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 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.- 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) doi:10.48550/arXiv.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
#> 0.8931122 1.0152501
#>
#> $se
#> [1] 0.11429185 0.09049167
#>
#> $lambda
#> [1] 0.6452418
#>
#> $rectifier_est
#> X(Intercept) XX1
#> 0.12340269 -0.02119519
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.29234541 -0.05370285
#> [2,] -0.05370285 0.34850964
#>
#> $var_l
#> [,1] [,2]
#> [1,] 3.6410544 0.5427797
#> [2,] 0.5427797 2.3822518
#>
#> $grads
#> [,1] [,2]
#> [1,] 0.185850086 -0.101969515
#> [2,] -0.537802143 -0.432387947
#> [3,] -0.953080666 0.058004109
#> [4,] 0.261338046 0.327210752
#> [5,] 0.023775774 -0.004887591
#> [6,] 0.493784543 0.759891383
#> [7,] 0.239432028 -0.224708704
#> [8,] 1.364818674 1.695363881
#> [9,] 0.524555528 -0.100341442
#> [10,] -0.243400272 0.126037166
#> [11,] 4.744173124 4.872710145
#> [12,] -0.944300901 -1.274991799
#> [13,] 0.577671476 0.236713403
#> [14,] -1.095856936 2.359528632
#> [15,] -2.711193687 -2.120147401
#> [16,] 0.898116469 0.004999821
#> [17,] -1.539522098 1.318869921
#> [18,] 1.754350534 0.458140661
#> [19,] 1.491895152 0.262400714
#> [20,] 0.756429061 -0.839782168
#> [21,] -0.090046196 0.088552736
#> [22,] -2.946640479 4.109307832
#> [23,] 0.703466041 0.482777205
#> [24,] -1.174565541 -0.104650147
#> [25,] 0.348149894 0.134725774
#> [26,] 2.687420942 1.353262775
#> [27,] -5.382839180 -4.868683254
#> [28,] -0.132207575 -0.002548527
#> [29,] 0.351674795 -0.061383051
#> [30,] 0.695594258 0.246662051
#> [31,] 1.085361196 0.238982985
#> [32,] 1.280932721 0.822849006
#> [33,] 2.382850454 -2.506475357
#> [34,] 0.651040661 0.351413466
#> [35,] 1.021211935 0.974091656
#> [36,] 1.082851913 -1.481452730
#> [37,] 1.746398788 -0.454186515
#> [38,] 0.302316594 0.102163317
#> [39,] 0.832509049 0.969883894
#> [40,] 2.793158302 1.277259582
#> [41,] -0.612390566 0.742185935
#> [42,] 0.227434527 -0.364038132
#> [43,] -0.377758803 -0.102829546
#> [44,] -7.404654168 -1.951268676
#> [45,] 1.851203707 0.636476151
#> [46,] -1.599293204 0.621883693
#> [47,] 1.388775255 0.142615751
#> [48,] 0.249525405 -0.451786516
#> [49,] 0.770369808 0.268171362
#> [50,] 0.015057926 0.002585219
#> [51,] -0.503920480 0.312113075
#> [52,] 0.085632176 0.009956438
#> [53,] -1.690025730 2.574940830
#> [54,] -3.476915004 3.149050535
#> [55,] 1.121204605 1.555348747
#> [56,] -0.510950980 0.195740796
#> [57,] 2.288742182 1.839591489
#> [58,] 0.401716902 0.661268281
#> [59,] -0.056676983 -0.101696207
#> [60,] 1.149537160 -0.378652115
#> [61,] -3.051467796 0.991746968
#> [62,] 0.272174794 -0.063099722
#> [63,] -2.596954567 -0.519127137
#> [64,] 0.228796054 0.033905952
#> [65,] 3.242654911 -0.417316970
#> [66,] 1.301109572 0.692851614
#> [67,] 1.713082859 -0.253540358
#> [68,] -1.100327633 -0.608175973
#> [69,] 0.200299376 -0.165053921
#> [70,] -0.220122325 0.145962352
#> [71,] -1.770384337 3.376883709
#> [72,] 1.534064218 3.163199793
#> [73,] 0.972545567 0.100460465
#> [74,] 1.185142040 -0.352126574
#> [75,] -0.045083612 0.030534450
#> [76,] -1.059321657 1.227985998
#> [77,] -2.994058114 -0.403556014
#> [78,] -1.392605973 1.120543800
#> [79,] -2.160507406 -3.493498954
#> [80,] -0.326678043 -0.185306974
#> [81,] -12.525413213 2.189122528
#> [82,] 0.374122053 -0.227901350
#> [83,] 3.353048956 -5.572303138
#> [84,] 0.385868045 -0.318588832
#> [85,] -1.417126694 -0.910238741
#> [86,] -0.844940339 0.709717443
#> [87,] 0.336494073 0.559098898
#> [88,] 0.288826766 -0.258134906
#> [89,] -0.406689133 0.144817125
#> [90,] -0.349056694 0.141539562
#> [91,] 1.675903105 -5.333529397
#> [92,] -5.661120308 1.350531878
#> [93,] -0.854453181 -0.521884994
#> [94,] 0.996159579 0.287494231
#> [95,] -3.251956517 -4.895223807
#> [96,] -2.114812420 2.212350967
#> [97,] 1.028723461 1.038477605
#> [98,] -0.232644227 0.247753883
#> [99,] -0.518103035 0.037548219
