848 -0.2123570347
#> [175,] -0.38653526 0.0135570579
#> [176,] -0.11956676 -0.2564217553
#> [177,] -0.27708141 -0.1859056429
#> [178,] -0.63409077 0.9333880740
#> [179,] -0.55731405 0.5674384849
#> [180,] -0.48903315 0.3075034450
#> [181,] -0.63683867 0.9481626357
#> [182,] -0.43318686 0.1339625107
#> [183,] -0.54124081 0.5011600486
#> [184,] -0.66044105 1.0804749274
#> [185,] -0.48696773 0.3004872582
#> [186,] 0.51023365 -0.3229606198
#> [187,] 0.37414724 -0.5319681198
#> [188,] 0.57158590 -0.1609607768
#> [189,] -0.47405175 0.2576771255
#> [190,] -0.24435361 -0.2229372564
#> [191,] -0.53638249 0.4817667757
#> [192,] -0.47216037 0.2515607676
#> [193,] 0.45200497 -0.4360164286
#> [194,] 0.62565473 0.0240162203
#> [195,] -0.42117672 0.1008861464
#> [196,] -0.57861084 0.6604591281
#> [197,] -0.29185768 -0.1657222308
#> [198,] -0.13584513 -0.2631042242
#> [199,] 0.43892878 -0.4564048885
#> [200,] -0.12703769 -0.2599703936
#> [201,] -0.38033407 -0.0008301041
#> [202,] -0.28746981 -0.1719378858
#> [203,] -0.50317276 0.3568196809
#> [204,] -0.48357775 0.2890739676
#> [205,] 0.55277525 -0.2156731403
#> [206,] -0.25415452 -0.2129707278
#> [207,] 0.49989777 -0.3457767278
#> [208,] 0.51220515 -0.3184686344
#> [209,] -0.44242687 0.1604107590
#> [210,] 0.49711599 -0.3517090658
#> [211,] 0.53959966 -0.2512735948
#> [212,] -0.33067426 -0.1026410208
#> [213,] -0.75320612 1.7142032468
#> [214,] -0.34351779 -0.0785828691
#> [215,] -0.33281992 -0.0987318611
#> [216,] 0.65949695 0.1634144385
#> [217,] 0.59211397 -0.0957735054
#> [218,] -0.51669601 0.4061247973
#> [219,] 0.62636895 0.0267575967
#> [220,] -0.42970363 0.1242189241
#> [221,] -0.52627099 0.4423325816
#> [222,] -0.21221843 -0.2486714809
#> [223,] -0.11427178 -0.2533908325
#> [224,] -0.69711588 1.3071525052
#> [225,] -0.27098945 -0.1936047670
#> [226,] -0.51586000 0.4030150010
#> [227,] -0.41077458 0.0734124153
#> [228,] -0.22078711 -0.2428698253
#> [229,] -0.47229416 0.2519921287
#> [230,] -0.31311519 -0.1329722168
#> [231,] 0.74112100 0.5949515095
#> [232,] -0.44933497 0.1807596366
#> [233,] -0.20584191 -0.2524757546
#> [234,] -0.24105965 -0.2260679907
#> [235,] 0.50385759 -0.3371803496
#> [236,] -0.32229741 -0.1174800937
#> [237,] -0.44478351 0.1672969880
#> [238,] -0.76020150 1.7711523237
#> [239,] -0.30540808 -0.1453498458
#> [240,] -0.22946154 -0.2362041204
#> [241,] 0.39440897 -0.5127973264
#> [242,] -0.38772814 0.0163676294
#> [243,] 0.63175868 0.0477137311
#> [244,] -0.27461863 -0.1890620829
#> [245,] -0.39099590 0.0241381395
#> [246,] -0.27707300 -0.1859165249
#> [247,] -0.28405803 -0.1766414175
#> [248,] -0.57279205 0.6344374055
#> [249,] -0.54996915 0.5367432750
#> [250,] -0.54851854 0.5307627798
#> [251,] -0.23947934 -0.2275306322
#> [252,] 0.73605023 0.5634243873
#> [253,] 0.59983628 -0.0696990012
#> [254,] -0.39780097 0.0406560982
#> [255,] 0.48129007 -0.3838092490
#> [256,] -0.71677975 1.4406800490
#> [257,] -0.29353683 -0.1632941678
