Helper function for PPI logistic regression
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.
- opts
(list, optional): Options to pass to the optimizer. See ?optim for details.
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.
- 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
Prediction Powered Inference (Angelopoulos et al., 2023) https://www.science.org/doi/10.1126/science.adi6000
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_logistic(X_l, Y_l, f_l, X_u, f_u)
#> $est
#> [,1]
#> [1,] 0.1793968
#> [2,] 0.6494671
#>
#> $se
#> [1] 0.1865305 0.1836512
#>
#> $rectifier_est
#> [,1]
#> [1,] 0.08682910
#> [2,] -0.03653341
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.18022109 -0.03753627
#> [2,] -0.03753627 0.16889304
#>
#> $var_l
#> [,1] [,2]
#> [1,] 0.35240803 -0.05586951
#> [2,] -0.05586951 0.26862173
#>
#> $grads
#> [,1] [,2]
#> [1,] 0.42955822 -0.3062621661
#> [2,] 0.47026499 -0.2161209596
#> [3,] -0.28823465 -0.3215683446
#> [4,] -0.52869416 0.2395726190
#> [5,] 0.40663873 -0.3489162529
#> [6,] 0.54168405 -0.0102359802
#> [7,] 0.47473951 -0.2050546250
#> [8,] -0.49655107 0.1266103949
#> [9,] -0.41601339 -0.1023359986
#> [10,] 0.70275582 0.7369394570
#> [11,] 0.47337208 -0.2084614412
#> [12,] 0.45496839 -0.2521981094
#> [13,] -0.69834289 1.0954872655
#> [14,] -0.39974297 -0.1398028443
#> [15,] 0.56786388 0.0819656549
#> [16,] 0.47268275 -0.2101705193
#> [17,] -0.40977186 -0.1170459478
#> [18,] -0.21500360 -0.3693242050
#> [19,] -0.30122878 -0.3070678449
#> [20,] 0.62388697 0.3138158094
#> [21,] -0.54481122 0.3012542763
#> [22,] -0.40371378 -0.1309216389
#> [23,] -0.41552125 -0.1035111881
#> [24,] 0.22746660 -0.4910545416
#> [25,] -0.52789747 0.2366130323
#> [26,] 0.41543254 -0.3332255917
#> [27,] 0.26242852 -0.4900453414
#> [28,] 0.20417069 -0.4840694030
#> [29,] -0.43903895 -0.0443904858
#> [30,] -0.22540331 -0.3661659456
#> [31,] -0.74325784 1.4217799995
#> [32,] 0.49657155 -0.1476492728
#> [33,] -0.44540390 -0.0273371619
#> [34,] -0.44912130 -0.0171669152
#> [35,] 0.33128806 -0.4497798060
#> [36,] -0.36520854 -0.2099879166
#> [37,] -0.31999497 -0.2830081595
#> [38,] -0.25536180 -0.3502586231
#> [39,] -0.53421251 0.2603017665
#> [40,] -0.22984003 -0.3644424100
#> [41,] -0.59161992 0.5010695125
#> [42,] -0.34911156 -0.2384237556
#> [43,] -0.49317424 0.1154914473
#> [44,] -0.35058332 -0.2359360554
#> [45,] 0.56037491 0.0546053985
#> [46,] 0.66150950 0.4997287493
#> [47,] -0.54218112 0.2909504267
#> [48,] 0.26076961 -0.4903958012
#> [49,] -0.67535260 0.9482368605
#> [50,] 0.68646300 0.6386575895
#> [51,] 0.33000641 -0.4509822331
#> [52,] 0.62635518 0.3252149320
#> [53,] -0.39511183 -0.1499480837
#> [54,] 0.41858905 -0.3273892949
#> [55,] -0.58982794 0.4928204598
#> [56,] 0.59601155 0.1922355336
#> [57,] 0.34957822 -0.4307594905
#> [58,] 0.49684555 -0.1468922539
#> [59,] 0.49457837 -0.1531284444
#> [60,] 0.61757229 0.2851337592
#> [61,] 0.41418110 -0.3355095322
#> [62,] -0.24145761 -0.3588802707
#> [63,] -0.33308289 -0.2640578702
#> [64,] -0.63218013 0.7017874325
#> [65,] -0.18954372 -0.3716881706
#> [66,] -0.34088551 -0.2519130923
#> [67,] -0.70552996 1.1440796185
#> [68,] -0.60056273 0.5429909155
#> [69,] 0.47248708 -0.2106546114
#> [70,] -0.28752708 -0.3223069547
#> [71,] -0.59455166 0.5146734180
#> [72,] 0.59110275 0.1721330470
#> [73,] -0.37138205 -0.1983638148
#> [74,] -0.54485573 0.3014294512
#> [75,] 0.52145637 -0.0750859870
#> [76,] -0.60849592 0.5812540974
#> [77,] -0.47273295 0.0511118399
#> [78,] 0.36182016 -0.4160833974
#> [79,] -0.26440101 -0.3435234949
#> [80,] -0.20169622 -0.3715279138
#> [81,] -0.46012310 0.0138502514
#> [82,] -0.61909550 0.6340046305
#> [83,] -0.41891766 -0.0953472839
#> [84,] -0.64963888 0.7970597728
#> [85,] 0.26252643 -0.4900237273
#> [86,] 0.28754790 -0.4811391526
#> [87,] -0.44245135 -0.0353041666
#> [88,] -0.44578022 -0.0263146907
#> [89,] 0.56812600 0.0829379073
#> [90,] 0.69219368 0.6725097114
#> [91,] 0.45621248 -0.2493643481
#> [92,] 0.30723463 -0.4694974136