#> [100,] -1.113934110 0.625712717
#> [101,] 2.912718103 -1.883575369
#> [102,] 0.705280504 -0.614336243
#> [103,] 1.497765669 0.338944892
#> [104,] -7.982321006 -13.747906914
#> [105,] -6.322993934 1.998898558
#> [106,] -1.596576537 -0.899382237
#> [107,] 0.845551325 -0.416038880
#> [108,] -1.413636978 0.801072577
#> [109,] 0.066015216 0.050496165
#> [110,] 0.718286259 1.962797768
#> [111,] -0.089133438 -0.008935741
#> [112,] -0.164393416 0.016083312
#> [113,] -0.348430684 0.375164146
#> [114,] 0.775681051 0.601713078
#> [115,] -2.036955805 0.399587320
#> [116,] -1.661275340 1.995097730
#> [117,] 0.815295131 0.698748650
#> [118,] 0.793535683 0.399791387
#> [119,] 0.480541824 -0.404412661
#> [120,] -0.158265887 -0.066440384
#> [121,] 1.527235865 0.151681065
#> [122,] 0.880152581 2.322107879
#> [123,] -3.402830726 1.708993065
#> [124,] -1.011497496 0.159943329
#> [125,] -1.035339382 0.690873606
#> [126,] 1.389018578 2.113706933
#> [127,] -0.643597880 0.483597490
#> [128,] 2.880225517 -0.476653482
#> [129,] 0.985343705 0.186001146
#> [130,] -2.091828411 0.325558867
#> [131,] -1.257992457 0.385948924
#> [132,] 1.256323760 -0.766514411
#> [133,] -2.402229241 0.576080568
#> [134,] 0.185185505 -0.164975488
#> [135,] 0.808564207 -0.136581254
#> [136,] 2.193688639 -2.258136520
#> [137,] 0.048470763 -0.073153373
#> [138,] -0.268891720 -0.151586056
#> [139,] -0.389529899 0.227463963
#> [140,] 0.035486131 0.017716215
#> [141,] -1.088454857 1.412305369
#> [142,] -1.148611983 -1.360788629
#> [143,] -1.478757993 -1.493537953
#> [144,] -1.074676743 1.394385909
#> [145,] -2.914384596 0.874338324
#> [146,] 0.756631291 0.825114014
#> [147,] 0.598885859 0.210177037
#> [148,] 1.629286805 2.101941179
#> [149,] -0.708371445 0.343949913
#> [150,] -1.336022099 -1.083415492
#> [151,] 0.645727191 0.996513367
#> [152,] -14.041094018 -11.307769991
#> [153,] 1.442881072 2.663705463
#> [154,] 3.025138246 0.567526665
#> [155,] -1.991513140 -0.606967699
#> [156,] 1.382859347 0.087185304
#> [157,] -0.009409962 -0.001627998
#> [158,] 2.430251864 0.313295444
#> [159,] 0.577660680 -0.114683144
#> [160,] 2.825699526 -3.486981273
#> [161,] -1.023062475 0.624310570
#> [162,] 0.423139967 -0.858092980
#> [163,] 0.583797002 0.445480066
#> [164,] 1.072097236 0.360574903
#> [165,] 0.564208136 0.110448017
#> [166,] 2.904556239 0.610144524
#> [167,] 0.189480531 0.012966179
#> [168,] -0.417533535 -0.020564541
#> [169,] -0.606563057 0.299823564
#> [170,] -2.130888379 -2.827064857
#> [171,] 1.689439608 -2.208380433
#> [172,] 1.176459080 0.879516208
#> [173,] 1.750023042 2.459585909
#> [174,] 1.581496918 1.838231934
#> [175,] -0.538860084 -0.993805587
#> [176,] 2.702999360 -2.570572459
#> [177,] 2.157644239 -0.010812071
#> [178,] -0.777934398 1.936441065
#> [179,] 0.749290148 -0.253020053
#> [180,] 1.600222107 0.403400628
#> [181,] -0.501401568 -0.313714966
#> [182,] 0.170299449 -0.103109533
#> [183,] -2.194009823 -0.322696747
#> [184,] -0.251078147 -0.444539257
#> [185,] 0.879084671 1.162086646
#> [186,] 0.891545694 -0.917760950
#> [187,] -0.207766802 -0.076138592
#> [188,] 1.710541761 -2.061639087
#> [189,] -0.108973852 -0.236996383
#> [190,] -1.776907155 1.242223815
#> [191,] 0.262192222 0.043372694
#> [192,] 12.546143699 -4.225695985
#> [193,] 3.692040246 -1.763328974
#> [194,] 0.733550624 -0.028618326
#> [195,] -0.049409108 0.034759662
#> [196,] -0.041497895 0.029599718
#> [197,] -0.315092283 0.126065329
#> [198,] -1.076229882 -0.135566654
#> [199,] -1.875019434 3.319618125
#> [200,] -1.201318798 -0.762818535
#> [201,] -1.703601040 -0.923922681
#> [202,] 0.836634595 0.670001372
#> [203,] -2.133968254 2.919329728
#> [204,] 1.412328738 0.188194154
#> [205,] -1.487218261 -1.839931786
#> [206,] -0.957723632 -2.451144021
#> [207,] -0.040749175 0.056551286
#> [208,] 0.570573685 1.057740074
#> [209,] -0.655224377 -0.946715486
#> [210,] -0.783275814 0.141294036
#> [211,] 0.825855578 1.479365981
#> [212,] 0.068675889 0.010680496
#> [213,] -0.088876449 -0.049099491
