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 doi:10.48550/arXiv.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) doi:10.48550/arXiv.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,] 0.4867633
#> [2,] 0.7831499
#>
#> $se
#> [1] 0.1287374 0.1380527
#>
#> $lambda
#> [1] 0.09444776
#>
#> $rectifier_est
#> [,1]
#> [1,] -0.02243142
#> [2,] 0.02812663
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.0014843434 -0.0005694134
#> [2,] -0.0005694134 0.0009791453
#>
#> $var_l
#> [,1] [,2]
#> [1,] 0.209228546 -0.003635673
#> [2,] -0.003635673 0.141135168
#>
#> $grads
#> [,1] [,2]
#> [1,] -0.16720832 -0.238867804
#> [2,] -0.44081164 0.140095311
#> [3,] 0.47130316 -0.362091929
#> [4,] -0.31116082 -0.122348484
#> [5,] -0.28883430 -0.152794344
#> [6,] -0.21026140 -0.224608348
#> [7,] 0.72476840 0.445583352
#> [8,] -0.52491663 0.393117779
#> [9,] 0.40998966 -0.445390471
#> [10,] 0.46747278 -0.368329219
#> [11,] -0.44392837 0.148247895
#> [12,] -0.17551201 -0.237622398
#> [13,] -0.27678238 -0.167421414
#> [14,] 0.59966885 -0.063307529
#> [15,] -0.60863221 0.721460063
#> [16,] 0.32109234 -0.506564481
#> [17,] 0.24943438 -0.505905983
#> [18,] 0.72164947 0.429308505
#> [19,] -0.45506480 0.178118986
#> [20,] 0.69796502 0.312698774
#> [21,] 0.51790029 -0.274528215
#> [22,] -0.34844334 -0.061900450
#> [23,] -0.18238969 -0.236031141
#> [24,] 0.46039219 -0.379487619
#> [25,] -0.43354375 0.121433335
#> [26,] 0.55999630 -0.175629456
#> [27,] -0.58682348 0.627632934
#> [28,] -0.34686794 -0.064697277
#> [29,] -0.57312097 0.571810711
#> [30,] 0.71078672 0.374334778
#> [31,] 0.25526151 -0.507657390
#> [32,] 0.57939337 -0.123165722
#> [33,] 0.41300581 -0.442098792
#> [34,] 0.76600335 0.683809815
#> [35,] 0.21855748 -0.491410696
#> [36,] -0.09152391 -0.211341801
#> [37,] -0.22394216 -0.216200360
#> [38,] -0.31227564 -0.120714937
#> [39,] -0.32508180 -0.101180039
#> [40,] -0.46124384 0.195197235
#> [41,] -0.44662265 0.155368246
#> [42,] -0.30435968 -0.132081321
#> [43,] -0.23558674 -0.207645498
#> [44,] 0.62302495 0.012445606
#> [45,] -0.09753506 -0.216473276
#> [46,] -0.33865647 -0.078928571
#> [47,] -0.39754535 0.036071689
#> [48,] -0.22886210 -0.212740907
#> [49,] -0.47943357 0.247599414
#> [50,] -0.28633827 -0.155928302
#> [51,] 0.58981795 -0.093050221
#> [52,] 0.48896836 -0.331471482
#> [53,] -0.52665914 0.399122537
#> [54,] -0.40195856 0.045912150
#> [55,] -0.20529629 -0.227211852
#> [56,] -0.11833712 -0.229906175
#> [57,] 0.36004718 -0.488210853
#> [58,] -0.48866734 0.275438880
#> [59,] -0.33175270 -0.090443649
#> [60,] -0.42100440 0.090373213
#> [61,] 0.35539978 -0.491088616
#> [62,] -0.45937076 0.189981912
#> [63,] -0.33840571 -0.079354011
#> [64,] -0.57706418 0.587635384
#> [65,] -0.26710121 -0.178243723
#> [66,] 0.80440016 0.952423317
#> [67,] -0.49000809 0.279551694
#> [68,] -0.32541581 -0.100651587
#> [69,] -0.52682266 0.399687720
#> [70,] -0.45161335 0.168736746
#> [71,] 0.50033723 -0.310120566
#> [72,] 0.49453374 -0.321182843
#> [73,] -0.23418449 -0.208742785
#> [74,] 0.60753271 -0.038640735
#> [75,] 0.78184714 0.788386306
#> [76,] 0.20502205 -0.482209684
#> [77,] -0.37576757 -0.009974738
#> [78,] -0.35160330 -0.056225949
#> [79,] 0.49916330 -0.312385890
#> [80,] 0.42584157 -0.427174542
#> [81,] 0.78657102 0.821187347
#> [82,] -0.62079529 0.776589570
#> [83,] 0.89661248 1.915814742
#> [84,] -0.32772171 -0.096977138
#> [85,] -0.30330398 -0.133556155
#> [86,] -0.28704393 -0.155047826
#> [87,] 0.21862929 -0.491454793
#> [88,] 0.53807288 -0.229600077
#> [89,] -0.35597612 -0.048231163
#> [90,] 0.49717671 -0.316187408
#> [91,] -0.61132313 0.733479107
#> [92,] -0.14013209 -0.237522284
#> [93,] 0.47765668 -0.351432014
#> [94,] -0.42819836 0.108017477
#> [95,] 0.73711923 0.512294692
#> [96,] 0.75223914 0.599204726
#> [97,] -0.31231978 -0.120650028
#> [98,] 0.75947124 0.642973025
#> [99,] -0.18262590 -0.235967630
#> [100,] 0.43357230 -0.417464167