#> [93,] 0.27801414 -0.4853102841
#> [94,] -0.42355218 -0.0840047125
#> [95,] -0.51094759 0.1755909041
#> [96,] -0.31979084 -0.2832896153
#> [97,] 0.30003983 -0.4742232099
#> [98,] 0.82888116 1.7846052481
#> [99,] 0.75398206 1.0919272518
#> [100,] -0.41198298 -0.1118830005
#> [101,] -0.48049046 0.0749581037
#> [102,] -0.49324381 0.1157191217
#> [103,] -0.34327921 -0.2480604410
#> [104,] 0.61063699 0.2544155626
#> [105,] -0.42616493 -0.0775067330
#> [106,] -0.45148135 -0.0106292084
#> [107,] 0.36769072 -0.4084904174
#> [108,] -0.26860303 -0.3400926154
#> [109,] -0.55805524 0.3545869697
#> [110,] 0.34661621 -0.4340758440
#> [111,] 0.42590837 -0.3134382525
#> [112,] -0.24745693 -0.3554205574
#> [113,] 0.60577066 0.2333401490
#> [114,] 0.75703818 1.1156387946
#> [115,] 0.56953006 0.0881630152
#> [116,] -0.30445939 -0.3031879209
#> [117,] -0.64949728 0.7962639524
#> [118,] -0.38061728 -0.1802278473
#> [119,] 0.49296603 -0.1575252761
#> [120,] -0.51437549 0.1876354783
#> [121,] 0.33075713 -0.4502799838
#> [122,] -0.44681935 -0.0234830652
#> [123,] -0.34791327 -0.2404325618
#> [124,] 0.59469699 0.1868150114
#> [125,] 0.30391882 -0.4717426050
#> [126,] -0.31148262 -0.2943768481
#> [127,] -0.45184258 -0.0096229792
#> [128,] 0.58898109 0.1635609539
#> [129,] -0.29937325 -0.3092469314
#> [130,] 0.27836567 -0.4851735889
#> [131,] 0.54492168 0.0006508340
#> [132,] 0.66034705 0.4935765218
#> [133,] -0.45345428 -0.0051154411
#> [134,] -0.28929718 -0.3204492809
#> [135,] 0.54488787 0.0005364103
#> [136,] 0.50557353 -0.1222948810
#> [137,] -0.23705804 -0.3611641047
#> [138,] 0.34289990 -0.4381076022
#> [139,] 0.56644077 0.0767043037
#> [140,] -0.16282806 -0.3655199389
#> [141,] 0.47879869 -0.1948119501
#> [142,] 0.77028187 1.2222028642
#> [143,] 0.31640329 -0.4626918878
#> [144,] 0.41245150 -0.3386382307
#> [145,] -0.41384562 -0.1074927542
#> [146,] -0.42694749 -0.0755459236
#> [147,] -0.20972712 -0.3704476015
#> [148,] 0.21786471 -0.4889373901
#> [149,] 0.59889784 0.2042331666
#> [150,] 0.67728300 0.5859827886
#> [151,] -0.29420254 -0.3151288477
#> [152,] -0.73629479 1.3674525172
#> [153,] 0.63793927 0.3801645635
#> [154,] -0.58317793 0.4626417233
#> [155,] -0.24159786 -0.3588039340
#> [156,] -0.41645367 -0.1012824189
#> [157,] -0.24246869 -0.3583250512
#> [158,] -0.57556025 0.4288966387
#> [159,] -0.41000981 -0.1164928746
#> [160,] 0.26686894 -0.4889613550
#> [161,] -0.26505930 -0.3429985410
#> [162,] -0.33408950 -0.2625266303
#> [163,] -0.22603145 -0.3659355052
#> [164,] 0.57572884 0.1115766511
#> [165,] 0.40273809 -0.3556094386
#> [166,] 0.47951526 -0.1929836045
#> [167,] -0.65544531 0.8300293862
#> [168,] -0.59101801 0.4982931331
#> [169,] 0.61269125 0.2634301443
#> [170,] 0.45396275 -0.2544758690
#> [171,] 0.36273205 -0.4149276126
#> [172,] 0.33416059 -0.4470228029
#> [173,] -0.43324539 -0.0595202715
#> [174,] -0.24509359 -0.3568308533
#> [175,] -0.79288471 1.8578479292
#> [176,] -0.60007789 0.5406855059
#> [177,] 0.55650686 0.0407871539
#> [178,] 0.41788800 -0.3286948659
#> [179,] -0.46257699 0.0209572067
#> [180,] -0.17307921 -0.3689779013
#> [181,] -0.54814230 0.3144398232
#> [182,] 0.52101565 -0.0764392920
#> [183,] 0.69245056 0.6740450905
#> [184,] -0.41499487 -0.1047652369
#> [185,] 0.50616297 -0.1205996788
#> [186,] 0.32093053 -0.4590085208
#> [187,] -0.45544864 0.0005031620
#> [188,] 0.46517009 -0.2284372571
#> [189,] -0.47576536 0.0603489001
#> [190,] -0.54307882 0.2944567625
#> [191,] 0.44700193 -0.2699274190
#> [192,] -0.57829563 0.4409137578
#> [193,] 0.34314244 -0.4378488553
#> [194,] -0.22595568 -0.3659635405
#> [195,] 0.77072818 1.2259062384
#> [196,] 0.64156509 0.3978662776
#> [197,] 0.45122455 -0.2606195462
#> [198,] -0.09197210 -0.2988561902
#> [199,] 0.70570878 0.7554410477
#> [200,] -0.34438549 -0.2462597536
#> [201,] -0.25094091 -0.3532296809
#> [202,] 0.63370554 0.3598010008
#> [203,] -0.32183614 -0.2804499536
#> [204,] 0.29120411 -0.4792847853
#> [205,] 0.33798675 -0.4432173443