#> [214,] -0.234896401 0.265132916
#> [215,] -2.047572501 -0.705168440
#> [216,] 2.383363834 -5.771399143
#> [217,] 1.267705431 -0.718380135
#> [218,] 1.473897455 0.910677690
#> [219,] 0.715573591 1.764358125
#> [220,] 0.746186147 1.427791853
#> [221,] 1.280427937 -2.300515882
#> [222,] -2.168156904 -2.324788172
#> [223,] 0.819382450 -0.436004591
#> [224,] 1.317854008 -0.216756640
#> [225,] -0.403164580 0.523690476
#> [226,] -1.625781684 -3.275907033
#> [227,] 0.561711550 -0.527118890
#> [228,] 1.047405749 0.849224967
#> [229,] 0.981919902 0.094546079
#> [230,] -0.502713859 -0.679273316
#> [231,] 0.242791630 -0.236535172
#> [232,] -0.136048262 -0.003671071
#> [233,] -0.181409208 0.003518503
#> [234,] -1.593115956 -0.941598903
#> [235,] -1.012200464 0.662405452
#> [236,] 1.132332199 -0.183527314
#> [237,] -1.601351180 0.377622868
#> [238,] 1.966044327 1.902764592
#> [239,] -0.780170871 -0.419588583
#> [240,] -1.107650699 -3.215633026
#> [241,] -1.213090752 -0.580635092
#> [242,] -0.518468756 -0.281671492
#> [243,] 0.873087373 -0.853822188
#> [244,] -0.827962253 -0.825257668
#> [245,] -1.286425629 -1.390288849
#> [246,] 0.333633712 -0.014427171
#> [247,] -4.634420275 3.479440300
#> [248,] -0.856205099 0.179034389
#> [249,] -1.105035870 -0.905338799
#> [250,] -0.033635571 -0.056372756
#> [251,] -2.651809558 -1.279983513
#> [252,] -0.973031086 -0.150430562
#> [253,] -1.064433684 -1.081908191
#> [254,] 0.198859512 0.421968407
#> [255,] -0.700298549 0.496395934
#> [256,] -1.056357241 0.287395215
#> [257,] 2.487270764 -1.710029887
#> [258,] 2.866051920 4.364967748
#> [259,] -1.283346907 -1.553713983
#> [260,] -0.890839068 -0.101519123
#> [261,] 2.828579975 1.167141592
#> [262,] -2.261533356 -2.419822278
#> [263,] 2.614397503 -1.403148921
#> [264,] 1.060558276 -0.529183672
#> [265,] -0.144520466 0.043004335
#> [266,] -0.401412764 0.204347051
#> [267,] -1.050673352 -0.815987017
#> [268,] 1.329727348 0.235632790
#> [269,] 2.260292804 2.212344896
#> [270,] 0.545771451 -0.812620517
#> [271,] 4.314005154 4.195793007
#> [272,] 0.710121491 0.267955090
#> [273,] 1.185548109 -0.591369320
#> [274,] 2.485581724 -0.976797994
#> [275,] 0.329997318 -0.257599135
#> [276,] 1.600151675 0.934501987
#> [277,] -1.171097480 0.609339558
#> [278,] -0.706117449 0.292129255
#> [279,] 1.766229668 -1.128230245
#> [280,] 0.682092862 0.096511858
#> [281,] -1.596116653 -1.706313456
#> [282,] -1.426971180 0.038238768
#> [283,] 1.972862090 1.246062896
#> [284,] -1.568172459 -2.428187572
#> [285,] 0.703438545 -0.744700535
#> [286,] 0.594974230 0.351037549
#> [287,] -0.361205193 0.489808248
#> [288,] 0.164608557 -0.164004689
#> [289,] 0.691045778 0.143716100
#> [290,] -0.944947108 -1.503952270
#> [291,] -0.176996878 0.295325904
#> [292,] 0.249459179 -0.078464560
#> [293,] 1.221134348 0.276736542
#> [294,] 1.224063157 2.845524042
#> [295,] -3.386574506 6.436071974
#> [296,] 0.310480131 0.062698935
#> [297,] 0.008780474 0.012941428
#> [298,] 3.200501853 0.617902380
#> [299,] 1.291808755 -0.399907201
#> [300,] -2.946027889 2.633400349
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] 1.221131630 -0.412186071
#> [2,] 0.921552983 1.309609564
#> [3,] -0.036402282 0.003897242
#> [4,] -0.987798990 0.651738058
#> [5,] -1.316067763 -1.824421252
#> [6,] -0.581039266 0.703857072
#> [7,] -0.251368405 0.113658168
#> [8,] -1.607192211 0.231952206
#> [9,] 1.648941600 2.467367056
#> [10,] -0.579173834 0.337896274
#> [11,] -0.525659907 -0.032571861
#> [12,] -0.231985031 -0.078058514
#> [13,] -1.011624618 1.749647467
#> [14,] 0.393307148 -0.209739588
#> [15,] -0.967698650 0.632227851
#> [16,] 0.236593951 0.110801256
#> [17,] 0.021169523 0.017361493
#> [18,] -0.133132942 -0.012967288
#> [19,] 1.450244335 -2.461305655
#> [20,] -0.441228509 0.513757534
#> [21,] 1.732932399 -1.120710458
#> [22,] 0.907859717 -0.024320732
#> [23,] 0.158787534 -0.134176805
#> [24,] 1.340320767 -1.571754127
#> [25,] -0.052771392 -0.081507749