#> [101,] -0.35333890 -0.053072542
#> [102,] 0.58142745 -0.117397147
#> [103,] -0.41781370 0.082697651
#> [104,] -0.39974058 0.040944982
#> [105,] -0.42824499 0.108133395
#> [106,] 0.72304140 0.436544035
#> [107,] -0.17681295 -0.237359891
#> [108,] 0.41956611 -0.434657018
#> [109,] 0.83152989 1.178301469
#> [110,] -0.21997805 -0.218821110
#> [111,] 0.80239975 0.937078463
#> [112,] 0.74929781 0.581821717
#> [113,] 0.23509022 -0.500274672
#> [114,] 0.61382024 -0.018312417
#> [115,] -0.37871084 -0.003994503
#> [116,] -0.37787533 -0.005699808
#> [117,] -0.44292032 0.145601232
#> [118,] 0.67970233 0.230551960
#> [119,] -0.58814170 0.633127778
#> [120,] -0.20560658 -0.227056232
#> [121,] 0.68070561 0.234901138
#> [122,] -0.30378982 -0.132878629
#> [123,] -0.13850433 -0.237164086
#> [124,] -0.42580137 0.102086253
#> [125,] 0.54339844 -0.216992782
#> [126,] 0.65553893 0.131177543
#> [127,] 0.63003157 0.036675608
#> [128,] -0.30721430 -0.128044969
#> [129,] 0.51054180 -0.289831762
#> [130,] 0.29400656 -0.511603853
#> [131,] -0.44415126 0.148834369
#> [132,] 0.39855494 -0.457132460
#> [133,] -0.54228186 0.454443319
#> [134,] 0.79836630 0.906631487
#> [135,] -0.11527526 -0.228326521
#> [136,] -0.31640384 -0.114572294
#> [137,] -0.18203414 -0.236125636
#> [138,] -0.49568764 0.297174282
#> [139,] -0.08754039 -0.207606548
#> [140,] -0.14471235 -0.238355794
#> [141,] -0.13528364 -0.236358111
#> [142,] 0.59120535 -0.088937731
#> [143,] 0.47797643 -0.350885113
#> [144,] 0.46161825 -0.377589778
#> [145,] 0.45391328 -0.389279271
#> [146,] 0.45905882 -0.381535294
#> [147,] -0.44726062 0.157064190
#> [148,] 0.45894336 -0.381711823
#> [149,] -0.65783443 0.957937655
#> [150,] 0.84745686 1.328866755
#> [151,] -0.40041744 0.042456178
#> [152,] 0.68330743 0.246266527
#> [153,] -0.64143106 0.875016544
#> [154,] 0.60634268 -0.042427186
#> [155,] 0.69100549 0.280638345
#> [156,] -0.43016286 0.112917596
#> [157,] 0.64096935 0.075960868
#> [158,] 0.47639568 -0.353579036
#> [159,] -0.30074226 -0.137094750
#> [160,] 0.63490797 0.053966817
#> [161,] -0.48149290 0.253735132
#> [162,] -0.30453657 -0.131833259
#> [163,] -0.37569140 -0.010128502
#> [164,] -0.40645334 0.056112306
#> [165,] 0.54388553 -0.215823807
#> [166,] -0.20994590 -0.224780951
#> [167,] -0.26379247 -0.181751598
#> [168,] -0.45893547 0.188774713
#> [169,] -0.59998260 0.683493720
#> [170,] 0.42061594 -0.433430109
#> [171,] 0.53881707 -0.227857389
#> [172,] -0.35746040 -0.045479878
#> [173,] -0.68973343 1.132282122
#> [174,] -0.43681337 0.129768579
#> [175,] 0.70242965 0.333775269
#> [176,] -0.13150476 -0.235245337
#> [177,] -0.30907787 -0.125371851
#> [178,] 0.56004515 -0.175503005
#> [179,] 0.62493180 0.018968882
#> [180,] 0.47430102 -0.357111131
#> [181,] -0.19197678 -0.232988515
#> [182,] 0.54905734 -0.203245966
#> [183,] -0.23934027 -0.204618649
#> [184,] -0.32620083 -0.099405807
#> [185,] 0.58973178 -0.093304822
#> [186,] -0.31471519 -0.117102750
#> [187,] -0.49410317 0.292225252
#> [188,] -0.17348231 -0.237995754
#> [189,] -0.22201891 -0.217490508
#> [190,] -0.28768099 -0.154249197
#> [191,] 0.42846276 -0.423943293
#> [192,] 0.63570928 0.056842350
#> [193,] -0.38948364 0.018540476
#> [194,] -0.46504125 0.205873351
#> [195,] 0.55875581 -0.178831283
#> [196,] 0.67801052 0.223259662
#> [197,] -0.27687598 -0.167312725
#> [198,] -0.39696822 0.034797571
#> [199,] -0.39456610 0.029526132
#> [200,] -0.41423738 0.074203509
#> [201,] -0.17427169 -0.237855808
#> [202,] 0.65549967 0.131024162
#> [203,] -0.55875231 0.515739699
#> [204,] 0.38947087 -0.465634591
#> [205,] -0.45292262 0.172282577
#> [206,] -0.41878533 0.085025253
#> [207,] -0.42747292 0.106216889
#> [208,] -0.34722170 -0.064071119
#> [209,] -0.22978706 -0.212065103
#> [210,] -0.43876968 0.134802890
#> [211,] 0.88270251 1.726194682
#> [212,] 0.37929591 -0.474296065
#> [213,] -0.26105086 -0.184584154
#> [214,] -0.17582502 -0.237560889
#> [215,] -0.48476419 0.263567903