#> [258,] -0.38439508 0.0085493604
#> [259,] -0.33763005 -0.0898079646
#> [260,] 0.63937143 0.0781363411
#> [261,] 0.46611487 -0.4120078675
#> [262,] -0.43415420 0.1366903791
#> [263,] 0.48951157 -0.3674812707
#> [264,] -0.34281865 -0.0799331812
#> [265,] -0.35293470 -0.0599378392
#> [266,] -0.45492636 0.1975939991
#> [267,] -0.60901748 0.8042519832
#> [268,] -0.31491435 -0.1300004880
#> [269,] -0.45138805 0.1869029860
#> [270,] -0.26846798 -0.1966844651
#> [271,] -0.29601659 -0.1596584144
#> [272,] 0.64122581 0.0856957338
#> [273,] -0.41820287 0.0929210620
#> [274,] -0.28278773 -0.1783636700
#> [275,] -0.39453418 0.0326698533
#> [276,] 0.26571521 -0.5672539432
#> [277,] -0.49862680 0.3407179681
#> [278,] 0.67393993 0.2292173040
#> [279,] 0.56550373 -0.1791631221
#> [280,] -0.19609114 -0.2574304854
#> [281,] -0.55282113 0.5485795940
#> [282,] 0.43694541 -0.4593431022
#> [283,] -0.37648006 -0.0095831511
#> [284,] -0.20633261 -0.2521987182
#> [285,] 0.66453525 0.1859161890
#> [286,] -0.46634919 0.2330096326
#> [287,] -0.36913187 -0.0258728215
#> [288,] -0.57678462 0.6522422359
#> [289,] -0.28716929 -0.1723567492
#> [290,] -0.32740706 -0.1085087103
#> [291,] -0.53563721 0.4788176147
#> [292,] -0.23563671 -0.2309801729
#> [293,] -0.07721231 -0.2186998766
#> [294,] 0.65443669 0.1412892605
#> [295,] -0.59865474 0.7537323352
#> [296,] 0.60001493 -0.0690854607
#> [297,] -0.25516262 -0.2118906508
#> [298,] -0.25169724 -0.2155606547
#> [299,] 0.57409621 -0.1533025729
#> [300,] -0.26330764 -0.2027911363
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] -0.35018621 -0.065467702
#> [2,] -0.34865053 -0.068525846
#> [3,] 0.55159384 -0.218955189
#> [4,] -0.67080193 1.141779389
#> [5,] -0.29953318 -0.154400444
#> [6,] -0.43602459 0.141991943
#> [7,] 0.54753800 -0.230087001
#> [8,] -0.28019681 -0.181827539
#> [9,] -0.36687570 -0.030769545
#> [10,] -0.53069983 0.459451715
#> [11,] -0.28516830 -0.175123236
#> [12,] 0.41170112 -0.493249199
#> [13,] -0.62406120 0.880524506
#> [14,] -0.49593066 0.331279047
#> [15,] 0.47235425 -0.400715374
#> [16,] 0.62060624 0.004873018
#> [17,] -0.21294580 -0.248209545
#> [18,] -0.17319888 -0.264822150
#> [19,] -0.47280342 0.253635917
#> [20,] 0.58752019 -0.110875203
#> [21,] 0.27273457 -0.568417965
#> [22,] 0.44671547 -0.444477496
#> [23,] -0.55146018 0.542918403
#> [24,] -0.31170860 -0.135273833
#> [25,] 0.65808180 0.157179463
#> [26,] -0.33408434 -0.096407560
#> [27,] 0.73882457 0.580583978
#> [28,] -0.38665732 0.013844019
#> [29,] -0.14390616 -0.265018904
#> [30,] -0.65956354 1.075376028
#> [31,] -0.46370100 0.224675780
#> [32,] 0.36566398 -0.538810162
#> [33,] -0.23376174 -0.232608103
#> [34,] -0.38185370 0.002660886
#> [35,] -0.29195904 -0.165576440
#> [36,] -0.40782411 0.065816400
#> [37,] -0.61226242 0.820405591
#> [38,] -0.19255699 -0.258964580
#> [39,] -0.50995497 0.381282181
#> [40,] 0.62559756 0.023797126
#> [41,] -0.37323324 -0.016845275
#> [42,] 0.62262536 0.012480199
#> [43,] 0.47317933 -0.399190709