#> [206,] -0.58516202 0.4715748464
#> [207,] -0.57110855 0.4095765499
#> [208,] -0.09801533 -0.3078813149
#> [209,] 0.65439382 0.4624945604
#> [210,] 0.69405418 0.6836647971
#> [211,] 0.48124264 -0.1885510664
#> [212,] -0.46277692 0.0215392939
#> [213,] 0.51099347 -0.1065436901
#> [214,] -0.27369036 -0.3356857579
#> [215,] -0.33608777 -0.2594556885
#> [216,] -0.35097745 -0.2352660361
#> [217,] 0.42048135 -0.3238385263
#> [218,] 0.35023853 -0.4300077457
#> [219,] -0.48541228 0.0904574690
#> [220,] 0.30395449 -0.4717190574
#> [221,] -0.27473091 -0.3347503499
#> [222,] 0.51017142 -0.1089563876
#> [223,] 0.39698531 -0.3651832847
#> [224,] 0.69078335 0.6641079815
#> [225,] 0.47868937 -0.1950903574
#> [226,] -0.36680190 -0.2070260044
#> [227,] -0.26380540 -0.3439944395
#> [228,] -0.32806045 -0.2715405446
#> [229,] 0.74103406 0.9948879209
#> [230,] -0.44976367 -0.0153936843
#> [231,] 0.65057197 0.4429096402
#> [232,] 0.76146124 1.1505278211
#> [233,] -0.30275985 -0.3052426618
#> [234,] 0.30363270 -0.4719309545
#> [235,] -0.42534093 -0.0795641162
#> [236,] -0.45846478 0.0090866013
#> [237,] -0.39950067 -0.1403393239
#> [238,] 0.50389900 -0.1270871318
#> [239,] 0.45700496 -0.2475500234
#> [240,] 0.57965516 0.1267015693
#> [241,] 0.38072634 -0.3903379562
#> [242,] -0.40188498 -0.1350329601
#> [243,] -0.36907960 -0.2027457945
#> [244,] -0.28356331 -0.3263468619
#> [245,] 0.30205507 -0.4729541209
#> [246,] -0.42986554 -0.0681749853
#> [247,] -0.52907328 0.2409839127
#> [248,] 0.53192141 -0.0422096547
#> [249,] 0.18733815 -0.4750168494
#> [250,] -0.48914719 0.1024126325
#> [251,] -0.71760466 1.2286716281
#> [252,] -0.49705567 0.1282838703
#> [253,] 0.45228900 -0.2582413525
#> [254,] -0.53272085 0.2546588250
#> [255,] 0.58842100 0.1613096931
#> [256,] -0.39485190 -0.1505107194
#> [257,] -0.47484038 0.0575198587
#> [258,] -0.66465306 0.8836867295
#> [259,] -0.50449371 0.1533149205
#> [260,] 0.52052423 -0.0779453463
#> [261,] -0.23931740 -0.3600181473
#> [262,] -0.37831715 -0.1848287198
#> [263,] -0.52178553 0.2141832404
#> [264,] 0.49618248 -0.1487226246
#> [265,] -0.34988277 -0.2371230104
#> [266,] -0.44655588 -0.0242021540
#> [267,] 0.35023332 -0.4300136955
#> [268,] -0.60228856 0.5512279095
#> [269,] 0.35194221 -0.4280471702
#> [270,] -0.68023645 0.9785169545
#> [271,] 0.52585015 -0.0614566848
#> [272,] -0.58915245 0.4897239267
#> [273,] -0.42596872 -0.0779973119
#> [274,] -0.45276051 -0.0070593614
#> [275,] 0.23088074 -0.4915543461
#> [276,] -0.41418865 -0.1066801359
#> [277,] -0.19189578 -0.3717978396
#> [278,] 0.61622773 0.2791149497
#> [279,] -0.27182178 -0.3373365423
#> [280,] 0.70927075 0.7780510559
#> [281,] -0.51824411 0.2014078191
#> [282,] -0.55011358 0.3223146234
#> [283,] 0.45347770 -0.2555703592
#> [284,] -0.65718457 0.8400341991
#> [285,] -0.51578848 0.1926436409
#> [286,] -0.47975096 0.0726542434
#> [287,] 0.59294321 0.1796257595
#> [288,] -0.45847254 0.0091088070
#> [289,] -0.52745884 0.2349871501
#> [290,] 0.66891045 0.5395476083
#> [291,] -0.28129361 -0.3285853522
#> [292,] -0.49031180 0.1061748189
#> [293,] 0.40226631 -0.3564079037
#> [294,] -0.46565799 0.0299786942
#> [295,] 0.64885268 0.4341918333
#> [296,] -0.24047138 -0.3594109327
#> [297,] -0.54213365 0.2907653427
#> [298,] -0.43937970 -0.0434889957
#> [299,] 0.61256470 0.2628727753
#> [300,] -0.23207536 -0.3634897193
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] -0.5371873 0.271643785
#> [2,] -0.2665503 -0.341792304
#> [3,] -0.3268345 -0.273327274
#> [4,] 0.3024587 -0.472694813
#> [5,] -0.7514194 1.487404139
#> [6,] -0.2863687 -0.323504727
#> [7,] 0.1762252 -0.467117095
#> [8,] -0.2410241 -0.359114880
#> [9,] -0.3694411 -0.202061493
#> [10,] 0.4029928 -0.355177306
#> [11,] 0.5910499 0.171918628
#> [12,] 0.4692606 -0.218572857
#> [13,] 0.3916991 -0.373669391
#> [14,] -0.5769333 0.434914990
#> [15,] 0.4272007 -0.310914145
#> [16,] -0.4493036 -0.016664231
#> [17,] 0.5436846 -0.003525789
#> [18,] -0.3079500 -0.298873160
#> [19,] 0.5523318 0.026108208