#> [216,] 0.50971048 -0.291524958
#> [217,] -0.29463133 -0.145305755
#> [218,] -0.29919456 -0.139205034
#> [219,] -0.57371383 0.574177743
#> [220,] 0.40957831 -0.445833087
#> [221,] -0.14061075 -0.237621392
#> [222,] 0.60591057 -0.043797277
#> [223,] -0.27172116 -0.173182784
#> [224,] -0.35935076 -0.041948291
#> [225,] 0.50090568 -0.309018583
#> [226,] -0.26504187 -0.180438478
#> [227,] -0.29052581 -0.150639570
#> [228,] -0.34186352 -0.073439634
#> [229,] 0.52952520 -0.249177402
#> [230,] 0.73084478 0.477944083
#> [231,] 0.38876193 -0.466267516
#> [232,] 0.68011370 0.232333004
#> [233,] -0.49195737 0.285563359
#> [234,] 0.18986396 -0.469758573
#> [235,] 0.76431715 0.673146221
#> [236,] -0.25464600 -0.190938731
#> [237,] -0.34870416 -0.061435355
#> [238,] 0.59653452 -0.072908870
#> [239,] -0.55201143 0.490275855
#> [240,] -0.19349298 -0.232420920
#> [241,] 0.45906049 -0.381532743
#> [242,] -0.32309376 -0.104305449
#> [243,] -0.18766712 -0.234473631
#> [244,] 0.39491634 -0.460625418
#> [245,] -0.35811071 -0.044268441
#> [246,] -0.44153622 0.141982599
#> [247,] -0.34437107 -0.069086140
#> [248,] 0.52883748 -0.250717709
#> [249,] -0.21958141 -0.219075101
#> [250,] -0.23407054 -0.208831152
#> [251,] 0.62489390 0.018838706
#> [252,] -0.19156146 -0.233139909
#> [253,] -0.26831110 -0.176936670
#> [254,] -0.59528756 0.663304015
#> [255,] -0.15839429 -0.239358340
#> [256,] -0.36121355 -0.038437930
#> [257,] 0.77073786 0.714217184
#> [258,] -0.22802586 -0.213344988
#> [259,] -0.16495127 -0.239076010
#> [260,] -0.16093692 -0.239306269
#> [261,] -0.23521910 -0.207934949
#> [262,] -0.10379840 -0.221202237
#> [263,] -0.51238989 0.350905521
#> [264,] 0.77337960 0.731489323
#> [265,] 0.73154101 0.481708227
#> [266,] 0.31921869 -0.507117387
#> [267,] -0.19366855 -0.232353684
#> [268,] -0.30776647 -0.127256066
#> [269,] 0.49073444 -0.328240251
#> [270,] 0.22728030 -0.496408125
#> [271,] 0.44968402 -0.395457166
#> [272,] 0.26789783 -0.510405988
#> [273,] -0.24660262 -0.198394092
#> [274,] -0.16278707 -0.239222542
#> [275,] -0.47705162 0.240554448
#> [276,] -0.14187437 -0.237869507
#> [277,] -0.41708202 0.080950455
#> [278,] -0.44967079 0.163505512
#> [279,] 0.46157851 -0.377651515
#> [280,] -0.21958365 -0.219073674
#> [281,] 0.56494778 -0.162668095
#> [282,] 0.66051252 0.150814075
#> [283,] -0.18698272 -0.234691914
#> [284,] -0.31886223 -0.110844365
#> [285,] -0.19135499 -0.233214517
#> [286,] -0.46212975 0.197675575
#> [287,] -0.33178043 -0.090398216
#> [288,] -0.28917327 -0.152364547
#> [289,] 0.50130164 -0.308249014
#> [290,] -0.52822658 0.404552081
#> [291,] -0.26938850 -0.175761792
#> [292,] -0.35490242 -0.050209503
#> [293,] -0.58980291 0.640084071
#> [294,] -0.20370922 -0.227992977
#> [295,] -0.35143071 -0.056538093
#> [296,] 0.30894267 -0.509610664
#> [297,] -0.42517668 0.100549076
#> [298,] -0.44826821 0.159750409
#> [299,] -0.28473704 -0.157910064
#> [300,] 0.67774310 0.222111798
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] 0.40405612 -0.451628702
#> [2,] -0.55879882 0.515917232
#> [3,] -0.21887250 -0.219525323
#> [4,] -0.14855347 -0.238859466
#> [5,] -0.29888669 -0.139622334
#> [6,] -0.44529132 0.151841413
#> [7,] -0.35304527 -0.053607859
#> [8,] -0.14996107 -0.239000082
#> [9,] 0.49606963 -0.318288508
#> [10,] -0.33842653 -0.079318703
#> [11,] 0.74765096 0.572191750
#> [12,] -0.25594237 -0.189682333
#> [13,] -0.52403994 0.390109031
#> [14,] -0.20988235 -0.224815603
#> [15,] 0.44147780 -0.406966850
#> [16,] 0.36907988 -0.482082294
#> [17,] -0.10622758 -0.222873018
#> [18,] -0.27248196 -0.172331201
#> [19,] 0.38392110 -0.470471344
#> [20,] 0.30452798 -0.510401314
#> [21,] -0.22114169 -0.218067296
#> [22,] -0.47250738 0.227268030
#> [23,] 0.36586444 -0.484345072
#> [24,] -0.47027648 0.220818841
#> [25,] -0.39600306 0.032673399
#> [26,] -0.27937906 -0.164377759
#> [27,] -0.37462607 -0.012273762
#> [28,] 0.48206194 -0.343808952
#> [29,] -0.43258090 0.118997455