#> [26,] -0.019292621 -0.006668394
#> [27,] -0.895920486 -1.133311427
#> [28,] -0.428094052 0.305002211
#> [29,] 1.199092752 -0.231968289
#> [30,] 0.815567632 2.042791518
#> [31,] -0.999041260 0.880037838
#> [32,] 1.280252306 -0.086679595
#> [33,] 0.052896996 -0.012462148
#> [34,] 0.091361601 -0.018931567
#> [35,] -0.058706657 -0.005510336
#> [36,] 0.356134276 -0.523412912
#> [37,] -0.359002257 0.160997199
#> [38,] -0.417702656 -0.112103714
#> [39,] 0.189309647 0.290560613
#> [40,] -0.931263675 0.548259727
#> [41,] -1.692775391 2.223069008
#> [42,] 0.742468639 0.549848265
#> [43,] 1.533804983 -1.375987132
#> [44,] 0.589035902 -0.898261161
#> [45,] 0.008295018 0.007855553
#> [46,] 0.460049797 0.555346533
#> [47,] 0.047960100 -0.004266489
#> [48,] -0.469091240 -0.452931975
#> [49,] -0.941120787 -1.004931571
#> [50,] 1.277287869 0.152955886
#> [51,] -0.136949562 -0.046672290
#> [52,] -0.534215841 0.190574557
#> [53,] 0.982062198 0.902705939
#> [54,] 0.500739675 -0.809066917
#> [55,] 0.101753357 -0.072788680
#> [56,] -0.318709773 -0.091300458
#> [57,] -0.348259719 0.226681253
#> [58,] 0.461328658 -0.708852275
#> [59,] -1.525705459 -0.046906952
#> [60,] 0.262253570 0.094743737
#> [61,] 0.247784521 -0.098404699
#> [62,] -0.223577776 0.139275223
#> [63,] 0.423804782 -0.228017985
#> [64,] 1.089747578 0.770151053
#> [65,] 0.518117143 1.244149718
#> [66,] -1.104323298 -2.228453491
#> [67,] 0.652522332 0.439048588
#> [68,] -1.653112431 1.094307308
#> [69,] -0.885510719 0.215584946
#> [70,] 0.211017318 0.171502797
#> [71,] 0.328987050 -0.232361203
#> [72,] 0.505832823 0.475547014
#> [73,] -0.490339622 0.072287095
#> [74,] 0.728758833 0.333916221
#> [75,] 0.371847058 -0.651656689
#> [76,] -0.352439715 0.327523608
#> [77,] -0.642894023 0.592230341
#> [78,] 0.271311224 0.483310628
#> [79,] 1.961289138 -0.585522480
#> [80,] 1.324785922 -1.053143143
#> [81,] 0.690759361 1.021309852
#> [82,] 0.403718644 -0.700921489
#> [83,] 0.728837615 -0.603394783
#> [84,] -0.185924602 0.111223264
#> [85,] 0.001286426 0.001813696
#> [86,] 0.340401544 -0.016274764
#> [87,] 0.646144934 -0.533113103
#> [88,] -0.249421560 0.001220306
#> [89,] 0.728167773 -1.665273871
#> [90,] 1.654967755 -0.422708842
#> [91,] 0.485672258 0.536588813
#> [92,] 0.501518321 -0.209947114
#> [93,] 0.679924846 1.134686602
#> [94,] -0.078663588 -0.105977134
#> [95,] 0.888957585 1.066902304
#> [96,] 1.089232628 -0.502277591
#> [97,] -0.514932181 -0.175059030
#> [98,] -0.847477949 1.608922063
#> [99,] 0.166667846 -0.165497113
#> [100,] 1.422266331 0.061545454
#> [101,] -0.861859852 1.660453685
#> [102,] 0.999110766 -1.065865123
#> [103,] 0.824525474 -0.052130407
#> [104,] -0.557937557 0.039566017
#> [105,] -0.061067904 -0.007210787
#> [106,] -0.250216764 -0.182824349
#> [107,] 0.418894985 -0.500384422
#> [108,] -0.534012782 0.061275978
#> [109,] 0.379813486 0.202044079
#> [110,] 0.647041185 0.089016852
#> [111,] 0.205055958 -0.237140932
#> [112,] -0.962719210 0.940461928
#> [113,] 1.757885205 0.737176472
#> [114,] -1.400836331 -0.128395394
#> [115,] -0.107463679 -0.239080646
#> [116,] 1.228125702 0.420689377
#> [117,] -0.889127775 1.266302998
#> [118,] 0.751900134 0.675289153
#> [119,] 0.436717911 0.679400483
#> [120,] -1.723484681 3.371816333
#> [121,] 0.465746053 -0.119858216
#> [122,] 0.848928221 0.785102960
#> [123,] 1.144049356 -0.790408767
#> [124,] 0.826149391 0.843473993
#> [125,] -0.212523036 -0.180184785
#> [126,] 0.085700861 -0.051714744
#> [127,] 0.004183594 0.006867658
#> [128,] -0.134392137 0.022748411
#> [129,] 0.364770506 -0.063914254
#> [130,] 0.192682203 -0.128158609
#> [131,] 0.308959839 0.200715830
#> [132,] 0.884470483 1.393522534
#> [133,] -0.631389241 0.180562919
#> [134,] -0.741337631 -0.249275539
#> [135,] 1.077614385 -0.564087040
#> [136,] 1.946024426 0.365353751
#> [137,] 1.808295413 -1.623731969
#> [138,] 1.090787079 -1.831873306
#> [139,] -0.271792203 -0.112049342
#> [140,] 1.516351743 0.183429436
#> [141,] 0.648015005 -0.566182683
