Helper function for PPI++ OLS estimation
Usage
ppi_plusplus_ols(
X_l,
Y_l,
f_l,
X_u,
f_u,
lhat = NULL,
coord = 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.- 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++ OLS 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 OLS 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 = "ols")
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_ols(X_l, Y_l, f_l, X_u, f_u)
#> $est
#> X(Intercept) XX1
#> 0.6544417 0.9111490
#>
#> $se
#> [1] 0.09422607 0.09331478
#>
#> $lambda
#> [1] 0.5148807
#>
#> $rectifier_est
#> X(Intercept) XX1
#> -0.02792686 -0.12413267
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.2706170 -0.0613664
#> [2,] -0.0613664 0.3545745
#>
#> $var_l
#> [,1] [,2]
#> [1,] 2.3769054 0.1485426
#> [2,] 0.1485426 2.7148554
#>
#> $grads
#> [,1] [,2]
#> [1,] 0.379868053 -0.8578402283
#> [2,] -0.247440619 -0.2555719703
#> [3,] 1.098087654 -0.3213743264
#> [4,] -0.832634591 -0.0302422846
#> [5,] 0.482010050 0.3545457105
#> [6,] -1.697369132 1.8925094610
#> [7,] 1.683521846 0.4344736388
#> [8,] -0.276875655 -0.4146972375
#> [9,] -1.676266828 -0.9483263109
#> [10,] 1.615360135 1.0177743507
#> [11,] 0.535657377 -0.7406377653
#> [12,] 1.275485727 0.3430037750
#> [13,] -0.053737786 -0.0405191043
#> [14,] -0.095195165 -0.1630775571
#> [15,] 4.469080965 -9.4397634519
#> [16,] 0.108908919 -0.0752808928
#> [17,] 1.185625682 -0.1191158265
#> [18,] 1.258399192 1.5045637398
#> [19,] -1.250471993 0.4093011194
#> [20,] -0.766697498 -0.0868170738
#> [21,] 0.650175534 0.4088765804
#> [22,] 1.026471362 -1.7288022647
#> [23,] -0.188502165 -0.0637660404
#> [24,] 1.189194389 -0.8456633088
#> [25,] -0.030183721 -0.0174548324
#> [26,] -1.803833110 0.8124805498
#> [27,] -0.379359998 -0.0859655935
#> [28,] -0.047662072 -0.1252966845
#> [29,] 2.751281185 2.3593842947
#> [30,] 0.936718201 0.5791464795
#> [31,] 0.129967890 0.1598360013
#> [32,] 1.483073477 -2.8050531505
#> [33,] -1.447952165 0.9192325385
#> [34,] -1.461022042 -1.0967095429
#> [35,] -1.360424539 0.5274867603
#> [36,] 2.985862959 2.4095812862
#> [37,] -0.572410970 -0.5214373001
#> [38,] 2.931228586 2.9626238569
#> [39,] -0.362851664 0.3492626008
#> [40,] 1.848497203 -2.0165552456
#> [41,] -0.356204932 0.1827133034
#> [42,] 0.409298018 0.0319884741
#> [43,] -1.405381318 -0.2539441194
#> [44,] -0.814548874 0.2871992728
#> [45,] -1.646016652 0.9646552248
#> [46,] 0.137162181 -0.1089119167
#> [47,] -2.590139377 -6.7558435195
#> [48,] -3.394286085 -4.6528412916
#> [49,] -1.018398986 -1.5529330091
#> [50,] 1.041757619 -0.1997309172
#> [51,] -0.526170991 0.7207630297
#> [52,] -2.871811140 -0.1222609453
#> [53,] 3.150625220 7.7620813721
#> [54,] 0.370897634 -0.5243634732
#> [55,] 0.816674627 0.2143042795
#> [56,] 0.479033955 -0.3290304869
#> [57,] -0.069625187 -0.0482740192
#> [58,] -0.587295426 -0.1229168483
#> [59,] -0.969874083 -1.2597517834
#> [60,] -1.239664929 -1.0357885005
#> [61,] -1.291613949 2.3697490855
#> [62,] -5.366052454 -2.5538996917
#> [63,] -0.004094896 0.0004328726
#> [64,] 3.633200455 6.3069639691
#> [65,] -0.731905691 0.4963351676
#> [66,] -2.801582709 -1.9046186091
#> [67,] 0.638237468 -0.1021512846
#> [68,] 0.056800382 0.0721820777
#> [69,] 1.659892571 -0.2600714493
#> [70,] 0.508703917 -0.4747920473
#> [71,] -0.650254103 -0.2047470439
#> [72,] -0.403970221 -0.8810034954
#> [73,] -1.066571672 -0.4225534759
#> [74,] -0.299042453 -0.1446475301
#> [75,] -0.753302434 0.8227743819
#> [76,] 2.163424541 -1.0604217384
#> [77,] -2.002198031 2.8670148588
#> [78,] 0.021006878 -0.0023676770
#> [79,] 1.040523197 -0.1257365151
#> [80,] -0.112670281 0.0839088844
#> [81,] 1.792735562 -1.0890736150
#> [82,] 0.763653521 0.1583078694
#> [83,] 0.771497212 0.9823834372
#> [84,] 2.977255537 -2.0230597770
#> [85,] -1.794756046 0.9051777974
#> [86,] 0.150735800 0.0802457422
#> [87,] -2.287740253 -0.6328976136
#> [88,] -0.756791839 0.4180230853
#> [89,] -1.266118987 -0.0944397613
#> [90,] 2.723568755 0.8452315495
#> [91,] 0.463069788 0.2276029715
#> [92,] -1.555485976 0.2339515693
#> [93,] -0.714452286 1.1024993352