#> [30,] -0.33901565 -0.078318236
#> [31,] -0.49152531 0.284227560
#> [32,] -0.27990210 -0.163757496
#> [33,] -0.52215117 0.383654999
#> [34,] -0.20133826 -0.229113503
#> [35,] 0.32632319 -0.504860591
#> [36,] 0.72532198 0.448495498
#> [37,] 0.65359547 0.123615283
#> [38,] -0.42821180 0.108050901
#> [39,] -0.17953303 -0.236752830
#> [40,] 0.36811014 -0.482774182
#> [41,] -0.45048860 0.165703515
#> [42,] -0.58146609 0.605528206
#> [43,] 0.55022607 -0.200361385
#> [44,] -0.23308451 -0.209590748
#> [45,] 0.80907841 0.988963010
#> [46,] -0.16666395 -0.238923171
#> [47,] -0.19685498 -0.231079283
#> [48,] -0.35696555 -0.046399257
#> [49,] 0.62372670 0.014840148
#> [50,] -0.56876449 0.554548461
#> [51,] 0.27008684 -0.510737298
#> [52,] -0.51383333 0.355684866
#> [53,] 0.51206514 -0.286710337
#> [54,] 0.23024794 -0.497944528
#> [55,] -0.38543889 0.009960228
#> [56,] 0.57313911 -0.140581026
#> [57,] -0.44399103 0.148412733
#> [58,] -0.44915600 0.162125130
#> [59,] -0.25670722 -0.188933966
#> [60,] -0.16885150 -0.238680918
#> [61,] -0.34897671 -0.060948704
#> [62,] 0.45400594 -0.389142035
#> [63,] 0.48312242 -0.341945296
#> [64,] -0.36699358 -0.027354262
#> [65,] -0.18664372 -0.234798245
#> [66,] -0.50124624 0.314737939
#> [67,] 0.35675831 -0.490266720
#> [68,] -0.17494396 -0.237731345
#> [69,] -0.49772703 0.303581678
#> [70,] -0.39867626 0.038576924
#> [71,] -0.17846913 -0.236999615
#> [72,] 0.56866193 -0.152753693
#> [73,] -0.35877991 -0.043018016
#> [74,] -0.08592405 -0.206012214
#> [75,] -0.09000099 -0.209945981
#> [76,] -0.19405521 -0.232204514
#> [77,] 0.62537931 0.020507443
#> [78,] -0.42509017 0.100336482
#> [79,] 0.71550517 0.397893192
#> [80,] -0.26662280 -0.178756960
#> [81,] -0.51388675 0.355862188
#> [82,] -0.27558985 -0.168799296
#> [83,] -0.62489394 0.795638932
#> [84,] 0.69761064 0.311043085
#> [85,] -0.18663223 -0.234801829
#> [86,] -0.70578359 1.227230401
#> [87,] 0.48554677 -0.337642960
#> [88,] -0.30011796 -0.137948494
#> [89,] -0.53900763 0.442625406
#> [90,] -0.21798580 -0.220081687
#> [91,] 0.46626247 -0.370270625
#> [92,] -0.39244836 0.024921009
#> [93,] -0.20906284 -0.225258908
#> [94,] -0.52662624 0.399008866
#> [95,] -0.21862891 -0.219678912
#> [96,] 0.42305728 -0.430538425
#> [97,] -0.43478430 0.124584292
#> [98,] -0.56616140 0.544343464
#> [99,] 0.51343309 -0.283886620
#> [100,] 0.55302897 -0.193379537
#> [101,] 0.28474948 -0.511864280
#> [102,] -0.06839590 -0.185570843
#> [103,] -0.29723048 -0.141853079
#> [104,] -0.18951556 -0.233859974
#> [105,] -0.35366515 -0.052476880
#> [106,] -0.23769507 -0.205961375
#> [107,] 0.49982240 -0.311115776
#> [108,] -0.38665362 0.012521983
#> [109,] 0.52310100 -0.263366320
#> [110,] -0.61741600 0.761064473
#> [111,] -0.22190975 -0.217562685
#> [112,] -0.56466014 0.538494968
#> [113,] -0.31161118 -0.121689862
#> [114,] -0.26778603 -0.177505508
#> [115,] -0.52537586 0.394697112
#> [116,] 0.63010447 0.036931470
#> [117,] 0.41308195 -0.442014637
#> [118,] -0.35230260 -0.054958518
#> [119,] -0.22003442 -0.218784888
#> [120,] 0.48548293 -0.337757001
#> [121,] 0.37397431 -0.478465436
#> [122,] -0.23751457 -0.206107170
#> [123,] -0.62048985 0.775179621
#> [124,] 0.41117672 -0.444104692
#> [125,] -0.36950147 -0.022454987
#> [126,] 0.63075121 0.039205056
#> [127,] 0.32303505 -0.505959213
#> [128,] -0.25442626 -0.191150190
#> [129,] -0.19031830 -0.233582550
#> [130,] 0.43506588 -0.415525043
#> [131,] 0.36023926 -0.488087891
#> [132,] 0.44724457 -0.398944255
#> [133,] -0.21706692 -0.220650301
#> [134,] 0.39215495 -0.463198225
#> [135,] 0.48732912 -0.334442715
#> [136,] -0.26693313 -0.178424270
#> [137,] -0.26267294 -0.182916385
#> [138,] -0.24172049 -0.202631878
#> [139,] 0.45645782 -0.385481110
#> [140,] -0.18973450 -0.233784963
#> [141,] -0.50634795 0.331136332
#> [142,] -0.15626098 -0.239345291
#> [143,] 0.33476179 -0.501614561
#> [144,] 0.56141402 -0.171948010
#> [145,] -0.25307567 -0.192440324