#> [142,] -0.073554146 -0.059220003
#> [143,] 1.691509501 -0.492275440
#> [144,] -0.524032302 -0.450428914
#> [145,] 0.873103855 0.861176768
#> [146,] 0.455018198 -0.809587933
#> [147,] 0.619387742 -0.029831436
#> [148,] 0.221294646 0.225234934
#> [149,] 0.733325846 1.156240305
#> [150,] 0.417995591 0.263972641
#> [151,] 0.141682727 0.302853583
#> [152,] 0.675851333 -0.935384778
#> [153,] -0.570408952 -0.368147807
#> [154,] -0.750318565 0.606952142
#> [155,] 0.943558905 0.130478410
#> [156,] 0.015855818 -0.012006073
#> [157,] -0.464789131 -0.333485072
#> [158,] 0.742211240 -0.064162558
#> [159,] -0.790026192 -1.805086076
#> [160,] -0.889602809 1.246707776
#> [161,] 0.038823595 -0.045613024
#> [162,] 0.481703681 0.120780425
#> [163,] 0.431437413 -0.230383553
#> [164,] 0.113397757 0.116799531
#> [165,] -0.075574658 -0.042000505
#> [166,] -0.305789554 0.414871915
#> [167,] 2.074087469 -1.178989240
#> [168,] 0.431829476 0.110573509
#> [169,] -1.037853156 -0.396904145
#> [170,] -1.120985628 -1.426230534
#> [171,] -0.143655925 -0.212093528
#> [172,] 0.337494634 0.020773799
#> [173,] -0.315864684 -0.098196380
#> [174,] 0.533981396 -0.230830084
#> [175,] -0.326398902 -0.433594992
#> [176,] 0.342670988 0.028581100
#> [177,] 1.049403175 -1.120529573
#> [178,] -1.426938611 -2.463945722
#> [179,] 1.302082040 -0.721476042
#> [180,] 0.677588893 -0.110433424
#> [181,] -0.247227617 0.315047983
#> [182,] 0.329649679 -0.219526763
#> [183,] -1.250106191 -0.630955021
#> [184,] -0.702338773 -0.164637434
#> [185,] 0.670071305 -0.174063203
#> [186,] 0.233063876 -0.124176063
#> [187,] 0.944973257 -0.112651169
#> [188,] -0.615109830 0.612490653
#> [189,] -0.727941409 0.918637536
#> [190,] -1.098313260 2.233513015
#> [191,] 0.207442855 -0.137400772
#> [192,] 0.743237762 0.493009491
#> [193,] -0.975678053 0.682839018
#> [194,] 1.874945841 -2.787902951
#> [195,] 0.894892958 -0.216155166
#> [196,] 0.715502431 0.562584087
#> [197,] -0.438097358 -0.114674127
#> [198,] 0.483045234 0.122845304
#> [199,] 0.607932519 0.337491730
#> [200,] -1.435804781 0.389782759
#> [201,] 0.312112543 0.704892639
#> [202,] -0.234128990 -0.124497050
#> [203,] 0.890842884 -0.592507647
#> [204,] -0.545488687 0.078409726
#> [205,] -1.871825088 -0.075319179
#> [206,] -0.941181274 -1.363021893
#> [207,] 0.964933527 1.469071651
#> [208,] 0.839414504 -0.369532212
#> [209,] -0.623808667 0.330902525
#> [210,] 0.116868999 0.047785558
#> [211,] -0.491568422 0.765944427
#> [212,] -0.378601283 0.287402735
#> [213,] 0.631012053 0.549206463
#> [214,] 0.355696842 -0.275531450
#> [215,] -0.142184900 0.037694039
#> [216,] -0.873104331 0.474326099
#> [217,] 0.029041151 0.020751844
#> [218,] 0.771887132 0.765646328
#> [219,] -0.622779192 -0.251094747
#> [220,] -1.019216368 1.802776031
#> [221,] 1.146641723 1.011736093
#> [222,] -0.686253275 -1.362175336
#> [223,] 0.574613754 0.080875404
#> [224,] 1.010735445 -1.167646702
#> [225,] 0.852289922 0.568106588
#> [226,] 1.510292700 -0.191848617
#> [227,] 0.326164475 -0.406296686
#> [228,] 0.346536678 0.144574102
#> [229,] -1.533334160 -2.415951214
#> [230,] -0.155088332 0.032757970
#> [231,] -0.082025195 0.033122151
#> [232,] 1.335379195 0.299827345
#> [233,] -0.430988524 -0.073359810
#> [234,] 0.551841789 0.363835308
#> [235,] -0.356629766 -0.543046109
#> [236,] -0.307850899 0.644023806
#> [237,] 0.872587429 -1.160131500
#> [238,] 0.660640182 1.363811969
#> [239,] 0.834433127 0.958844621
#> [240,] -0.028350180 -0.008179543
#> [241,] 0.709523687 -0.813695524
#> [242,] -0.535448275 0.695348942
#> [243,] 0.292484639 -0.263606963
#> [244,] 0.070662181 0.075317860
#> [245,] -1.494781617 0.195648013
#> [246,] 1.348617875 0.261099149
#> [247,] 0.464069545 -0.394464331
#> [248,] 0.141910902 0.143572033
#> [249,] 1.338874083 -0.090771513
#> [250,] 0.685951576 0.391147531
#> [251,] 0.111465680 0.011549084
#> [252,] -0.378028098 0.249992197
#> [253,] 0.448178079 0.108191920
#> [254,] -0.883415483 1.061310868
#> [255,] 0.472595879 -0.554282038