#> [44,] 0.72715771 0.509842328
#> [45,] -0.27444274 -0.189285242
#> [46,] -0.34057269 -0.084239323
#> [47,] -0.45818541 0.207557627
#> [48,] -0.84031222 2.562627966
#> [49,] -0.23638102 -0.230323878
#> [50,] -0.37260416 -0.018240529
#> [51,] 0.58130119 -0.130842726
#> [52,] -0.29874979 -0.155582121
#> [53,] 0.43045030 -0.468683339
#> [54,] -0.44430724 0.165900673
#> [55,] -0.18877292 -0.260450470
#> [56,] -0.42230443 0.103929847
#> [57,] -0.50369016 0.358667240
#> [58,] -0.14569671 -0.265324239
#> [59,] -0.39765028 0.040285390
#> [60,] -0.61519590 0.835148469
#> [61,] -0.36688923 -0.030740320
#> [62,] 0.69949658 0.355979698
#> [63,] -0.61131051 0.815650136
#> [64,] 0.49317236 -0.359969437
#> [65,] -0.11389706 -0.253159685
#> [66,] -0.51348100 0.394210375
#> [67,] -0.58434978 0.686582168
#> [68,] -0.26336085 -0.202729518
#> [69,] 0.50842562 -0.327040492
#> [70,] -0.24103058 -0.226095128
#> [71,] -0.20686027 -0.251897877
#> [72,] -0.25371565 -0.213437745
#> [73,] 0.50809014 -0.327793344
#> [74,] -0.18238363 -0.262586825
#> [75,] -0.39308637 0.029163980
#> [76,] -0.41163940 0.075655318
#> [77,] 0.42479247 -0.476469698
#> [78,] 0.71688530 0.450522602
#> [79,] -0.14631699 -0.265420010
#> [80,] -0.38556395 0.011278797
#> [81,] -0.21262401 -0.248414606
#> [82,] -0.39866588 0.042788108
#> [83,] -0.29954920 -0.154376221
#> [84,] -0.36067404 -0.043976254
#> [85,] 0.44982345 -0.439541324
#> [86,] 0.42796296 -0.472146417
#> [87,] -0.08646345 -0.229768931
#> [88,] -0.39857539 0.042564691
#> [89,] -0.28153058 -0.180052586
#> [90,] -0.31965343 -0.122023970
#> [91,] -0.25121153 -0.216065383
#> [92,] -0.26292888 -0.203228951
#> [93,] 0.66148373 0.172230789
#> [94,] -0.48060775 0.279178658
#> [95,] -0.58056018 0.669280865
#> [96,] 0.55726894 -0.203024990
#> [97,] 0.65693971 0.152174527
#> [98,] 0.66275415 0.177907030
#> [99,] 0.58592528 -0.116047985
#> [100,] 0.44734784 -0.443481287
#> [101,] -0.12503347 -0.259100241
#> [102,] -0.56034047 0.580289531
#> [103,] -0.27401322 -0.189828891
#> [104,] -0.38694108 0.014511677
#> [105,] -0.59808372 0.750995038
#> [106,] -0.17385918 -0.264694851
#> [107,] -0.25352276 -0.213642394
#> [108,] -0.38591101 0.012091783
#> [109,] -0.52013995 0.419022628
#> [110,] 0.72259351 0.483153554
#> [111,] 0.74094662 0.593855268
#> [112,] -0.47150297 0.249443921
#> [113,] -0.11032917 -0.250846303
#> [114,] -0.30124772 -0.151793529
#> [115,] 0.67766747 0.246860662
#> [116,] -0.52646695 0.443085042
#> [117,] -0.45899729 0.210057143
#> [118,] -0.33841607 -0.088328625
#> [119,] 0.67655171 0.241550314
#> [120,] 0.46760590 -0.409347298
#> [121,] 0.56370812 -0.184442457
#> [122,] -0.65468810 1.047305767
#> [123,] -0.40391423 0.055883541
#> [124,] -0.21559247 -0.246480706
#> [125,] -0.40102259 0.048634803
#> [126,] -0.35316936 -0.059462340
#> [127,] -0.33157234 -0.101010195
#> [128,] -0.20369820 -0.253655114
#> [129,] 0.24268172 -0.559981640
#> [130,] 0.50257979 -0.339973891