#> [94,] 1.723817047 1.4679573460
#> [95,] 0.056697408 -0.0137536409
#> [96,] 1.683263394 0.3989202593
#> [97,] 1.204343196 -2.0775018868
#> [98,] 1.773120949 -1.9043203327
#> [99,] 0.061583525 -0.0139895803
#> [100,] -1.170029077 3.0515005895
#> [101,] 0.990442686 -0.4114379040
#> [102,] -1.105594195 1.5580605477
#> [103,] -0.202765980 0.0067244131
#> [104,] 0.770685894 0.1583927539
#> [105,] 5.584282933 6.9041986828
#> [106,] -2.356572304 -1.7351001061
#> [107,] -2.469005678 -2.5226506376
#> [108,] -2.952117839 2.5802385257
#> [109,] 0.650130366 0.7102491644
#> [110,] -3.083043266 -9.6295099734
#> [111,] 2.339718777 -0.2363766669
#> [112,] 0.205483335 -0.1954353583
#> [113,] -0.130870542 0.0943251610
#> [114,] 3.153692605 0.7396122617
#> [115,] 0.711622704 0.6094816696
#> [116,] -2.182344641 1.3054640978
#> [117,] -2.615959151 1.3990735231
#> [118,] 0.895877712 0.6548668325
#> [119,] 0.683652567 -1.1082945084
#> [120,] -0.413866212 -0.2458835600
#> [121,] -0.918752869 1.2140519772
#> [122,] -3.633182972 -1.4449624373
#> [123,] 0.207331300 0.0206628613
#> [124,] 0.906552710 1.7253037768
#> [125,] 3.927021233 -0.7514045272
#> [126,] -0.702390101 -0.4341252373
#> [127,] -0.373791248 -0.3016043183
#> [128,] -1.997224818 -0.5540765900
#> [129,] 0.358813015 0.1677765628
#> [130,] -1.270120597 1.3932406481
#> [131,] -0.640829769 -0.5941367253
#> [132,] -0.350134767 -0.5808079658
#> [133,] 0.853634981 0.3520312523
#> [134,] 0.078315442 0.1108900956
#> [135,] 2.085483719 -2.9326243417
#> [136,] -0.841035246 0.5690570875
#> [137,] -2.163061307 -4.8219385788
#> [138,] 0.034948057 -0.0163876304
#> [139,] -1.042014854 -0.3926622831
#> [140,] 1.093351120 -0.5821207367
#> [141,] -0.812144781 -0.6016563927
#> [142,] -0.156906088 -0.1567559698
#> [143,] -0.548840822 0.0699086463
#> [144,] 0.803460815 -0.9317074270
#> [145,] 0.044301448 -0.0222029974
#> [146,] 0.853680691 1.0001477976
#> [147,] -0.394606924 0.5007588108
#> [148,] 3.115431876 2.5373379734
#> [149,] 1.093502964 -0.5718515554
#> [150,] -1.635894165 1.2726447784
#> [151,] 0.368596352 0.7958645113
#> [152,] -1.914395627 0.3899563408
#> [153,] -0.447790937 0.8081022036
#> [154,] 2.546802175 -4.1361718115
#> [155,] 2.015175112 1.0609575916
#> [156,] -1.031027064 -0.0248413414
#> [157,] 0.610367058 -0.6936175496
#> [158,] -0.455172512 0.2190857795
#> [159,] 0.296087942 0.3144693626
#> [160,] 1.219722749 -0.5731384371
#> [161,] 2.312551053 -1.8278738131
#> [162,] 0.135876017 0.0531426256
#> [163,] 5.728887565 9.3165843934
#> [164,] 0.116208707 0.1629669625
#> [165,] 1.004154364 1.1010014497
#> [166,] -5.195732898 0.7558618108
#> [167,] -0.044290276 -0.0073771976
#> [168,] 0.163409209 0.1638469169
#> [169,] 1.094208968 -0.5936587080
#> [170,] 0.986967244 2.3845587948
#> [171,] 4.495095759 -1.8358906701
#> [172,] -1.785484990 -3.3696133426
#> [173,] -0.453622366 0.4424508817
#> [174,] 0.450631691 0.3399256762
#> [175,] -0.091145693 0.0704224827
#> [176,] -2.392134313 3.5165958415
#> [177,] -1.356381547 -0.7327037407
#> [178,] 0.803167041 -0.8757466765
#> [179,] -1.856941601 1.0386411142
#> [180,] 1.721382379 0.6782931706
#> [181,] 1.908416548 -1.0098282994
#> [182,] -0.558111405 -0.2584594830
#> [183,] 2.937944782 -2.4258494266
#> [184,] 1.013156726 -0.1705396075
#> [185,] -0.196416881 0.1092233448
#> [186,] 1.233514990 -0.9121702131
#> [187,] -1.670076156 0.0569475290
#> [188,] -1.389085378 1.1158622903
#> [189,] 0.477963088 0.1493941168
#> [190,] -1.875969756 -3.1771842131
#> [191,] -0.135627034 0.1939013701
#> [192,] -1.203572768 2.8248436220
#> [193,] 0.041644967 0.0649559427
#> [194,] 0.003729231 -0.0007100393
#> [195,] -2.922331370 -2.5619983104
#> [196,] -0.010158337 -0.0110338473
#> [197,] 1.320916632 -0.8429989862
#> [198,] 1.520019924 -1.0957875959
#> [199,] 0.989600162 -0.4754438857
#> [200,] 0.123999979 0.0442453839
#> [201,] 0.995994367 -0.5569338864
#> [202,] 1.315955988 2.2040200595
#> [203,] 0.419548860 -0.2026320944
#> [204,] -2.072415314 0.2946345616
#> [205,] -0.305130338 -0.6243533659
#> [206,] 0.046631528 -0.0131668254