#> [20,] -0.2402483 -0.359529460
#> [21,] -0.3541993 -0.229728165
#> [22,] -0.3652297 -0.209948765
#> [23,] -0.5690053 0.400549922
#> [24,] -0.3333112 -0.263711426
#> [25,] -0.2646930 -0.343291222
#> [26,] -0.7599091 1.558012649
#> [27,] 0.5953090 0.189335193
#> [28,] -0.6094767 0.586056406
#> [29,] -0.5392802 0.279694433
#> [30,] -0.6433790 0.762244359
#> [31,] -0.3939221 -0.152517552
#> [32,] -0.4878337 0.098188974
#> [33,] 0.4386250 -0.287799203
#> [34,] -0.2844447 -0.325462923
#> [35,] -0.3612474 -0.217236214
#> [36,] 0.4198984 -0.324936465
#> [37,] 0.3087711 -0.468418112
#> [38,] -0.6195974 0.636549800
#> [39,] -0.4182388 -0.096989136
#> [40,] -0.2420912 -0.358533686
#> [41,] 0.3833104 -0.386527203
#> [42,] -0.5607127 0.365583974
#> [43,] -0.5370684 0.271188026
#> [44,] 0.2935477 -0.478022149
#> [45,] 0.8293826 1.790204431
#> [46,] -0.2974156 -0.311506852
#> [47,] -0.3596760 -0.220066246
#> [48,] -0.1885525 -0.371621486
#> [49,] -0.3035500 -0.304291069
#> [50,] 0.5210774 -0.076249826
#> [51,] -0.6064898 0.571481229
#> [52,] 0.6029373 0.221247832
#> [53,] 0.5104144 -0.108244046
#> [54,] 0.4695439 -0.217882517
#> [55,] -0.2988668 -0.309835395
#> [56,] -0.5448917 0.301570902
#> [57,] -0.3764591 -0.188504542
#> [58,] -0.4121867 -0.111404577
#> [59,] 0.4324356 -0.300501536
#> [60,] 0.4489897 -0.265570873
#> [61,] -0.1464398 -0.357021133
#> [62,] 0.4144113 -0.335090722
#> [63,] 0.8682466 2.280879053
#> [64,] -0.3464839 -0.242809278
#> [65,] -0.5988711 0.534963838
#> [66,] -0.1574185 -0.363126889
#> [67,] -0.2106229 -0.370279844
#> [68,] -0.6515004 0.807557778
#> [69,] -0.2217520 -0.367416351
#> [70,] -0.2021159 -0.371490459
#> [71,] -0.5175263 0.198837843
#> [72,] -0.5466110 0.308359657
#> [73,] -0.2678214 -0.340745188
#> [74,] 0.4751806 -0.203950973
#> [75,] -0.5764576 0.432826612
#> [76,] 0.5873858 0.157161315
#> [77,] -0.3899056 -0.161080318
#> [78,] -0.3507880 -0.235588336
#> [79,] -0.1884554 -0.371614296
#> [80,] -0.5392887 0.279726993
#> [81,] 0.4671832 -0.223606924
#> [82,] -0.6333333 0.707907026
#> [83,] -0.2437254 -0.357619170
#> [84,] 0.2848032 -0.482438577
#> [85,] -0.5051118 0.155425620
#> [86,] 0.3535505 -0.426168640
#> [87,] -0.2512248 -0.353045276
#> [88,] 0.3448477 -0.436012407
#> [89,] -0.3972999 -0.145183364
#> [90,] -0.4357771 -0.052954698
#> [91,] -0.3196176 -0.283528070
#> [92,] -0.2406286 -0.359327042
#> [93,] -0.2248462 -0.366366569
#> [94,] 0.2490575 -0.492020677
#> [95,] -0.4717960 0.048279717
#> [96,] -0.2325236 -0.363291922
#> [97,] -0.3342137 -0.262336897
#> [98,] 0.2027365 -0.483431580
#> [99,] -0.3788345 -0.183798802
#> [100,] 0.2047241 -0.484308958
#> [101,] -0.6091424 0.584417545
#> [102,] -0.4623832 0.020393337
#> [103,] 0.4951618 -0.151529763
#> [104,] 0.5872291 0.156534916
#> [105,] 0.4998811 -0.138444063
#> [106,] -0.4652902 0.028895964
#> [107,] -0.4640279 0.025191807
#> [108,] -0.2359031 -0.361727907
#> [109,] 0.7093651 0.778654197
#> [110,] -0.4951735 0.122057724
#> [111,] -0.2146205 -0.369416716
#> [112,] -0.3761448 -0.189122847
#> [113,] -0.5626079 0.373487942
#> [114,] -0.3352097 -0.260810234
#> [115,] 0.7166056 0.825643514
#> [116,] -0.5015376 0.143285067
#> [117,] -0.1628415 -0.365525411
#> [118,] 0.3878696 -0.379631594
#> [119,] 0.4334237 -0.298502029
#> [120,] -0.1876918 -0.371553700
#> [121,] -0.4274740 -0.074222977
#> [122,] -0.4543598 -0.002570022
#> [123,] -0.4676068 0.035741733
#> [124,] -0.6453987 0.773395982
#> [125,] -0.3091351 -0.297379268
#> [126,] 0.3482953 -0.432207050
#> [127,] -0.4138186 -0.107556797
#> [128,] -0.3212867 -0.281217045
#> [129,] -0.5211963 0.212046547
#> [130,] 0.4213489 -0.322197624
#> [131,] 0.3511423 -0.428971457
#> [132,] 0.6226966 0.308356434
#> [133,] -0.1844389 -0.371214139
#> [134,] -0.4524541 -0.007916166
#> [135,] 0.4024154 -0.356155889
#> [136,] -0.4149344 -0.104909208
#> [137,] 0.5484429 0.012653564
#> [138,] 0.4100810 -0.342873668
#> [139,] -0.3890536 -0.162874830
#> [140,] -0.5717781 0.412463640
#> [141,] -0.4021835 -0.134364371