#> [256,] 1.543002322 -1.950492298
#> [257,] 0.912818481 -0.433637582
#> [258,] 0.304517948 0.075210820
#> [259,] -0.957291884 -0.450625210
#> [260,] -1.352936308 1.364751755
#> [261,] 0.935147845 -0.179133190
#> [262,] -1.828994013 -1.552237115
#> [263,] 1.159548447 -1.821239431
#> [264,] 0.573598167 -0.069531373
#> [265,] 0.662620001 -0.935778703
#> [266,] 0.203040314 0.103291774
#> [267,] 1.306969341 -0.307278513
#> [268,] -0.592945142 0.623787289
#> [269,] 0.005285423 -0.014087230
#> [270,] -0.397981094 -0.711376033
#> [271,] 1.464352147 -3.928857695
#> [272,] -1.875368764 0.112419123
#> [273,] 0.656781187 -0.153392139
#> [274,] 0.766906944 0.313550666
#> [275,] 1.067406734 -2.271061842
#> [276,] -0.117730538 -0.048239372
#> [277,] 0.578994328 0.285471554
#> [278,] 1.078197604 0.045804685
#> [279,] -1.293099723 -1.097666634
#> [280,] 1.000823133 0.251538040
#> [281,] -0.803330171 -2.073786760
#> [282,] 1.789307189 -0.729264638
#> [283,] 0.076639247 0.082236074
#> [284,] 0.042294184 -0.010955591
#> [285,] 1.632775890 -3.763919595
#> [286,] -1.090336560 1.097844602
#> [287,] -0.251674021 -0.071430738
#> [288,] 1.748540269 4.188146712
#> [289,] -1.061928391 -1.782032236
#> [290,] -1.484220370 2.849011429
#> [291,] -0.800791932 0.239529991
#> [292,] -1.316228969 0.471553746
#> [293,] -0.056494587 -0.066246350
#> [294,] 0.627026129 0.341407334
#> [295,] 0.228360513 0.339989209
#> [296,] -0.015088233 0.015759561
#> [297,] 0.175723264 -0.502053399
#> [298,] 0.616229737 0.072661607
#> [299,] 0.791377585 0.024013557
#> [300,] -0.259083486 -0.059484306
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] -0.5413647935 0.2970281411
#> [2,] -1.2961803859 -1.0421170393
#> [3,] 0.4273962134 -0.0260111628
#> [4,] -0.2249547827 -0.2816567460
#> [5,] -0.2770255650 0.0569482001
#> [6,] 0.7107179245 1.0937329548
#> [7,] -0.7490750971 0.7030124415
#> [8,] 1.0539203496 1.3091691429
#> [9,] 0.3552140618 -0.0679483665
#> [10,] 0.1196447319 -0.0619542567
#> [11,] 1.7739481164 1.8220108663
#> [12,] -0.0053053023 -0.0071632007
#> [13,] -0.3838953663 -0.1573094435
#> [14,] -0.1499689769 0.3229035499
#> [15,] 0.2881853209 0.2253602765
#> [16,] 0.0431372279 0.0002401452
#> [17,] 0.5335477364 -0.4570769472
#> [18,] 1.1925614629 0.3114320011
#> [19,] -0.5137258625 -0.0903562379
#> [20,] 0.9385888165 -1.0420146335
#> [21,] -0.0009563219 0.0009404608
#> [22,] -1.2155638334 1.6951935656
#> [23,] -0.4745650039 -0.3256861781
#> [24,] 1.8760014605 0.1671459125
#> [25,] 0.9273029559 0.3588443092
#> [26,] 0.9309433202 0.4687806518
#> [27,] -1.7391533582 -1.5730335883
#> [28,] -0.4004701624 -0.0077197474
#> [29,] 0.4715629002 -0.0823089111
#> [30,] 1.3528973833 0.4797458277
#> [31,] 0.4952000573 0.1090368702
#> [32,] 0.7103398948 0.4563100519
#> [33,] 0.0121723463 -0.0128038610
#> [34,] 0.4290777683 0.2316041300
#> [35,] 0.4645671202 0.4431312832
#> [36,] -0.0231942012 0.0317320516
#> [37,] -0.7118381873 0.1851279947
#> [38,] 0.9774780958 0.3303239285
#> [39,] 0.1448642875 0.1687687832
#> [40,] 0.3287277558 0.1503211170
#> [41,] -0.4977958732 0.6033030485
#> [42,] 0.9384928239 -1.5021781395
#> [43,] -0.4205862378 -0.1144875819
#> [44,] -0.1888495968 -0.0497654981
#> [45,] -0.6845024003 -0.2353438747
#> [46,] 0.2203296086 -0.0856749658
#> [47,] -0.0414121115 -0.0042526819
#> [48,] -0.3096143335 0.5605825242
#> [49,] 0.9505617113 0.3308974812
#> [50,] -0.4598797337 -0.0789544333
#> [51,] -0.4189594892 0.2594908117
#> [52,] -0.0589826096 -0.0068578975
#> [53,] -0.7109505798 1.0832117191
#> [54,] 0.0101162125 -0.0091622788
#> [55,] -0.7455688918 -1.0342622897
#> [56,] 0.0192808816 -0.0073863350
#> [57,] -0.0687094278 -0.0552256517
#> [58,] 0.3337638575 0.5494104215
#> [59,] -0.2817240065 -0.5055008434
#> [60,] -0.2745197530 0.0904255109
#> [61,] 0.3803281219 -0.1236091241
#> [62,] 0.3877158212 -0.0898862090
#> [63,] -1.3315961686 -0.2661839816
#> [64,] 0.5402629063 0.0800631300
#> [65,] 1.5638063121 -0.2012557396