#> [131,] -0.13787624 -0.263670990
#> [132,] -0.11388179 -0.253150218
#> [133,] -0.64014023 0.966083019
#> [134,] 0.80482131 1.063205913
#> [135,] 0.41536852 -0.488721838
#> [136,] 0.39229172 -0.514988193
#> [137,] 0.74896654 0.645188462
#> [138,] -0.48908110 0.307666876
#> [139,] -0.36251580 -0.040092858
#> [140,] 0.57786331 -0.141648806
#> [141,] -0.27057984 -0.194109336
#> [142,] -0.27209871 -0.192230059
#> [143,] -0.24983776 -0.217480078
#> [144,] -0.33919524 -0.086856331
#> [145,] 0.48448278 -0.377557155
#> [146,] 0.32901680 -0.560406785
#> [147,] -0.38961974 0.020852987
#> [148,] -0.15203039 -0.266062943
#> [149,] 0.54470045 -0.237750969
#> [150,] 0.49221315 -0.361952190
#> [151,] -0.37879876 -0.004334323
#> [152,] -0.18648518 -0.261269500
#> [153,] -0.17875979 -0.263587726
#> [154,] 0.58037591 -0.133767210
#> [155,] -0.51765223 0.409691902
#> [156,] 0.63008197 0.041143281
#> [157,] -0.14367857 -0.264977007
#> [158,] 0.64691811 0.109272012
#> [159,] -0.33434752 -0.095921850
#> [160,] -0.43918904 0.151043120
#> [161,] 0.58166437 -0.129691569
#> [162,] 0.80485517 1.063497083
#> [163,] -0.22866073 -0.236852544
#> [164,] -0.29594209 -0.159768508
#> [165,] 0.58221755 -0.127934652
#> [166,] -0.27362095 -0.190323819
#> [167,] 0.48224806 -0.381945043
#> [168,] -0.31011931 -0.137851532
#> [169,] 0.31979636 -0.563811579
#> [170,] -0.34559822 -0.074536980
#> [171,] -0.60995549 0.808904898
#> [172,] -0.31389277 -0.131691671
#> [173,] -0.24258359 -0.224633333
#> [174,] 0.70401368 0.379859496
#> [175,] -0.17852453 -0.263647377
#> [176,] 0.56106903 -0.192124283
#> [177,] -0.31878455 -0.123502561
#> [178,] -0.23053175 -0.235327127
#> [179,] -0.28438566 -0.176194660
#> [180,] -0.19155804 -0.259372646
#> [181,] -0.59137209 0.719178308
#> [182,] 0.60207101 -0.061989900
#> [183,] -0.61414558 0.829854507
#> [184,] -0.31714229 -0.126277423
#> [185,] -0.24464094 -0.222658922
#> [186,] -0.41015369 0.071806731
#> [187,] 0.53023581 -0.275259894
#> [188,] -0.13917820 -0.264004182
#> [189,] -0.44141153 0.157461569
#> [190,] 0.45128365 -0.437187439
#> [191,] -0.58271560 0.679096748
#> [192,] -0.68369222 1.220990787
#> [193,] -0.57458088 0.642387764
#> [194,] -0.38928001 0.020044848
#> [195,] -0.10214638 -0.244750087
#> [196,] -0.25895505 -0.207736216
#> [197,] 0.49090436 -0.364640956
#> [198,] -0.52899202 0.452822300
#> [199,] 0.74025318 0.589504409
#> [200,] -0.29668307 -0.158671087
#> [201,] -0.39554593 0.035132056
#> [202,] 0.32381920 -0.562426369
#> [203,] -0.54321127 0.509109327
#> [204,] 0.55609332 -0.206359164
#> [205,] -0.22095903 -0.242745411
#> [206,] -0.25322617 -0.213956325
#> [207,] -0.22268499 -0.241479106
#> [208,] 0.75571425 0.689874210
#> [209,] -0.50845181 0.375814748
#> [210,] 0.65562502 0.146442876
#> [211,] -0.34723896 -0.071316877
#> [212,] -0.47622592 0.264755847
#> [213,] 0.45491478 -0.431237203
#> [214,] -0.34018333 -0.084980904
#> [215,] 0.55474542 -0.210159919
#> [216,] -0.68736814 1.244202561
#> [217,] 0.37170464 -0.534009535