#> [142,] 0.2971110 -0.475991770
#> [143,] 0.1890628 -0.476103763
#> [144,] -0.3080192 -0.298786222
#> [145,] 0.5127718 -0.101295273
#> [146,] -0.2748497 -0.334642816
#> [147,] -0.5389523 0.278428935
#> [148,] -0.4902147 0.105860390
#> [149,] 0.7643717 1.173863448
#> [150,] 0.3415599 -0.439526091
#> [151,] -0.5066796 0.160801308
#> [152,] -0.3262818 -0.274127570
#> [153,] -0.4365403 -0.050961400
#> [154,] 0.2589446 -0.490747008
#> [155,] -0.5053280 0.156165102
#> [156,] -0.5386716 0.277347064
#> [157,] -0.4832619 0.083650112
#> [158,] 0.2479179 -0.492098859
#> [159,] -0.2336495 -0.362785073
#> [160,] -0.1825243 -0.370952268
#> [161,] -0.2808232 -0.329042475
#> [162,] -0.3214925 -0.280930153
#> [163,] -0.5771382 0.435815252
#> [164,] -0.2728197 -0.336459601
#> [165,] -0.5219803 0.214890703
#> [166,] -0.4051906 -0.127575466
#> [167,] 0.4083252 -0.345971769
#> [168,] -0.3612276 -0.217272134
#> [169,] -0.5193912 0.205528119
#> [170,] -0.1983851 -0.371749235
#> [171,] -0.6492628 0.794947134
#> [172,] 0.4822333 -0.185992953
#> [173,] -0.2018607 -0.371513488
#> [174,] 0.4473377 -0.269194744
#> [175,] -0.1246349 -0.339641197
#> [176,] 0.3218481 -0.458236001
#> [177,] 0.3409757 -0.440138659
#> [178,] -0.4299339 -0.068001204
#> [179,] -0.3962753 -0.147420892
#> [180,] 0.5437814 -0.003199841
#> [181,] -0.6441145 0.766296438
#> [182,] 0.4261853 -0.312898822
#> [183,] 0.4518854 -0.259144533
#> [184,] 0.5690267 0.086286610
#> [185,] 0.4326369 -0.300094997
#> [186,] -0.3928169 -0.154890786
#> [187,] -0.4607511 0.015662521
#> [188,] 0.6441425 0.410598572
#> [189,] -0.3820469 -0.177340174
#> [190,] 0.4463770 -0.271287752
#> [191,] -0.4954137 0.122849929
#> [192,] -0.3909949 -0.158774952
#> [193,] -0.2377028 -0.360842878
#> [194,] -0.2713769 -0.337724053
#> [195,] -0.2076491 -0.370800414
#> [196,] -0.6550389 0.827700303
#> [197,] -0.2889765 -0.320788241
#> [198,] 0.5084756 -0.113906762
#> [199,] -0.3898070 -0.161288565
#> [200,] 0.3021836 -0.472871738
#> [201,] -0.6029094 0.554202570
#> [202,] -0.1941850 -0.371839300
#> [203,] 0.3142594 -0.464361701
#> [204,] 0.4356214 -0.294016411
#> [205,] -0.4888978 0.101609268
#> [206,] -0.5054544 0.156597681
#> [207,] 0.5832820 0.140878678
#> [208,] -0.2858176 -0.324069667
#> [209,] -0.3913148 -0.158095589
#> [210,] -0.5923286 0.504345591
#> [211,] 0.4676792 -0.222409398
#> [212,] -0.3005091 -0.307917293
#> [213,] 0.3989209 -0.362001558
#> [214,] -0.5896037 0.491791796
#> [215,] 0.3053779 -0.470768814
#> [216,] -0.3586776 -0.221851042
#> [217,] -0.4601415 0.013903426
#> [218,] 0.4480185 -0.267704944
#> [219,] -0.5673488 0.393485808
#> [220,] 0.4985691 -0.142109421
#> [221,] -0.3621240 -0.215646259
#> [222,] -0.4747136 0.057132961
#> [223,] -0.4329309 -0.060330884
#> [224,] 0.4410227 -0.282763980
#> [225,] 0.3080236 -0.468946297
#> [226,] -0.2846293 -0.325276729
#> [227,] 0.5422292 -0.008412900
#> [228,] 0.3093395 -0.468012572
#> [229,] -0.3252572 -0.275602940
#> [230,] -0.6333739 0.708122582
#> [231,] -0.5680595 0.396511588
#> [232,] -0.1808146 -0.370679326
#> [233,] 0.5724167 0.098995874
#> [234,] -0.1536342 -0.361213685
#> [235,] -0.3221445 -0.280018025
#> [236,] 0.4800322 -0.191660784
#> [237,] 0.5318309 -0.042500058
#> [238,] 0.2183543 -0.489070954
#> [239,] -0.7049472 1.140091816
#> [240,] 0.4699155 -0.216975460
#> [241,] -0.3882240 -0.164614688
#> [242,] -0.3603105 -0.218926714
#> [243,] 0.3421271 -0.438927928
#> [244,] -0.4579667 0.007662013
#> [245,] 0.4077855 -0.346917409
#> [246,] -0.4972516 0.128934529
#> [247,] -0.2233641 -0.366883051
#> [248,] 0.4871672 -0.173078221
#> [249,] -0.3830042 -0.175394420
#> [250,] -0.5947103 0.515413533
#> [251,] -0.4420163 -0.036469872
#> [252,] -0.3763759 -0.188668432
#> [253,] -0.3212964 -0.281203535
#> [254,] 0.3236088 -0.456729033
#> [255,] 0.3134017 -0.465016315
#> [256,] -0.2061199 -0.371027470
#> [257,] -0.4112641 -0.113567291
#> [258,] -0.2583373 -0.348139209
#> [259,] 0.3899170 -0.376463356
#> [260,] -0.3387227 -0.255342855
#> [261,] -0.5452636 0.303035949