#> [207,] -3.161071177 2.4030334494
#> [208,] 3.571124588 3.5801135785
#> [209,] 0.279572412 -0.2399954902
#> [210,] -1.103346490 -0.4780123395
#> [211,] -1.159600614 -3.7864040724
#> [212,] 0.579176631 -0.3195383801
#> [213,] -0.978986746 0.1751548393
#> [214,] 1.211400412 2.2648091377
#> [215,] -1.940936801 -1.2525262267
#> [216,] 0.565604216 0.7040780783
#> [217,] 1.326807464 0.8836519804
#> [218,] -1.404355965 0.8532030162
#> [219,] -0.414296450 -0.2618776923
#> [220,] -3.312654753 0.6479682418
#> [221,] -1.404439286 -1.7863749536
#> [222,] -0.037471166 -0.0252637303
#> [223,] -0.769144693 -1.9542503052
#> [224,] -0.820877204 0.4072500593
#> [225,] 1.409172308 1.3044369357
#> [226,] 1.626803595 -0.4874897912
#> [227,] 0.473848201 -0.5772489205
#> [228,] 0.746095637 -0.5831758435
#> [229,] -0.668016314 -0.0502741067
#> [230,] 0.623046841 0.0948806973
#> [231,] 0.980033073 1.7043085059
#> [232,] -0.171468003 -0.0308058380
#> [233,] 0.351149103 -0.1612242129
#> [234,] 1.131854397 -0.8251913390
#> [235,] 1.006992774 -0.7010010017
#> [236,] -0.341828506 -0.1225670230
#> [237,] -1.923131259 0.3859212105
#> [238,] -5.565484427 0.2215817032
#> [239,] 1.223937781 1.1564606222
#> [240,] -0.402685085 -0.4655516615
#> [241,] 0.064195919 0.0180347783
#> [242,] -0.413204920 -0.1938476223
#> [243,] -0.042329542 -0.0516280547
#> [244,] 2.043960572 -2.2033239485
#> [245,] -1.726774655 1.9994688664
#> [246,] -0.991582891 -0.1069203415
#> [247,] -1.369634574 -1.1233162860
#> [248,] 1.709498840 1.4424068068
#> [249,] -0.756459611 0.3709866138
#> [250,] -2.291185115 -3.3183755772
#> [251,] 0.330182552 -0.3342089569
#> [252,] -0.673354167 0.7667508709
#> [253,] 1.626811854 -0.2412241892
#> [254,] 0.613265643 0.5828391059
#> [255,] 0.041668714 0.0189042125
#> [256,] 1.419648195 1.3571990982
#> [257,] -2.059827738 -4.3525605142
#> [258,] 0.047238290 0.0207549321
#> [259,] -1.263770506 -1.6879306274
#> [260,] 1.183903024 0.1388306582
#> [261,] 1.683962536 -0.1238054913
#> [262,] 1.145420753 1.8095117280
#> [263,] -0.544473794 0.3425930833
#> [264,] 1.029986747 -1.2256676105
#> [265,] -1.346614905 -0.5312215537
#> [266,] 0.698322476 -0.4750619260
#> [267,] -1.345749455 -0.1344520657
#> [268,] 1.346596804 0.8960220614
#> [269,] -0.173173892 0.0189528661
#> [270,] -0.906703990 0.3299221066
#> [271,] 2.458711363 4.4775317383
#> [272,] -0.908167138 0.2611813173
#> [273,] 1.709002724 -1.0788246143
#> [274,] 1.816457098 -3.4832197948
#> [275,] -0.064736257 0.1437904842
#> [276,] 1.223748796 0.2645470238
#> [277,] -1.563878705 -0.4985993841
#> [278,] 0.929231046 1.8258979204
#> [279,] 0.324571883 -0.2859815030
#> [280,] -2.016216269 -3.5532560696
#> [281,] 0.670503503 -1.0159961719
#> [282,] 1.171763731 0.3212482042
#> [283,] 0.697694963 -0.1377215678
#> [284,] -6.672623397 13.0192501186
#> [285,] -0.059812745 0.0523776365
#> [286,] 3.578860759 1.9405211722
#> [287,] -3.259445573 2.7774074288
#> [288,] -1.857966123 -2.1250533688
#> [289,] 3.126634670 0.1903127610
#> [290,] -0.796155137 -0.0239903362
#> [291,] 10.522541876 -5.5854585017
#> [292,] -0.143259110 0.0374545320
#> [293,] -0.726028078 0.3768531716
#> [294,] 0.013337179 0.0134419385
#> [295,] 1.506159314 1.0125537428
#> [296,] 0.620302577 0.7599263569
#> [297,] -1.405620549 3.5230808891
#> [298,] -2.286704856 -0.0778841634
#> [299,] -3.615727792 2.2937944913
#> [300,] 3.019675290 -1.7289470389
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] 0.183157343 0.1336966511
#> [2,] -0.840917483 -0.3264281527
#> [3,] 0.852378483 0.7100090438
#> [4,] -2.276547082 -2.1045784849
#> [5,] -0.116942788 0.0442690606
#> [6,] 0.555681364 -0.4615951381
#> [7,] -1.070813607 -0.0014801812
#> [8,] 0.566661862 -1.0068621104
#> [9,] 0.495801593 -0.1463274333
#> [10,] 0.845202712 0.3770070061
#> [11,] -1.971748454 0.6048374105
#> [12,] 0.080928621 0.0765922751
#> [13,] 1.817533429 2.2195696232
#> [14,] -0.216987032 0.0587415867
#> [15,] 0.657139431 1.1353205272
#> [16,] -0.596752584 0.4746963634
#> [17,] 0.662807745 -0.0058994388
#> [18,] -0.438100986 0.5972101151