#> [66,] -0.5025786110 -0.2676272694
#> [67,] 0.9118423527 -0.1349548481
#> [68,] 0.6609573636 0.3653260860
#> [69,] -0.1213289576 0.0999794438
#> [70,] -0.9986885250 0.6622269047
#> [71,] -0.7579977241 1.4458273903
#> [72,] 0.1942020566 0.4004394977
#> [73,] -0.4346764275 -0.0449005144
#> [74,] -0.4994973870 0.1484094713
#> [75,] 0.4340507404 -0.2939760123
#> [76,] -0.0687217843 0.0796636113
#> [77,] 0.2277367986 0.0306956482
#> [78,] -0.9220616676 0.7419259324
#> [79,] 0.7187053397 1.1621327220
#> [80,] 0.1510683596 0.0856929971
#> [81,] -2.4496053024 0.4281284825
#> [82,] 0.8831872056 -0.5380050569
#> [83,] 1.0348842477 -1.7198343409
#> [84,] 0.8488993428 -0.7008868793
#> [85,] -0.5140936055 -0.3302089491
#> [86,] -0.9557647086 0.8028056581
#> [87,] 0.9343624741 1.5524821130
#> [88,] 0.9573498358 -0.8556181030
#> [89,] 0.3205093331 -0.1141295317
#> [90,] -0.3974871802 0.1611777184
#> [91,] 1.4293128194 -4.5487605545
#> [92,] -0.5957602335 0.1421261417
#> [93,] 0.1106353196 0.0675741098
#> [94,] 0.4168850071 0.1203140912
#> [95,] -1.4353732654 -2.1606910619
#> [96,] -1.3795619058 1.4431895176
#> [97,] 0.4753668712 0.4798742020
#> [98,] 0.0215270546 -0.0229251825
#> [99,] 0.2263971193 -0.0164075637
#> [100,] 0.4056479030 -0.2278582272
#> [101,] 1.1874260539 -0.7678760486
#> [102,] 0.9940335048 -0.8658552240
#> [103,] 0.6280751165 0.1421336175
#> [104,] -2.1655023402 -3.7296325932
#> [105,] -2.0409370838 0.6452048251
#> [106,] -0.1312646497 -0.0739438990
#> [107,] 0.2552703008 -0.1256013289
#> [108,] 0.3097658549 -0.1755365313
#> [109,] 1.0337496198 0.7907327240
#> [110,] 0.2966606935 0.8106586195
#> [111,] 0.8291128277 0.0831196227
#> [112,] 1.2204597388 -0.1194028074
#> [113,] 0.1213203419 -0.1306286863
#> [114,] 1.1768500473 0.9129088093
#> [115,] -1.0411643420 0.2042440333
#> [116,] 0.1129705273 -0.1356712142
#> [117,] -1.3187050367 -1.1301960845
#> [118,] -0.7886915045 -0.3973508404
#> [119,] 1.4703578664 -1.2374184863
#> [120,] -1.0932396228 -0.4589445137
#> [121,] 0.2957194920 0.0293700852
#> [122,] -0.0404160594 -0.1066297502
#> [123,] -1.2735627047 0.6396174261
#> [124,] -0.1145097393 0.0181068850
#> [125,] 0.6416976643 -0.4281996678
#> [126,] 0.9774811973 1.4874594311
#> [127,] -0.1499364415 0.1126617862
#> [128,] -0.8425578993 0.1394363581
#> [129,] -0.3877317064 -0.0731912545
#> [130,] -0.4658134934 0.0724962489
#> [131,] 0.4185232209 -0.1284018724
#> [132,] -0.0814710294 0.0497075038
#> [133,] -0.2627070954 0.0630000044
#> [134,] 0.5337534028 -0.4755028114
#> [135,] -0.5463186180 0.0922831873
#> [136,] 1.3037101931 -1.3420115991
#> [137,] 0.6763767852 -1.0208059547
#> [138,] -0.0055240746 -0.0031141631
#> [139,] 0.7165245211 -0.4184107758
#> [140,] 0.3244475553 0.1619782852
#> [141,] -0.4209288999 0.5461688568
#> [142,] -0.2803679502 -0.3321587483
#> [143,] 0.4494484885 0.4539406576
#> [144,] -0.1227278047 0.1592385084
#> [145,] -1.2779255183 0.3833877165
#> [146,] 0.5886008945 0.6418751812
#> [147,] 0.5647127437 0.1981840933
#> [148,] 0.8947164079 1.1542726891
#> [149,] -0.2432279171 0.1180993694
#> [150,] 0.2365030075 0.1917865150
#> [151,] 0.0099417606 0.0153425431
#> [152,] -2.6130687539 -2.1043930339
#> [153,] 1.4471688831 2.6716211989
#> [154,] 1.1364261566 0.2131975778
#> [155,] 0.4110499706 0.1252786386
#> [156,] -0.1378239741 -0.0086894051
#> [157,] -0.4873034866 -0.0843073904
#> [158,] 1.0324907591 0.1331033445
#> [159,] -0.4029494238 0.0799976672
#> [160,] 1.8163710494 -2.2414456235
#> [161,] 0.8906511803 -0.5435082992
#> [162,] 0.1286018192 -0.2607938908
#> [163,] 1.1391943927 0.8692891384
#> [164,] 0.6875357156 0.2312366042
#> [165,] 1.3411190653 0.2625342171
#> [166,] -0.3370294974 -0.0707979758
#> [167,] 0.0837891010 0.0057336998
#> [168,] -0.8079918954 -0.0397955641
#> [169,] -0.0879670579 0.0434820362
#> [170,] 0.8319721278 1.1037833741
#> [171,] -0.7541503744 0.9858008079
#> [172,] 0.9356056344 0.6994551135
#> [173,] 0.0500103289 0.0702874747