#> [262,] 0.4735912 -0.207917117
#> [263,] -0.6208480 0.642909713
#> [264,] 0.4432916 -0.277940220
#> [265,] -0.1904687 -0.371739436
#> [266,] -0.3610599 -0.217575323
#> [267,] 0.4285525 -0.308254211
#> [268,] -0.2872203 -0.322625513
#> [269,] -0.4276999 -0.073654210
#> [270,] -0.3313174 -0.266717993
#> [271,] -0.3030196 -0.304930534
#> [272,] -0.3662575 -0.208040913
#> [273,] -0.3231825 -0.278556916
#> [274,] -0.6102077 0.589645734
#> [275,] 0.4220828 -0.320802931
#> [276,] -0.3366689 -0.258554821
#> [277,] 0.4326564 -0.300055546
#> [278,] 0.6373453 0.377287730
#> [279,] 0.3623158 -0.415456259
#> [280,] 0.4429070 -0.278761965
#> [281,] -0.1617601 -0.365079051
#> [282,] -0.3226141 -0.279358403
#> [283,] -0.4544104 -0.002427478
#> [284,] 0.4660762 -0.226268965
#> [285,] -0.2126759 -0.369859887
#> [286,] -0.1521160 -0.360389943
#> [287,] -0.4898854 0.104795612
#> [288,] -0.3964954 -0.146941181
#> [289,] 0.8292105 1.788281338
#> [290,] -0.5876438 0.482833731
#> [291,] -0.3907903 -0.159208994
#> [292,] -0.5191140 0.204530833
#> [293,] -0.4286527 -0.071249970
#> [294,] -0.2708843 -0.338150714
#> [295,] -0.5380001 0.274763271
#> [296,] 0.3337495 -0.447422639
#> [297,] -0.4172275 -0.099425527
#> [298,] -0.4762981 0.061982933
#> [299,] -0.5322688 0.252954362
#> [300,] 0.4135180 -0.336712940
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] 0.42955822 -0.3062621661
#> [2,] -0.52973501 0.2434517576
#> [3,] -0.28823465 -0.3215683446
#> [4,] -0.52869416 0.2395726190
#> [5,] 0.40663873 -0.3489162529
#> [6,] 0.54168405 -0.0102359802
#> [7,] -0.52526049 0.2268761906
#> [8,] 0.50344893 -0.1283692101
#> [9,] -0.41601339 -0.1023359986
#> [10,] -0.29724418 -0.3117028140
#> [11,] 0.47337208 -0.2084614412
#> [12,] -0.54503161 0.3021219582
#> [13,] -0.69834289 1.0954872655
#> [14,] -0.39974297 -0.1398028443
#> [15,] -0.43213612 -0.0623746657
#> [16,] 0.47268275 -0.2101705193
#> [17,] -0.40977186 -0.1170459478
#> [18,] -0.21500360 -0.3693242050
#> [19,] -0.30122878 -0.3070678449
#> [20,] -0.37611303 -0.1891852565
#> [21,] -0.54481122 0.3012542763
#> [22,] -0.40371378 -0.1309216389
#> [23,] -0.41552125 -0.1035111881
#> [24,] 0.22746660 -0.4910545416
#> [25,] -0.52789747 0.2366130323
#> [26,] -0.58456746 0.4688916240
#> [27,] 0.26242852 -0.4900453414
#> [28,] -0.79582931 1.8868360742
#> [29,] -0.43903895 -0.0443904858
#> [30,] -0.22540331 -0.3661659456
#> [31,] -0.74325784 1.4217799995
#> [32,] -0.50342845 0.1496880858
#> [33,] -0.44540390 -0.0273371619
#> [34,] -0.44912130 -0.0171669152
#> [35,] 0.33128806 -0.4497798060
#> [36,] -0.36520854 -0.2099879166
#> [37,] -0.31999497 -0.2830081595
#> [38,] -0.25536180 -0.3502586231
#> [39,] -0.53421251 0.2603017665
#> [40,] -0.22984003 -0.3644424100
#> [41,] -0.59161992 0.5010695125
#> [42,] -0.34911156 -0.2384237556
#> [43,] -0.49317424 0.1154914473
#> [44,] -0.35058332 -0.2359360554
#> [45,] -0.43962509 -0.0428390037
#> [46,] -0.33849050 -0.2557082429
#> [47,] 0.45781888 -0.2456791564
#> [48,] 0.26076961 -0.4903958012
#> [49,] 0.32464740 -0.4558250499
#> [50,] -0.31353700 -0.2917022239
#> [51,] -0.66999359 0.9156040383
#> [52,] -0.37364482 -0.1940031476
#> [53,] 0.60488817 0.2295598786
#> [54,] -0.58141095 0.4547365098
#> [55,] 0.41017206 -0.3427121211
#> [56,] -0.40398845 -0.1303010591
#> [57,] -0.65042178 0.8014668339
#> [58,] -0.50315445 0.1487574773
#> [59,] 0.49457837 -0.1531284444
#> [60,] 0.61757229 0.2851337592
#> [61,] -0.58581890 0.4745456129
#> [62,] -0.24145761 -0.3588802707
#> [63,] -0.33308289 -0.2640578702
#> [64,] 0.36781987 -0.4083193219
#> [65,] -0.18954372 -0.3716881706
#> [66,] -0.34088551 -0.2519130923
#> [67,] 0.29447004 -0.4775093856
#> [68,] 0.39943727 -0.3611459674
#> [69,] 0.47248708 -0.2106546114
#> [70,] -0.28752708 -0.3223069547
#> [71,] -0.59455166 0.5146734180
#> [72,] -0.40889725 -0.1190735936
#> [73,] -0.37138205 -0.1983638148
#> [74,] -0.54485573 0.3014294512
#> [75,] -0.47854363 0.0689068606
#> [76,] 0.39150408 -0.3739767963
#> [77,] -0.47273295 0.0511118399