#> [19,] 0.340404340 0.0986944498
#> [20,] 1.019335796 -0.9689978049
#> [21,] 0.608947174 0.3494751762
#> [22,] -1.423596862 -1.0035757237
#> [23,] -1.139455423 0.3552356769
#> [24,] -1.273239151 -1.5529942543
#> [25,] 0.199294893 0.0231694755
#> [26,] -0.037520002 -0.0164920263
#> [27,] -0.786796418 -0.4788113692
#> [28,] -1.190741904 2.0564978447
#> [29,] -0.576121423 -0.3240397672
#> [30,] 1.487410002 -2.3311698585
#> [31,] 0.869239877 -0.7828958867
#> [32,] -2.460757983 0.9783126714
#> [33,] -1.459496787 -0.2418011211
#> [34,] -0.242806380 0.1972581388
#> [35,] -0.603670059 -0.6618173110
#> [36,] -0.761513902 -1.4824032256
#> [37,] -1.346618412 -1.2862982511
#> [38,] 1.057592843 -1.0458072171
#> [39,] -1.361496639 0.2633144696
#> [40,] -1.284973036 -2.3152346912
#> [41,] 0.212936864 -0.0958479268
#> [42,] 0.189146197 0.0573557572
#> [43,] 0.841413629 -0.4866945987
#> [44,] 0.602131662 -0.2876527285
#> [45,] 0.137029646 -0.1295255799
#> [46,] -0.755859119 0.0977946408
#> [47,] -1.353508797 -3.0815670226
#> [48,] -0.612820075 0.5847256346
#> [49,] 0.204722698 0.0704219912
#> [50,] -1.204010525 -0.2776054358
#> [51,] 0.038871182 -0.0008558238
#> [52,] -1.885959131 3.1902301323
#> [53,] -0.309065380 0.1914323470
#> [54,] -0.348236855 0.1798824868
#> [55,] -0.342612734 -0.5699825376
#> [56,] 0.110650434 -0.0195531612
#> [57,] 1.376424089 -1.5292410453
#> [58,] -0.717529329 0.0525965750
#> [59,] 0.392344910 -0.2187852427
#> [60,] 0.936827281 1.5103790720
#> [61,] 1.422933468 -0.4887909774
#> [62,] -0.480389351 -0.7534409226
#> [63,] -1.112198872 -1.9688049363
#> [64,] -0.676923643 1.2946047858
#> [65,] 0.086490343 -0.1427454360
#> [66,] 0.406170898 0.1148851358
#> [67,] -0.715730380 -1.7648413292
#> [68,] -0.284437829 0.0857614301
#> [69,] 0.196054794 -0.1003652253
#> [70,] 1.515461313 -2.2348535171
#> [71,] 2.097439751 -3.2259576931
#> [72,] -0.540171773 -0.7211582902
#> [73,] -0.024528464 -0.0147956717
#> [74,] -0.751664310 -0.0442345455
#> [75,] -0.087618123 -0.0440283815
#> [76,] -0.730311681 -0.6824360480
#> [77,] 1.646802215 -0.1154836306
#> [78,] 0.081400181 0.0230210409
#> [79,] -0.839187366 0.2687343149
#> [80,] -1.314530644 1.1532588262
#> [81,] 0.720373416 -0.1876152667
#> [82,] 1.480035191 -0.4503246811
#> [83,] 0.978712262 0.9768164760
#> [84,] -1.185520927 -1.0904368805
#> [85,] 0.390356845 -0.0679198782
#> [86,] 0.413413603 -0.2490551224
#> [87,] -0.740508200 -0.4058216449
#> [88,] 0.204308042 0.2296672928
#> [89,] -0.227284643 0.0931479849
#> [90,] -0.038368413 0.0073610395
#> [91,] 0.172970180 0.0954560798
#> [92,] -0.572077895 -0.0968356243
#> [93,] 1.786234992 -2.1406237810
#> [94,] 0.448086097 0.1504024466
#> [95,] -0.526798217 -0.2407084932
#> [96,] -1.757819073 0.2191334094
#> [97,] 1.029323262 2.0906191236
#> [98,] -1.026650668 -0.4127166744
#> [99,] -0.624693424 -0.7112039927
#> [100,] -1.739754117 -0.1456524420
#> [101,] -0.231701505 0.0335787930
#> [102,] 0.430185572 -0.9604049143
#> [103,] -0.499092738 0.0835262535
#> [104,] 1.009451710 0.1060638889
#> [105,] -1.113472762 -1.0777790492
#> [106,] -0.734609879 0.3072090388
#> [107,] 1.001547251 0.0883912949
#> [108,] -0.823690822 0.8485469715
#> [109,] 0.733619128 0.0720205035
#> [110,] 0.173201198 0.0989259275
#> [111,] 1.681821611 1.0999368354
#> [112,] 0.968324810 -0.7350883176
#> [113,] 1.800198189 -1.5099306469
#> [114,] 0.961552924 -2.5662074048
#> [115,] 1.073656964 -0.9933555899
#> [116,] 0.510043732 -0.0832152312
#> [117,] -0.602415934 -0.3864673491
#> [118,] 0.496101054 0.5232294075
#> [119,] -0.906256605 -0.1326292389
#> [120,] -0.505195254 -0.2854096924
#> [121,] -0.495398021 0.3652018978
#> [122,] 0.396758660 0.1882644767
#> [123,] 0.827571129 0.2435088923
#> [124,] 0.894555518 0.0518535917
#> [125,] 1.131239666 1.3250338465
#> [126,] -0.694611302 -0.4360725617
#> [127,] -1.206967713 -0.5330525609
#> [128,] -0.940543606 -0.4566951846
#> [129,] 0.826986325 0.1479312482
#> [130,] -0.445976232 -0.1585934196
#> [131,] -1.704533009 -0.8213375017
#> [132,] -0.590393677 0.3623558615
#> [133,] -1.165369111 -0.8817123356