#> [218,] -0.74652494 1.661227957
#> [219,] -0.72551104 1.502973046
#> [220,] -0.48032046 0.278226622
#> [221,] -0.48523417 0.294634885
#> [222,] 0.30361435 -0.567812804
#> [223,] -0.24794139 -0.219401600
#> [224,] -0.15160669 -0.266029952
#> [225,] -0.39247431 0.027688047
#> [226,] -0.26076148 -0.205706801
#> [227,] -0.28792610 -0.171300248
#> [228,] -0.26360168 -0.202450269
#> [229,] -0.19928784 -0.255922252
#> [230,] -0.40721240 0.064252372
#> [231,] -0.39031313 0.022505928
#> [232,] -0.41573792 0.086386286
#> [233,] 0.63266185 0.051272271
#> [234,] -0.56759791 0.611598019
#> [235,] 0.25394992 -0.564205413
#> [236,] 0.31962163 -0.563868226
#> [237,] 0.52924976 -0.277723389
#> [238,] -0.31391648 -0.131652536
#> [239,] -0.47217013 0.251592225
#> [240,] -0.52223738 0.426946491
#> [241,] -0.36548763 -0.033757718
#> [242,] -0.31066969 -0.136961609
#> [243,] -0.49031965 0.311897631
#> [244,] -0.47341144 0.255602139
#> [245,] -0.27644817 -0.186722924
#> [246,] -0.30636924 -0.143837436
#> [247,] -0.30596844 -0.144469194
#> [248,] 0.55923755 -0.197401520
#> [249,] -0.27324921 -0.190791434
#> [250,] 0.62855486 0.035199335
#> [251,] -0.28429713 -0.176315489
#> [252,] -0.35042070 -0.064998759
#> [253,] -0.43342496 0.134633062
#> [254,] 0.73489465 0.556340101
#> [255,] 0.33138567 -0.559400397
#> [256,] -0.59242521 0.724127417
#> [257,] -0.43466018 0.138121047
#> [258,] -0.48605827 0.297412845
#> [259,] -0.12498879 -0.259080167
#> [260,] -0.50399080 0.359742197
#> [261,] 0.57736781 -0.143192804
#> [262,] -0.54295223 0.508061553
#> [263,] -0.52616695 0.441933272
#> [264,] -0.22534902 -0.239462867
#> [265,] 0.53042433 -0.274787562
#> [266,] -0.42208370 0.103333093
#> [267,] 0.58286987 -0.125857354
#> [268,] -0.35535691 -0.055004221
#> [269,] -0.39034161 0.022573915
#> [270,] -0.53610961 0.480686160
#> [271,] -0.40568818 0.060371377
#> [272,] 0.51506095 -0.311881418
#> [273,] 0.63071263 0.043609165
#> [274,] -0.36262860 -0.039853953
#> [275,] -0.27285122 -0.191290564
#> [276,] 0.48524828 -0.376041440
#> [277,] -0.37132390 -0.021068200
#> [278,] 0.57456501 -0.151862894
#> [279,] -0.13453565 -0.262708237
#> [280,] 0.57424249 -0.152853689
#> [281,] -0.60204920 0.770103871
#> [282,] -0.57319830 0.636239157
#> [283,] 0.49055524 -0.365354955
#> [284,] 0.54341146 -0.241198938
#> [285,] -0.59323253 0.727932284
#> [286,] 0.58911431 -0.105668839
#> [287,] -0.61063756 0.812296691
#> [288,] -0.49028842 0.311790725
#> [289,] -0.45337911 0.192902835
#> [290,] 0.70471404 0.383603722
#> [291,] -0.50200962 0.352677462
#> [292,] -0.26626540 -0.199323247
#> [293,] 0.51518687 -0.311588767
#> [294,] -0.13982851 -0.264161855
#> [295,] -0.55319116 0.550122950
#> [296,] 0.39246739 -0.514808090
#> [297,] -0.09351397 -0.237064475
#> [298,] -0.16823942 -0.265610292
#> [299,] 0.38602672 -0.521213889
#> [300,] -0.63441507 0.935125158
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 4.809733 1.142218
#> [2,] 1.142218 5.838868
#>