#> [78,] 0.36182016 -0.4160833974
#> [79,] -0.26440101 -0.3435234949
#> [80,] -0.20169622 -0.3715279138
#> [81,] 0.53987690 -0.0162509355
#> [82,] 0.38090450 -0.3900774851
#> [83,] -0.41891766 -0.0953472839
#> [84,] -0.64963888 0.7970597728
#> [85,] 0.26252643 -0.4900237273
#> [86,] 0.28754790 -0.4811391526
#> [87,] -0.44245135 -0.0353041666
#> [88,] -0.44578022 -0.0263146907
#> [89,] 0.56812600 0.0829379073
#> [90,] -0.30780632 -0.2990532134
#> [91,] -0.54378752 0.2972326024
#> [92,] 0.30723463 -0.4694974136
#> [93,] -0.72198586 1.2603213739
#> [94,] -0.42355218 -0.0840047125
#> [95,] -0.51094759 0.1755909041
#> [96,] 0.68020916 0.6025694540
#> [97,] 0.30003983 -0.4742232099
#> [98,] 0.82888116 1.7846052481
#> [99,] -0.24601794 -0.3562865805
#> [100,] -0.41198298 -0.1118830005
#> [101,] -0.48049046 0.0749581037
#> [102,] -0.49324381 0.1157191217
#> [103,] -0.34327921 -0.2480604410
#> [104,] -0.38936301 -0.1622240549
#> [105,] -0.42616493 -0.0775067330
#> [106,] -0.45148135 -0.0106292084
#> [107,] 0.36769072 -0.4084904174
#> [108,] -0.26860303 -0.3400926154
#> [109,] 0.44194476 -0.2808106494
#> [110,] 0.34661621 -0.4340758440
#> [111,] 0.42590837 -0.3134382525
#> [112,] -0.24745693 -0.3554205574
#> [113,] -0.39422934 -0.1518553757
#> [114,] -0.24296182 -0.3580501484
#> [115,] -0.43046994 -0.0666365668
#> [116,] -0.30445939 -0.3031879209
#> [117,] -0.64949728 0.7962639524
#> [118,] 0.61938272 0.2932867748
#> [119,] -0.50703397 0.1620206293
#> [120,] 0.48562451 -0.1771476088
#> [121,] 0.33075713 -0.4502799838
#> [122,] -0.44681935 -0.0234830652
#> [123,] 0.65208673 0.4506378314
#> [124,] -0.40530301 -0.1273197749
#> [125,] 0.30391882 -0.4717426050
#> [126,] -0.31148262 -0.2943768481
#> [127,] 0.54815742 0.0116742150
#> [128,] 0.58898109 0.1635609539
#> [129,] -0.29937325 -0.3092469314
#> [130,] 0.27836567 -0.4851735889
#> [131,] -0.45507832 -0.0005435284
#> [132,] -0.33965295 -0.2538736609
#> [133,] -0.45345428 -0.0051154411
#> [134,] -0.28929718 -0.3204492809
#> [135,] -0.45511213 -0.0004480313
#> [136,] 0.50557353 -0.1222948810
#> [137,] -0.23705804 -0.3611641047
#> [138,] 0.34289990 -0.4381076022
#> [139,] 0.56644077 0.0767043037
#> [140,] -0.16282806 -0.3655199389
#> [141,] -0.52120131 0.2120645787
#> [142,] 0.77028187 1.2222028642
#> [143,] 0.31640329 -0.4626918878
#> [144,] 0.41245150 -0.3386382307
#> [145,] -0.41384562 -0.1074927542
#> [146,] -0.42694749 -0.0755459236
#> [147,] -0.20972712 -0.3704476015
#> [148,] 0.21786471 -0.4889373901
#> [149,] 0.59889784 0.2042331666
#> [150,] -0.32271700 -0.2792135728
#> [151,] 0.70579746 0.7560000566
#> [152,] -0.73629479 1.3674525172
#> [153,] -0.36206073 -0.2157613799
#> [154,] -0.58317793 0.4626417233
#> [155,] -0.24159786 -0.3588039340
#> [156,] -0.41645367 -0.1012824189
#> [157,] -0.24246869 -0.3583250512
#> [158,] 0.42443975 -0.3162844977
#> [159,] -0.41000981 -0.1164928746
#> [160,] 0.26686894 -0.4889613550
#> [161,] -0.26505930 -0.3429985410
#> [162,] -0.33408950 -0.2625266303
#> [163,] -0.22603145 -0.3659355052
#> [164,] -0.42427116 -0.0822240465
#> [165,] 0.40273809 -0.3556094386
#> [166,] 0.47951526 -0.1929836045
#> [167,] 0.34455469 -0.4363300958
#> [168,] 0.40898199 -0.3448167691
#> [169,] -0.38730875 -0.1665256329
#> [170,] -0.54603725 0.3060896655
#> [171,] 0.36273205 -0.4149276126
#> [172,] 0.33416059 -0.4470228029
#> [173,] -0.43324539 -0.0595202715
#> [174,] -0.24509359 -0.3568308533
#> [175,] 0.20711529 -0.4853022321
#> [176,] 0.39992211 -0.3603400366
#> [177,] 0.55650686 0.0407871539
#> [178,] -0.58211200 0.4578672432
#> [179,] 0.53742301 -0.0243481308
#> [180,] -0.17307921 -0.3689779013
#> [181,] -0.54814230 0.3144398232
#> [182,] -0.47898435 0.0702727928
#> [183,] -0.30754944 -0.2993747115
#> [184,] 0.58500513 0.1476842390
#> [185,] -0.49383703 0.1176628686
#> [186,] 0.32093053 -0.4590085208
#> [187,] -0.45544864 0.0005031620
#> [188,] 0.46517009 -0.2284372571
#> [189,] 0.52423464 -0.0664970300
#> [190,] -0.54307882 0.2944567625