#> [134,] -0.498052167 0.7739060188
#> [135,] 0.547191216 -0.6293547724
#> [136,] -0.106645182 -0.1246577065
#> [137,] -1.840553648 -0.0571151591
#> [138,] -0.250640571 0.3422473281
#> [139,] -1.294846380 -1.5294752883
#> [140,] 0.259882371 0.5433762387
#> [141,] 2.481426295 -5.3161070918
#> [142,] -0.646344348 0.8301675231
#> [143,] 0.395956465 0.6659807499
#> [144,] 0.851563115 -0.1229363808
#> [145,] -0.634858553 -1.1675005168
#> [146,] -0.607907108 0.2822787779
#> [147,] 0.269755360 0.0126159773
#> [148,] 0.785307361 -0.3363056928
#> [149,] -0.707825080 -2.0221999704
#> [150,] 0.701694460 0.1343380654
#> [151,] -0.483452954 -0.0839570083
#> [152,] -0.334325424 -0.1911918993
#> [153,] -0.032532714 -0.0069325788
#> [154,] -0.061481678 0.0226659468
#> [155,] 0.241878080 0.2498404087
#> [156,] -0.582158995 -0.4221530417
#> [157,] 0.239495715 -0.0011270883
#> [158,] -0.006305324 -0.0072443939
#> [159,] 0.428708349 0.6573940578
#> [160,] 0.141311705 -0.0063262474
#> [161,] -0.524584988 -0.5622407125
#> [162,] 1.036230055 -1.0766582529
#> [163,] -0.180964802 0.2381260861
#> [164,] -1.263008679 0.5418811393
#> [165,] -1.414988810 0.0794523816
#> [166,] -0.124518934 0.0388088767
#> [167,] 1.610283681 -1.5884573850
#> [168,] -1.104705556 -0.7804455721
#> [169,] 0.204857640 0.1665820694
#> [170,] -0.078258885 -0.1332604927
#> [171,] -0.780999652 0.3496859313
#> [172,] -0.456690492 -0.1308984420
#> [173,] 0.289778026 0.4177058435
#> [174,] -0.086377844 -0.0898095645
#> [175,] 0.608314295 -0.4459250487
#> [176,] -0.578195649 1.3151153598
#> [177,] 0.276027186 0.0593480694
#> [178,] 1.181333546 -0.9049869727
#> [179,] 1.108652910 0.2659837946
#> [180,] -0.851771061 -0.8203631989
#> [181,] 0.598460400 0.1153879915
#> [182,] -1.599690823 -0.4523781141
#> [183,] 0.800839672 -0.3383932932
#> [184,] -0.860295060 0.5314132869
#> [185,] -1.319291477 2.5461687917
#> [186,] 0.852590419 -0.4063705007
#> [187,] -2.611801718 -0.8130863258
#> [188,] -0.190557432 -0.0760335132
#> [189,] 0.347802369 0.2557180928
#> [190,] -1.240529758 -1.0132860580
#> [191,] 2.259540834 2.3844147979
#> [192,] -1.061763528 -0.9023738069
#> [193,] -0.124768516 -0.1963880305
#> [194,] 1.240492265 -0.5581303747
#> [195,] -0.400057348 0.2402342841
#> [196,] -0.644357761 0.4641943544
#> [197,] 0.715266132 -0.1889326417
#> [198,] -0.986718973 -0.6763751164
#> [199,] 0.695494204 0.0116925824
#> [200,] 1.491968314 0.9610875799
#> [201,] 0.801862574 -0.5028251583
#> [202,] -1.744407991 3.4982313830
#> [203,] -1.715979698 2.1809412341
#> [204,] 0.322007000 0.0624815528
#> [205,] -1.071430545 1.4450015700
#> [206,] -1.318764296 2.8298138583
#> [207,] 0.956375066 -1.4909070341
#> [208,] 1.898715808 -4.0025337511
#> [209,] -1.279980267 1.5600911481
#> [210,] -1.092241244 -1.0899966552
#> [211,] -1.374020047 -0.5620167553
#> [212,] -0.450234395 -0.5117264719
#> [213,] 1.268476866 -0.6870347337
#> [214,] -0.591236164 0.8589855420
#> [215,] -0.346921506 -0.1904386159
#> [216,] 0.415972119 -0.0374380335
#> [217,] 0.357424339 0.2574933045
#> [218,] -0.701620424 -0.3782729782
#> [219,] -2.043753334 2.9809504774
#> [220,] 1.117434716 -0.2431978457
#> [221,] 1.186594512 -0.7043854455
#> [222,] -0.182837980 0.0742077438
#> [223,] -0.479187879 0.1107029723
#> [224,] -2.186028012 2.5957661441
#> [225,] -0.604512647 -0.4464631998
#> [226,] -0.681999287 -0.1408686279
#> [227,] 0.679627145 -0.1031038832
#> [228,] 1.208781992 -2.5729795694
#> [229,] 0.869924672 -0.4862823424
#> [230,] 1.299704552 -1.3738773788
#> [231,] -0.140625929 -0.0243617899
#> [232,] 1.077617340 1.1339437172
#> [233,] 0.178700226 -0.0483084824
#> [234,] 0.095084205 -0.0306279432
#> [235,] -0.257137118 -0.4781908119
#> [236,] -1.166779636 1.4247385899
#> [237,] -0.211474646 -0.0777230399
#> [238,] -0.979815542 -1.0111078148
#> [239,] 0.841247249 0.0847631632
#> [240,] -1.678900939 -2.3866625878
#> [241,] -0.308861792 -0.0806195877
#> [242,] -0.656094433 0.6855939204
#> [243,] -0.628496847 -0.0479084855
#> [244,] -0.822474477 -0.7100890398
#> [245,] -1.723123243 -2.1154618103
#> [246,] 1.002150744 -2.0313506290