#> [146,] 0.47775492 -0.351264086
#> [147,] 0.39822204 -0.457456905
#> [148,] -0.16480952 -0.239087203
#> [149,] -0.54702389 0.471772928
#> [150,] 0.37395984 -0.478476435
#> [151,] -0.31939421 -0.110030795
#> [152,] -0.38761501 0.014558637
#> [153,] -0.40847199 0.060751894
#> [154,] -0.19851884 -0.230373194
#> [155,] 0.33589765 -0.501130767
#> [156,] 0.26830281 -0.510470492
#> [157,] -0.19864276 -0.230319497
#> [158,] 0.68073538 0.235030500
#> [159,] 0.40287242 -0.452835654
#> [160,] -0.33048478 -0.092513811
#> [161,] 0.54839027 -0.204885392
#> [162,] -0.20287369 -0.228394210
#> [163,] -0.16031164 -0.239325858
#> [164,] 0.48306378 -0.342048626
#> [165,] 0.29569154 -0.511475999
#> [166,] 0.48018066 -0.347087765
#> [167,] -0.47414620 0.232036393
#> [168,] 0.56111288 -0.172731980
#> [169,] -0.32737166 -0.097537900
#> [170,] -0.49870220 0.306660422
#> [171,] -0.29275251 -0.147764925
#> [172,] -0.43034700 0.113378728
#> [173,] -0.38737912 0.014058156
#> [174,] -0.59468295 0.660725105
#> [175,] 0.41785061 -0.436640452
#> [176,] 0.24486236 -0.504315897
#> [177,] 0.43567444 -0.414729062
#> [178,] -0.31584690 -0.115409594
#> [179,] 0.58088155 -0.118950393
#> [180,] 0.35390166 -0.491976464
#> [181,] -0.32645628 -0.098999280
#> [182,] 0.64014907 0.072951518
#> [183,] 0.42191370 -0.431899666
#> [184,] -0.36212239 -0.036714349
#> [185,] -0.41813439 0.083464932
#> [186,] 0.71530304 0.396874000
#> [187,] -0.22564592 -0.215028200
#> [188,] 0.50002429 -0.310725822
#> [189,] -0.36613405 -0.029020816
#> [190,] -0.26393243 -0.181605194
#> [191,] -0.33242496 -0.089340411
#> [192,] -0.62250579 0.784510044
#> [193,] 0.37241971 -0.479636839
#> [194,] 0.34554217 -0.496574724
#> [195,] -0.40608534 0.055270429
#> [196,] -0.43329845 0.120811942
#> [197,] -0.28545107 -0.157029114
#> [198,] 0.52347960 -0.262542481
#> [199,] -0.52018977 0.376993218
#> [200,] -0.32305149 -0.104371538
#> [201,] -0.36221309 -0.036541958
#> [202,] -0.40029466 0.042181755
#> [203,] 0.17254096 -0.452637990
#> [204,] -0.34352069 -0.070568611
#> [205,] -0.21948871 -0.219134251
#> [206,] -0.14235982 -0.237959633
#> [207,] 0.54363869 -0.216416528
#> [208,] 0.35421307 -0.491793511
#> [209,] 0.59335063 -0.082529726
#> [210,] -0.34298915 -0.071492083
#> [211,] 0.25775908 -0.508314171
#> [212,] -0.26977356 -0.175339399
#> [213,] -0.41265350 0.070478390
#> [214,] -0.21368095 -0.222675516
#> [215,] -0.29556333 -0.144074467
#> [216,] -0.49250203 0.287249933
#> [217,] 0.42046226 -0.433610335
#> [218,] 0.41069068 -0.444632678
#> [219,] -0.31547084 -0.115973454
#> [220,] -0.50408124 0.323817387
#> [221,] -0.35238586 -0.054807332
#> [222,] 0.71682371 0.404563638
#> [223,] -0.26807279 -0.177195142
#> [224,] -0.20764301 -0.226011412
#> [225,] -0.29996006 -0.138163875
#> [226,] -0.46011643 0.192054137
#> [227,] -0.34052080 -0.075748522
#> [228,] 0.57275875 -0.141624653
#> [229,] -0.15563525 -0.239331575
#> [230,] 0.32624975 -0.504886146
#> [231,] -0.40788805 0.059406052
#> [232,] 0.56268124 -0.168637255
#> [233,] 0.54266180 -0.218755596
#> [234,] -0.38074938 0.000191842
#> [235,] -0.42398262 0.097620734
#> [236,] -0.38418464 0.007328722
#> [237,] -0.82027865 2.100055956
#> [238,] 0.26347612 -0.509606683
#> [239,] -0.32192031 -0.106134244
#> [240,] -0.25833719 -0.187321540
#> [241,] 0.49053052 -0.328614950
#> [242,] -0.45501232 0.177975477
#> [243,] -0.09172617 -0.211524219
#> [244,] -0.58818061 0.633290303
#> [245,] -0.25034997 -0.194993737
#> [246,] 0.30643956 -0.510079682
#> [247,] 0.74457117 0.554377248
#> [248,] -0.32478366 -0.101650920
#> [249,] -0.15889950 -0.239353828
#> [250,] -0.47401441 0.231651974
#> [251,] -0.26153472 -0.184089139
#> [252,] -0.46566881 0.207650991
#> [253,] 0.55212741 -0.195635134
#> [254,] -0.48913570 0.276873575
#> [255,] -0.57024585 0.560392402
#> [256,] -0.11015626 -0.225387874
#> [257,] 0.62431462 0.016851755
#> [258,] 0.44076330 -0.407939340
#> [259,] -0.22711945 -0.213992329