#> [191,] 0.44700193 -0.2699274190
#> [192,] 0.42170437 -0.3215228455
#> [193,] 0.34314244 -0.4378488553
#> [194,] -0.22595568 -0.3659635405
#> [195,] -0.22927182 -0.3646755930
#> [196,] -0.35843491 -0.2222832354
#> [197,] 0.45122455 -0.2606195462
#> [198,] -0.09197210 -0.2988561902
#> [199,] -0.29429122 -0.3150303337
#> [200,] -0.34438549 -0.2462597536
#> [201,] -0.25094091 -0.3532296809
#> [202,] 0.63370554 0.3598010008
#> [203,] -0.32183614 -0.2804499536
#> [204,] 0.29120411 -0.4792847853
#> [205,] -0.66201325 0.8681279739
#> [206,] -0.58516202 0.4715748464
#> [207,] 0.42889145 -0.3075840548
#> [208,] -0.09801533 -0.3078813149
#> [209,] -0.34560618 -0.2442580757
#> [210,] -0.30594582 -0.3013660751
#> [211,] 0.48124264 -0.1885510664
#> [212,] -0.46277692 0.0215392939
#> [213,] -0.48900653 0.1019593478
#> [214,] -0.27369036 -0.3356857579
#> [215,] -0.33608777 -0.2594556885
#> [216,] -0.35097745 -0.2352660361
#> [217,] -0.57951865 0.4463229310
#> [218,] -0.64976147 0.7977490814
#> [219,] -0.48541228 0.0904574690
#> [220,] -0.69604551 1.0802206759
#> [221,] -0.27473091 -0.3347503499
#> [222,] -0.48982858 0.1046118029
#> [223,] 0.39698531 -0.3651832847
#> [224,] -0.30921665 -0.2972759065
#> [225,] -0.52131063 0.2124606967
#> [226,] -0.36680190 -0.2070260044
#> [227,] -0.26380540 -0.3439944395
#> [228,] -0.32806045 -0.2715405446
#> [229,] -0.25896594 -0.3476791473
#> [230,] -0.44976367 -0.0153936843
#> [231,] 0.65057197 0.4429096402
#> [232,] -0.23853876 -0.3604194982
#> [233,] -0.30275985 -0.3052426618
#> [234,] 0.30363270 -0.4719309545
#> [235,] -0.42534093 -0.0795641162
#> [236,] -0.45846478 0.0090866013
#> [237,] 0.60049933 0.2109475053
#> [238,] -0.49610100 0.1251204181
#> [239,] -0.54299504 0.2941290474
#> [240,] -0.42034484 -0.0918793689
#> [241,] -0.61927366 0.6349075104
#> [242,] -0.40188498 -0.1350329601
#> [243,] -0.36907960 -0.2027457945
#> [244,] -0.28356331 -0.3263468619
#> [245,] 0.30205507 -0.4729541209
#> [246,] -0.42986554 -0.0681749853
#> [247,] -0.52907328 0.2409839127
#> [248,] 0.53192141 -0.0422096547
#> [249,] 0.18733815 -0.4750168494
#> [250,] -0.48914719 0.1024126325
#> [251,] 0.28239534 -0.4835129463
#> [252,] -0.49705567 0.1282838703
#> [253,] 0.45228900 -0.2582413525
#> [254,] 0.46727915 -0.2233754484
#> [255,] -0.41157900 -0.1128302382
#> [256,] -0.39485190 -0.1505107194
#> [257,] -0.47484038 0.0575198587
#> [258,] 0.33534694 -0.4458591464
#> [259,] 0.49550629 -0.1505836578
#> [260,] -0.47947577 0.0717985896
#> [261,] -0.23931740 -0.3600181473
#> [262,] -0.37831715 -0.1848287198
#> [263,] 0.47821447 -0.1962981302
#> [264,] 0.49618248 -0.1487226246
#> [265,] -0.34988277 -0.2371230104
#> [266,] -0.44655588 -0.0242021540
#> [267,] 0.35023332 -0.4300136955
#> [268,] -0.60228856 0.5512279095
#> [269,] 0.35194221 -0.4280471702
#> [270,] -0.68023645 0.9785169545
#> [271,] 0.52585015 -0.0614566848
#> [272,] -0.58915245 0.4897239267
#> [273,] 0.57403128 0.1051084153
#> [274,] -0.45276051 -0.0070593614
#> [275,] 0.23088074 -0.4915543461
#> [276,] -0.41418865 -0.1066801359
#> [277,] -0.19189578 -0.3717978396
#> [278,] 0.61622773 0.2791149497
#> [279,] 0.72817822 0.9036844907
#> [280,] -0.29072925 -0.3189222126
#> [281,] -0.51824411 0.2014078191
#> [282,] 0.44988642 -0.2635909721
#> [283,] -0.54652230 0.3080083056
#> [284,] 0.34281543 -0.4381975749
#> [285,] -0.51578848 0.1926436409
#> [286,] -0.47975096 0.0726542434
#> [287,] -0.40705679 -0.1233134723
#> [288,] -0.45847254 0.0091088070
#> [289,] -0.52745884 0.2349871501
#> [290,] -0.33108955 -0.2670590233
#> [291,] -0.28129361 -0.3285853522
#> [292,] -0.49031180 0.1061748189
#> [293,] 0.40226631 -0.3564079037
#> [294,] -0.46565799 0.0299786942
#> [295,] -0.35114732 -0.2349767584
#> [296,] -0.24047138 -0.3594109327
#> [297,] -0.54213365 0.2907653427
#> [298,] -0.43937970 -0.0434889957
#> [299,] -0.38743530 -0.1662619326
#> [300,] -0.23207536 -0.3634897193
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 4.4654436 0.2523231
#> [2,] 0.2523231 4.8551413
#>