#> [247,] -0.323546293 -0.6717041920
#> [248,] 1.916327146 -0.1785483636
#> [249,] -0.234019756 -0.2656904929
#> [250,] 1.562086334 1.8269517032
#> [251,] 0.781450698 -0.7906077071
#> [252,] -0.337411631 0.0386211336
#> [253,] 3.048703090 -8.2227290789
#> [254,] -0.551832507 0.6638281866
#> [255,] 1.907657945 1.8256458438
#> [256,] -1.293206984 -0.8644598327
#> [257,] -0.035503240 -0.0467297992
#> [258,] -1.554438046 2.4898572794
#> [259,] -0.076302056 -0.0046271839
#> [260,] -1.709243023 -1.1900378402
#> [261,] -1.544911579 1.6651659521
#> [262,] -2.752111595 -0.4844487926
#> [263,] -0.328369079 -0.1243307025
#> [264,] -0.336742244 -0.1752120761
#> [265,] -0.025203475 0.0450013679
#> [266,] -0.018674984 0.0336146214
#> [267,] 0.259444014 -0.2305589827
#> [268,] -0.750983231 0.8210842647
#> [269,] 1.536253076 -2.1339945960
#> [270,] 1.603867398 0.8422943662
#> [271,] 0.325173764 0.6892817021
#> [272,] 1.575966749 -0.7147329408
#> [273,] 0.747337125 -0.3338090637
#> [274,] -0.866280177 0.9731128084
#> [275,] 0.025765448 -0.0137581928
#> [276,] -0.689083353 -1.0983575351
#> [277,] 1.142311244 0.8528223147
#> [278,] 0.528220048 0.1894524928
#> [279,] -0.728637599 -0.5024216193
#> [280,] -0.546089915 -0.8978908705
#> [281,] 1.221396321 0.1033604479
#> [282,] 1.382183573 1.2836856931
#> [283,] 1.345246054 1.3627240089
#> [284,] -0.265892868 -0.3009428518
#> [285,] -1.716412089 0.0961390995
#> [286,] -0.037718832 0.1021533981
#> [287,] 1.717572176 0.0589552510
#> [288,] -1.609991181 1.9245798469
#> [289,] 1.032767900 -0.0812115868
#> [290,] 0.315957699 0.3943468914
#> [291,] -0.841133287 -1.1213348607
#> [292,] 0.432015144 -0.7269336792
#> [293,] 0.805500753 1.8065400581
#> [294,] -0.962389844 -0.6312531267
#> [295,] -0.178035280 -0.1883119481
#> [296,] 1.126132416 0.7103560493
#> [297,] -0.866825900 -1.2441509602
#> [298,] 1.328341587 0.6272447749
#> [299,] 1.283350304 -2.0190542822
#> [300,] -0.824154534 -0.5573438239
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] -0.005839403 0.01318688
#> [2,] -0.897842063 -0.92734680
#> [3,] 0.622608561 -0.18221715
#> [4,] -0.560900199 -0.02037257
#> [5,] -1.174772249 -0.86411157
#> [6,] -2.140623709 2.38672340
#> [7,] 0.255418712 0.06591699
#> [8,] -0.346658000 -0.51921544
#> [9,] -2.398903625 -1.35714875
#> [10,] -1.074650175 -0.67709445
#> [11,] 0.897085899 -1.24037440
#> [12,] -0.776610834 -0.20884628
#> [13,] -0.245822801 -0.18535411
#> [14,] 0.865667526 1.48296340
#> [15,] 2.834558083 -5.98726181
#> [16,] 1.688363711 -1.16704425
#> [17,] 1.981747029 -0.19909946
#> [18,] 1.151994806 1.37734482
#> [19,] 0.231845999 -0.07588721
#> [20,] -0.838352403 -0.09493092
#> [21,] 1.255349241 0.78945281
#> [22,] -0.069568687 0.11716888
#> [23,] 1.275787893 0.43157033
#> [24,] -1.045239507 0.74329370
#> [25,] -0.389777806 -0.22540317
#> [26,] -0.802183338 0.36131855
#> [27,] 0.227775084 0.05161541
#> [28,] 0.272444932 0.71621826
#> [29,] -0.596668821 -0.51167836
#> [30,] -0.462442089 -0.28591492
#> [31,] -0.206921818 -0.25447483
#> [32,] -0.211205313 0.39946917
#> [33,] -0.335756972 0.21315534
#> [34,] -0.192113855 -0.14420939
#> [35,] -1.186788422 0.46016163
#> [36,] 1.246089778 1.00559023
#> [37,] 1.073698944 0.97808517
#> [38,] 2.076449341 2.09868940
#> [39,] 0.658822113 -0.63414874
#> [40,] 1.946691173 -2.12367662
#> [41,] 0.504986276 -0.25902985
#> [42,] -0.459639100 -0.03592286
#> [43,] 0.502094723 0.09072556
#> [44,] -0.516839221 0.18223075
#> [45,] -0.613798845 0.35971949
#> [46,] -0.754297244 0.59894031
#> [47,] -2.520289643 -6.57365492
#> [48,] -0.520136331 -0.71299582
#> [49,] -0.353295998 -0.53873288
#> [50,] -1.155860710 0.22160733
#> [51,] 1.167800366 -1.59968403
#> [52,] -1.778918563 -0.07573348
#> [53,] 0.853773388 2.10341061
#> [54,] 0.506456452 -0.71601229
#> [55,] 0.480077408 0.12597752
#> [56,] 1.160298629 -0.79696568
#> [57,] -1.035717336 -0.71810563
#> [58,] -2.122687197 -0.44426367
#> [59,] 0.139292719 0.18092478
#> [60,] 1.415664732 1.18284322
#> [61,] 0.506233299 -0.92879602
#> [62,] 0.247316888 0.11770711
#> [63,] 0.773142393 -0.08172910