#> [260,] -0.38629427 0.011762800
#> [261,] 0.72119860 0.426974288
#> [262,] -0.22270826 -0.217032116
#> [263,] 0.35633411 -0.490525072
#> [264,] -0.11471560 -0.228023518
#> [265,] -0.23399006 -0.208893483
#> [266,] -0.43611340 0.127975861
#> [267,] -0.14123397 -0.237746210
#> [268,] -0.47658028 0.239166982
#> [269,] -0.40368227 0.049802570
#> [270,] -0.36297862 -0.035083996
#> [271,] 0.28272277 -0.511821232
#> [272,] -0.48445260 0.262626775
#> [273,] -0.33223367 -0.089654732
#> [274,] 0.16593477 -0.445264244
#> [275,] -0.25978966 -0.185864599
#> [276,] -0.12742171 -0.233836242
#> [277,] -0.30895536 -0.125548504
#> [278,] -0.47453756 0.233178991
#> [279,] -0.36035712 -0.040055574
#> [280,] -0.28458308 -0.158099427
#> [281,] -0.17016962 -0.238509627
#> [282,] -0.30833165 -0.126445840
#> [283,] 0.51692870 -0.276581551
#> [284,] -0.52719816 0.400986696
#> [285,] -0.20899055 -0.225297696
#> [286,] -0.47406761 0.231807132
#> [287,] -0.26007125 -0.185579946
#> [288,] 0.40178293 -0.453935529
#> [289,] 0.63444853 0.052322508
#> [290,] 0.36447674 -0.485293900
#> [291,] -0.27758997 -0.166481174
#> [292,] -0.38409755 0.007146514
#> [293,] -0.08493610 -0.205014881
#> [294,] -0.36861165 -0.024199539
#> [295,] -0.29004421 -0.151255609
#> [296,] 0.48535615 -0.337983352
#> [297,] -0.20932299 -0.225118890
#> [298,] 0.16097598 -0.439412701
#> [299,] -0.55873351 0.515667971
#> [300,] 0.50529989 -0.300388099
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] -0.16720832 -0.238867804
#> [2,] 0.55918836 -0.177716877
#> [3,] 0.47130316 -0.362091929
#> [4,] 0.68883918 0.270851673
#> [5,] -0.28883430 -0.152794344
#> [6,] -0.21026140 -0.224608348
#> [7,] -0.27523160 -0.169210771
#> [8,] 0.47508337 -0.355796918
#> [9,] 0.40998966 -0.445390471
#> [10,] -0.53252722 0.419586652
#> [11,] -0.44392837 0.148247895
#> [12,] -0.17551201 -0.237622398
#> [13,] -0.27678238 -0.167421414
#> [14,] 0.59966885 -0.063307529
#> [15,] 0.39136779 -0.463919307
#> [16,] 0.32109234 -0.506564481
#> [17,] 0.24943438 -0.505905983
#> [18,] -0.27835053 -0.165590434
#> [19,] 0.54493520 -0.213295564
#> [20,] 0.69796502 0.312698774
#> [21,] 0.51790029 -0.274528215
#> [22,] -0.34844334 -0.061900450
#> [23,] -0.18238969 -0.236031141
#> [24,] -0.53960781 0.444782710
#> [25,] -0.43354375 0.121433335
#> [26,] -0.44000370 0.137996642
#> [27,] 0.41317652 -0.441910049
#> [28,] -0.34686794 -0.064697277
#> [29,] -0.57312097 0.571810711
#> [30,] -0.28921328 -0.152313747
#> [31,] 0.25526151 -0.507657390
#> [32,] -0.42060663 0.089411308
#> [33,] 0.41300581 -0.442098792
#> [34,] -0.23399665 -0.208888386
#> [35,] 0.21855748 -0.491410696
#> [36,] -0.09152391 -0.211341801
#> [37,] 0.77605784 0.749229084
#> [38,] -0.31227564 -0.120714937
#> [39,] 0.67491820 0.210064822
#> [40,] -0.46124384 0.195197235
#> [41,] -0.44662265 0.155368246
#> [42,] -0.30435968 -0.132081321
#> [43,] -0.23558674 -0.207645498
#> [44,] 0.62302495 0.012445606
#> [45,] -0.09753506 -0.216473276
#> [46,] -0.33865647 -0.078928571
#> [47,] -0.39754535 0.036071689
#> [48,] -0.22886210 -0.212740907
#> [49,] 0.52056643 -0.268842133
#> [50,] -0.28633827 -0.155928302
#> [51,] -0.41018205 0.064710696
#> [52,] 0.48896836 -0.331471482
#> [53,] 0.47334086 -0.358715899
#> [54,] -0.40195856 0.045912150
#> [55,] -0.20529629 -0.227211852
#> [56,] -0.11833712 -0.229906175
#> [57,] 0.36004718 -0.488210853
#> [58,] -0.48866734 0.275438880
#> [59,] -0.33175270 -0.090443649
#> [60,] 0.57899560 -0.124287756
#> [61,] 0.35539978 -0.491088616
#> [62,] 0.54062924 -0.223587975
#> [63,] -0.33840571 -0.079354011
#> [64,] 0.42293582 -0.430683559
#> [65,] -0.26710121 -0.178243723
#> [66,] 0.80440016 0.952423317
#> [67,] 0.50999191 -0.290952551
#> [68,] -0.32541581 -0.100651587
#> [69,] 0.47317734 -0.358988306
#> [70,] -0.45161335 0.168736746
#> [71,] -0.49966277 0.309702516
#> [72,] 0.49453374 -0.321182843
#> [73,] -0.23418449 -0.208742785
#> [74,] -0.39246729 0.024961988
#> [75,] -0.21815286 -0.219977439
#> [76,] 0.20502205 -0.482209684
#> [77,] -0.37576757 -0.009974738
#> [78,] -0.35160330 -0.056225949
#> [79,] 0.49916330 -0.312385890
#> [80,] 0.42584157 -0.427174542
#> [81,] -0.21342898 -0.222821805
#> [82,] 0.37920471 -0.474369606
#> [83,] -0.10338752 -0.220910738
#> [84,] -0.32772171 -0.096977138
#> [85,] -0.30330398 -0.133556155
#> [86,] -0.28704393 -0.155047826
#> [87,] 0.21862929 -0.491454793
#> [88,] -0.46192712 0.197108065
#> [89,] -0.35597612 -0.048231163
#> [90,] 0.49717671 -0.316187408
#> [91,] 0.38867687 -0.466343163
#> [92,] -0.14013209 -0.237522284
#> [93,] 0.47765668 -0.351432014
#> [94,] -0.42819836 0.108017477
#> [95,] -0.26288077 -0.182701004
#> [96,] -0.24776086 -0.197356753
#> [97,] 0.68768022 0.265652843
#> [98,] -0.24052876 -0.203633130
#> [99,] -0.18262590 -0.235967630
#> [100,] 0.43357230 -0.417464167
#> [101,] 0.64666110 0.097130400
#> [102,] 0.58142745 -0.117397147
#> [103,] -0.41781370 0.082697651
#> [104,] -0.39974058 0.040944982
#> [105,] -0.42824499 0.108133395
#> [106,] 0.72304140 0.436544035
#> [107,] -0.17681295 -0.237359891
#> [108,] 0.41956611 -0.434657018
#> [109,] 0.83152989 1.178301469
#> [110,] -0.21997805 -0.218821110
#> [111,] -0.19760025 -0.230766448
#> [112,] -0.25070219 -0.194667566
#> [113,] 0.23509022 -0.500274672
#> [114,] -0.38617976 0.011521101
#> [115,] -0.37871084 -0.003994503
#> [116,] -0.37787533 -0.005699808
#> [117,] -0.44292032 0.145601232
#> [118,] 0.67970233 0.230551960
#> [119,] 0.41185830 -0.443360732
#> [120,] -0.20560658 -0.227056232
#> [121,] -0.31929439 -0.110183632
#> [122,] -0.30378982 -0.132878629
#> [123,] -0.13850433 -0.237164086
#> [124,] -0.42580137 0.102086253
#> [125,] -0.45660156 0.182332583
#> [126,] -0.34446107 -0.068928870
#> [127,] 0.63003157 0.036675608
#> [128,] -0.30721430 -0.128044969
#> [129,] 0.51054180 -0.289831762
#> [130,] 0.29400656 -0.511603853
#> [131,] -0.44415126 0.148834369
#> [132,] 0.39855494 -0.457132460
#> [133,] 0.45771814 -0.383577180
#> [134,] -0.20163370 -0.228976921
#> [135,] -0.11527526 -0.228326521
#> [136,] -0.31640384 -0.114572294
#> [137,] -0.18203414 -0.236125636
#> [138,] 0.50431236 -0.302344970
#> [139,] -0.08754039 -0.207606548
#> [140,] -0.14471235 -0.238355794
#> [141,] -0.13528364 -0.236358111
#> [142,] -0.40879465 0.061496853
#> [143,] 0.47797643 -0.350885113
#> [144,] 0.46161825 -0.377589778
#> [145,] 0.45391328 -0.389279271
#> [146,] 0.45905882 -0.381535294
#> [147,] -0.44726062 0.157064190
#> [148,] 0.45894336 -0.381711823
#> [149,] 0.34216557 -0.498261066
#> [150,] -0.15254314 -0.239197427
#> [151,] 0.59958256 -0.063573615
#> [152,] -0.31669257 -0.114137173
#> [153,] 0.35856894 -0.489146485
#> [154,] -0.39365732 0.027545105
#> [155,] -0.30899451 -0.125492067
#> [156,] -0.43016286 0.112917596
#> [157,] -0.35903065 -0.042548493
#> [158,] -0.52360432 0.388617105
#> [159,] -0.30074226 -0.137094750
#> [160,] 0.63490797 0.053966817
#> [161,] -0.48149290 0.253735132
#> [162,] -0.30453657 -0.131833259
#> [163,] 0.62430860 0.016831130
#> [164,] 0.59354666 -0.081941196
#> [165,] 0.54388553 -0.215823807
#> [166,] -0.20994590 -0.224780951
#> [167,] -0.26379247 -0.181751598
#> [168,] 0.54106453 -0.222556996
#> [169,] 0.40001740 -0.455695521
#> [170,] -0.57938406 0.597035146
#> [171,] 0.53881707 -0.227857389
#> [172,] -0.35746040 -0.045479878
#> [173,] 0.31026657 -0.509340671
#> [174,] 0.56318663 -0.167311563
#> [175,] -0.29757035 -0.141397254
#> [176,] -0.13150476 -0.235245337
#> [177,] -0.30907787 -0.125371851
#> [178,] -0.43995485 0.137869951
#> [179,] -0.37506820 -0.011384641
#> [180,] 0.47430102 -0.357111131
#> [181,] -0.19197678 -0.232988515
#> [182,] -0.45094266 0.166926602
#> [183,] -0.23934027 -0.204618649
#> [184,] -0.32620083 -0.099405807
#> [185,] 0.58973178 -0.093304822
#> [186,] -0.31471519 -0.117102750
#> [187,] 0.50589683 -0.299200330
#> [188,] -0.17348231 -0.237995754
#> [189,] -0.22201891 -0.217490508