#> [64,] 2.042664692 3.54591297
#> [65,] 0.654787518 -0.44403818
#> [66,] 0.334875307 0.22766051
#> [67,] 1.403065385 -0.22456364
#> [68,] 0.329177031 0.41831906
#> [69,] 1.366815835 -0.21415228
#> [70,] -1.300557820 1.21385837
#> [71,] -0.823919434 -0.25942946
#> [72,] 0.643566048 1.40352905
#> [73,] -1.471056669 -0.58280200
#> [74,] -0.385465242 -0.18645043
#> [75,] 1.161139411 -1.26822338
#> [76,] 1.646721264 -0.80715504
#> [77,] -1.699301635 2.43328730
#> [78,] 1.148582973 -0.12945634
#> [79,] 1.521286605 -0.18383182
#> [80,] 1.133958075 -0.84449206
#> [81,] 1.757966165 -1.06795146
#> [82,] 1.573918550 0.32627846
#> [83,] -0.494262685 -0.62936776
#> [84,] 1.444216624 -0.98135230
#> [85,] -1.965119345 0.99109982
#> [86,] 0.397212091 0.21145991
#> [87,] -2.402631904 -0.66468210
#> [88,] 0.360975425 -0.19938912
#> [89,] 1.081883941 0.08069768
#> [90,] -0.556622418 -0.17274204
#> [91,] 0.428646597 0.21068366
#> [92,] -1.545448193 0.23244184
#> [93,] -1.425917785 2.20038964
#> [94,] -0.375676520 -0.31991626
#> [95,] 1.277025735 -0.30978053
#> [96,] 0.977453872 0.23164892
#> [97,] 0.047127933 -0.08129607
#> [98,] 1.519417211 -1.63184417
#> [99,] -0.589548177 0.13392432
#> [100,] -0.263773365 0.68793553
#> [101,] -0.926093208 0.38470661
#> [102,] -1.951997865 2.75085640
#> [103,] -0.304933056 0.01011262
#> [104,] 1.051276217 0.21606018
#> [105,] 2.870734471 3.54926879
#> [106,] -0.771632480 -0.56813856
#> [107,] -1.710771053 -1.74794158
#> [108,] -0.972716060 0.85018268
#> [109,] 0.838335439 0.91585792
#> [110,] 0.077250431 0.24128231
#> [111,] 1.731556046 -0.17493532
#> [112,] 0.811598709 -0.77191216
#> [113,] 1.763768675 -1.27123921
#> [114,] 1.577841804 0.37003960
#> [115,] 1.265351330 1.08373220
#> [116,] -0.066540814 0.03980427
#> [117,] -1.844750682 0.98661397
#> [118,] 0.044987250 0.03288469
#> [119,] 2.024920168 -3.28267312
#> [120,] 0.890994077 0.52935173
#> [121,] 0.750608660 -0.99186404
#> [122,] -0.680555618 -0.27066551
#> [123,] 0.176337328 0.01757397
#> [124,] -1.097585682 -2.08886775
#> [125,] 2.068627637 -0.39581558
#> [126,] 0.528303248 0.32652763
#> [127,] -0.389319240 -0.31413353
#> [128,] -2.353910129 -0.65302939
#> [129,] -0.201521791 -0.09422912
#> [130,] 0.586385198 -0.64322687
#> [131,] -0.134241689 -0.12446038
#> [132,] 0.417168271 0.69200399
#> [133,] 0.312992029 0.12907505
#> [134,] -0.506746160 -0.71752299
#> [135,] 0.200225514 -0.28155876
#> [136,] 0.104626904 -0.07079214
#> [137,] -1.480895794 -3.30124187
#> [138,] -0.928941488 0.43559360
#> [139,] 0.470174172 0.17717565
#> [140,] 0.821046590 -0.43714067
#> [141,] 0.548062494 0.40601788
#> [142,] 1.152989470 1.15188636
#> [143,] 0.300468851 -0.03827225
#> [144,] 0.557425626 -0.64640065
#> [145,] 0.711961333 -0.35682074
#> [146,] -0.561863015 -0.65826258
#> [147,] 1.096173125 -1.39105098
#> [148,] -0.811519396 -0.66093533
#> [149,] -0.866002652 0.45287940
#> [150,] 0.353116257 -0.27470699
#> [151,] -0.753820998 -1.62763244
#> [152,] -1.807031433 0.36808659
#> [153,] -0.327336047 0.59072428
#> [154,] 2.500510083 -4.06099045
#> [155,] -0.075609907 -0.03980741
#> [156,] 1.018960767 0.02455062
#> [157,] -0.444224398 0.50481400
#> [158,] -1.150928571 0.55397037
#> [159,] -0.844571504 -0.89700330
#> [160,] 1.526844114 -0.71745243
#> [161,] -1.041019711 0.82283704
#> [162,] -1.181229906 -0.46199219
#> [163,] 2.530082554 4.11453836
#> [164,] -0.578954918 -0.81190581
#> [165,] 0.526648137 0.57744146
#> [166,] -2.516874557 0.36614841
#> [167,] 0.330929996 0.05512126
#> [168,] -0.909966725 -0.91240416
#> [169,] -0.364578501 0.19780061
#> [170,] 1.015069765 2.45245579
#> [171,] 2.286160569 -0.93371556
#> [172,] -2.020057061 -3.81230381
#> [173,] -0.435230332 0.42451179
#> [174,] -0.382645546 -0.28864159
#> [175,] -0.504011227 0.38941743
#> [176,] 0.328330818 -0.48266804
#> [177,] -0.767446408 -0.41456687
#> [178,] 0.968652569 -1.05618660
#> [179,] 0.046849389 -0.02620422
#> [180,] 1.173995471 0.46260094
#> [181,] -0.488488729 0.25848117
#> [182,] 0.189778747 0.08788589
#> [183,] 2.068989230 -1.70835625
#> [184,] -0.518359280 0.08725283