#> [190,] -0.28768099 -0.154249197
#> [191,] 0.42846276 -0.423943293
#> [192,] 0.63570928 0.056842350
#> [193,] 0.61051636 -0.029062232
#> [194,] 0.53495875 -0.236825768
#> [195,] -0.44124419 0.141221378
#> [196,] 0.67801052 0.223259662
#> [197,] -0.27687598 -0.167312725
#> [198,] -0.39696822 0.034797571
#> [199,] 0.60543390 -0.045305770
#> [200,] -0.41423738 0.074203509
#> [201,] -0.17427169 -0.237855808
#> [202,] -0.34450033 -0.068860245
#> [203,] 0.44124769 -0.407280560
#> [204,] 0.38947087 -0.465634591
#> [205,] -0.45292262 0.172282577
#> [206,] -0.41878533 0.085025253
#> [207,] 0.57252708 -0.142259413
#> [208,] 0.65277830 0.120453983
#> [209,] -0.22978706 -0.212065103
#> [210,] 0.56123032 -0.172426381
#> [211,] -0.11729749 -0.229384534
#> [212,] 0.37929591 -0.474296065
#> [213,] -0.26105086 -0.184584154
#> [214,] -0.17582502 -0.237560889
#> [215,] 0.51523581 -0.280135423
#> [216,] 0.50971048 -0.291524958
#> [217,] -0.29463133 -0.145305755
#> [218,] -0.29919456 -0.139205034
#> [219,] 0.42628617 -0.426630872
#> [220,] 0.40957831 -0.445833087
#> [221,] -0.14061075 -0.237621392
#> [222,] 0.60591057 -0.043797277
#> [223,] -0.27172116 -0.173182784
#> [224,] -0.35935076 -0.041948291
#> [225,] -0.49909432 0.307901123
#> [226,] -0.26504187 -0.180438478
#> [227,] -0.29052581 -0.150639570
#> [228,] -0.34186352 -0.073439634
#> [229,] 0.52952520 -0.249177402
#> [230,] -0.26915522 -0.176017053
#> [231,] 0.38876193 -0.466267516
#> [232,] -0.31988630 -0.109276058
#> [233,] 0.50804263 -0.294900265
#> [234,] 0.18986396 -0.469758573
#> [235,] -0.23568285 -0.207569619
#> [236,] -0.25464600 -0.190938731
#> [237,] -0.34870416 -0.061435355
#> [238,] 0.59653452 -0.072908870
#> [239,] 0.44798857 -0.397886647
#> [240,] -0.19349298 -0.232420920
#> [241,] 0.45906049 -0.381532743
#> [242,] -0.32309376 -0.104305449
#> [243,] -0.18766712 -0.234473631
#> [244,] 0.39491634 -0.460625418
#> [245,] -0.35811071 -0.044268441
#> [246,] -0.44153622 0.141982599
#> [247,] -0.34437107 -0.069086140
#> [248,] -0.47116252 0.223374462
#> [249,] -0.21958141 -0.219075101
#> [250,] -0.23407054 -0.208831152
#> [251,] 0.62489390 0.018838706
#> [252,] -0.19156146 -0.233139909
#> [253,] -0.26831110 -0.176936670
#> [254,] 0.40471244 -0.450954126
#> [255,] -0.15839429 -0.239358340
#> [256,] -0.36121355 -0.038437930
#> [257,] -0.22926214 -0.212449611
#> [258,] -0.22802586 -0.213344988
#> [259,] -0.16495127 -0.239076010
#> [260,] -0.16093692 -0.239306269
#> [261,] -0.23521910 -0.207934949
#> [262,] -0.10379840 -0.221202237
#> [263,] 0.48761011 -0.333935315
#> [264,] -0.22662040 -0.214345450
#> [265,] -0.26845899 -0.176776012
#> [266,] 0.31921869 -0.507117387
#> [267,] -0.19366855 -0.232353684
#> [268,] -0.30776647 -0.127256066
#> [269,] -0.50926556 0.340635258
#> [270,] 0.22728030 -0.496408125
#> [271,] -0.55031598 0.483954044
#> [272,] 0.26789783 -0.510405988
#> [273,] -0.24660262 -0.198394092
#> [274,] -0.16278707 -0.239222542
#> [275,] -0.47705162 0.240554448
#> [276,] -0.14187437 -0.237869507
#> [277,] -0.41708202 0.080950455
#> [278,] -0.44967079 0.163505512
#> [279,] 0.46157851 -0.377651515
#> [280,] -0.21958365 -0.219073674
#> [281,] 0.56494778 -0.162668095
#> [282,] -0.33948748 -0.077514791
#> [283,] -0.18698272 -0.234691914
#> [284,] -0.31886223 -0.110844365
#> [285,] -0.19135499 -0.233214517
#> [286,] 0.53787025 -0.230073503
#> [287,] -0.33178043 -0.090398216
#> [288,] -0.28917327 -0.152364547
#> [289,] 0.50130164 -0.308249014
#> [290,] 0.47177342 -0.361316387
#> [291,] -0.26938850 -0.175761792
#> [292,] -0.35490242 -0.050209503
#> [293,] -0.58980291 0.640084071
#> [294,] -0.20370922 -0.227992977
#> [295,] -0.35143071 -0.056538093
#> [296,] 0.30894267 -0.509610664
#> [297,] -0.42517668 0.100549076
#> [298,] 0.55173179 -0.196621973
#> [299,] -0.28473704 -0.157910064
#> [300,] 0.67774310 0.222111798
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 4.821593 0.875322
#> [2,] 0.875322 6.278641
#>