#> [185,] -0.597259851 0.33212379
#> [186,] -0.656290981 0.48531967
#> [187,] -1.222517174 0.04168632
#> [188,] 1.208682939 -0.97094371
#> [189,] 0.282997331 0.08845481
#> [190,] 0.292833623 0.49594956
#> [191,] 0.700641195 -1.00168295
#> [192,] -0.333838092 0.78353418
#> [193,] -0.949697830 -1.48129588
#> [194,] -0.600390533 0.11431334
#> [195,] -0.184540464 -0.16178602
#> [196,] 0.118661113 0.12888809
#> [197,] 0.778987371 -0.49714384
#> [198,] -0.432420235 0.31173324
#> [199,] 1.660123638 -0.79759045
#> [200,] -1.064219976 -0.37973250
#> [201,] 1.929745127 -1.07906278
#> [202,] 1.655902028 2.77337640
#> [203,] 2.290035970 -1.10603276
#> [204,] -2.530500366 0.35976035
#> [205,] 0.357248834 0.73099749
#> [206,] -0.432628207 0.12215641
#> [207,] -1.378704238 1.04808535
#> [208,] 0.065710655 0.06587606
#> [209,] 0.323357781 -0.27758250
#> [210,] -0.136264920 -0.05903523
#> [211,] -1.479778531 -4.83187003
#> [212,] 0.886911857 -0.48931943
#> [213,] 0.080191993 -0.01434750
#> [214,] 1.120094859 2.09410617
#> [215,] -0.917887294 -0.59233145
#> [216,] -1.401795708 -1.74498987
#> [217,] 1.542384331 1.02722588
#> [218,] -1.731174311 1.05175837
#> [219,] -1.815545354 -1.14761019
#> [220,] -1.260613465 0.24658093
#> [221,] 0.223060635 0.28372172
#> [222,] -0.647190331 -0.43634730
#> [223,] -2.015031462 -5.11981150
#> [224,] -1.403543868 0.69632013
#> [225,] -1.373151661 -1.27109349
#> [226,] -0.958983232 0.28736999
#> [227,] -1.492046627 1.81763338
#> [228,] -0.626417924 0.48963133
#> [229,] 0.671047230 0.05050221
#> [230,] -0.570683526 -0.08690655
#> [231,] 0.812890512 1.41364231
#> [232,] -0.726218634 -0.13047200
#> [233,] -0.649857586 0.29837120
#> [234,] 1.141917221 -0.83252776
#> [235,] -0.236639554 0.16473263
#> [236,] -0.470741586 -0.16879047
#> [237,] -0.858407742 0.17225957
#> [238,] -2.797345801 0.11137227
#> [239,] -1.634070017 -1.54398178
#> [240,] -1.299604680 -1.50249696
#> [241,] -1.203314457 -0.33805123
#> [242,] -1.798461697 -0.84371581
#> [243,] 0.637279782 0.77727075
#> [244,] -0.651253826 0.70203074
#> [245,] -0.595053590 0.68902513
#> [246,] 1.026882372 0.11072661
#> [247,] 0.060597112 0.04969919
#> [248,] 1.585049038 1.33740104
#> [249,] 0.525966502 -0.25794706
#> [250,] 0.255070432 0.36942432
#> [251,] 0.488872016 -0.49483356
#> [252,] 1.005478275 -1.14494182
#> [253,] 1.735172223 -0.25729190
#> [254,] 0.468948288 0.44568191
#> [255,] -0.786226306 -0.35669421
#> [256,] -0.188835726 -0.18052901
#> [257,] -1.591738195 -3.36345447
#> [258,] 0.340445932 0.14958061
#> [259,] -0.981822302 -1.31135196
#> [260,] 1.007744322 0.11817337
#> [261,] -1.013328904 0.07450028
#> [262,] -1.350357936 -2.13326720
#> [263,] 0.374712422 -0.23577606
#> [264,] 0.918579178 -1.09309440
#> [265,] 0.085289528 0.03364558
#> [266,] 1.701839206 -1.15774451
#> [267,] -2.197345755 -0.21953394
#> [268,] -0.280466812 -0.18662190
#> [269,] -1.161961726 0.12716989
#> [270,] -1.178943311 0.42898175
#> [271,] 1.744968819 3.17774318
#> [272,] -0.298763644 0.08592194
#> [273,] 0.097435452 -0.06150708
#> [274,] 0.941469214 -1.80535186
#> [275,] 0.541325996 -1.20237917
#> [276,] 0.995166190 0.21513259
#> [277,] -1.595219033 -0.50859138
#> [278,] -0.802784587 -1.57743622
#> [279,] -1.254116187 1.10500647
#> [280,] -2.461351651 -4.33773541
#> [281,] 0.839311560 -1.27178654
#> [282,] 0.698020929 0.19136790
#> [283,] -0.569787930 0.11247335
#> [284,] -2.777488108 5.41927968
#> [285,] -1.428182496 1.25065023
#> [286,] 2.168007820 1.17553192
#> [287,] 0.227948285 -0.19423710
#> [288,] -1.518611758 -1.73691597
#> [289,] 2.112961448 0.12861225
#> [290,] 0.405471488 0.01221797
#> [291,] 3.568261861 -1.89406503
#> [292,] 0.438482072 -0.11463942
#> [293,] 0.648251169 -0.33648218
#> [294,] 1.108711549 1.11742014
#> [295,] 0.952876371 0.64059527
#> [296,] -0.935534786 -1.14611412
#> [297,] -0.604457270 1.51502613
#> [298,] 1.496227557 0.05096085
#> [299,] -2.160024317 1.37030556
#> [300,] 1.800437180 -1.03085936
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 1.00312675 -0.05371302
#> [2,] -0.05371302 0.92271068
#>