#> [174,] -0.1946546085 -0.2262541984
#> [175,] -0.7920862166 -1.4608239319
#> [176,] 1.5777157683 -1.5004194087
#> [177,] 1.1581790309 -0.0058036974
#> [178,] 0.9811894981 -2.4423854286
#> [179,] -0.5399481826 0.1823295262
#> [180,] 0.7672073338 0.1934056022
#> [181,] 0.9141055386 0.5719339680
#> [182,] -0.0609494993 0.0369024940
#> [183,] -0.4003343047 -0.0588814947
#> [184,] 0.7031438785 1.2449313532
#> [185,] -0.6607247008 -0.8734304868
#> [186,] 0.4801114942 -0.4942288252
#> [187,] -0.0486519989 -0.0178290981
#> [188,] 0.1791038059 -0.2158657656
#> [189,] -0.4267434982 -0.9280819537
#> [190,] 0.1876543466 -0.1311878889
#> [191,] 0.0380807856 0.0062994480
#> [192,] 2.6092790624 -0.8788373801
#> [193,] 1.3816371654 -0.6598738593
#> [194,] -0.3984015491 0.0155430110
#> [195,] -0.2790375205 0.1963048980
#> [196,] -1.0697851730 0.7630589394
#> [197,] 1.0128764642 -0.4052419303
#> [198,] -0.9569853333 -0.1205460857
#> [199,] -0.7596591653 1.3449345050
#> [200,] 0.5496132229 0.3489957492
#> [201,] 0.3954490150 0.2144658905
#> [202,] 1.6177309219 1.2955260804
#> [203,] -1.0186807342 1.3935844386
#> [204,] -0.5272599978 -0.0702578986
#> [205,] -1.6706674918 -2.0668884330
#> [206,] -0.1050592856 -0.2688828293
#> [207,] 0.4770569993 -0.6620547956
#> [208,] 0.9971009910 1.8484443005
#> [209,] -1.3883636722 -2.0060080691
#> [210,] 0.4905626751 -0.0884919197
#> [211,] 0.3933275518 0.7045728275
#> [212,] -0.7702387924 -0.1197877840
#> [213,] -0.1472966578 -0.0813735358
#> [214,] -0.1427856448 0.1611654085
#> [215,] -0.7091499702 -0.2442258714
#> [216,] 1.3517027146 -3.2731955463
#> [217,] -0.2132928217 0.1208682414
#> [218,] 1.1944755368 0.7380311426
#> [219,] -0.7887779584 -1.9448548933
#> [220,] -0.2022720645 -0.3870380159
#> [221,] 1.5622000955 -2.8067695394
#> [222,] -0.4712233235 -0.5052652819
#> [223,] -0.5074609951 0.2700269253
#> [224,] -0.0697662942 0.0114749490
#> [225,] 0.6485718130 -0.8424621063
#> [226,] 0.0041748334 0.0084121788
#> [227,] 0.4317843385 -0.4051931662
#> [228,] 0.6350713262 0.5149087892
#> [229,] -0.6688318928 -0.0643997873
#> [230,] -0.1011401139 -0.1366617994
#> [231,] -0.0463793619 0.0451842197
#> [232,] -0.2823600088 -0.0076190886
#> [233,] 0.4505455867 -0.0087385093
#> [234,] -0.5603260033 -0.3311763643
#> [235,] 0.5384638648 -0.3523821738
#> [236,] 0.7484194725 -0.1213031082
#> [237,] 0.1565896175 -0.0369262040
#> [238,] 0.8702067343 0.8421979806
#> [239,] 0.5632750104 0.3029384618
#> [240,] -0.5691978581 -1.6524446137
#> [241,] 0.7547751532 0.3612664098
#> [242,] -0.4602397825 -0.2500371036
#> [243,] -0.8692932477 0.8501117818
#> [244,] -0.8963702584 -0.8934422154
#> [245,] 0.0485563335 0.0524766668
#> [246,] 0.3901199621 -0.0168697797
#> [247,] 0.6090594745 -0.4572710187
#> [248,] -0.1896761249 0.0396616993
#> [249,] 0.8215836452 0.6731107747
#> [250,] 0.0287735035 0.0482239985
#> [251,] -1.2278969851 -0.5926850561
#> [252,] -0.6181002871 -0.0955582767
#> [253,] 1.1623468555 1.1814287752
#> [254,] -0.3719782495 -0.7893163747
#> [255,] -0.8881710570 0.6295664926
#> [256,] -0.4293471579 0.1168092706
#> [257,] -0.3703486370 0.2546193391
#> [258,] -0.3364561099 -0.5124192126
#> [259,] -0.3172406171 -0.3840747816
#> [260,] 0.5558494089 0.0633440387
#> [261,] -0.4929321466 -0.2033959144
#> [262,] -0.7597874768 -0.8129664141
#> [263,] 0.1622893388 -0.0871007987
#> [264,] 0.0631440104 -0.0315067829
#> [265,] 1.0609131035 -0.3156913570
#> [266,] -0.4302519428 0.2190282020
#> [267,] -0.5897364744 -0.4580084811
#> [268,] -0.4447106170 -0.0788044282
#> [269,] 0.8194293104 0.8020466416
#> [270,] -0.7841767618 1.1675915336
#> [271,] 1.9562465693 1.9026416018
#> [272,] -0.1493266423 -0.0563464623
#> [273,] 1.7507014158 -0.8732763335
#> [274,] 0.7473524573 -0.2936988046
#> [275,] 0.1909145646 -0.1490297773
#> [276,] 0.0728041579 0.0425182383
#> [277,] -0.9978909321 0.5192175974
#> [278,] 0.3888725122 -0.1608812210
#> [279,] 0.0223319400 -0.0142651721
#> [280,] 1.1049853907 0.1563484958
#> [281,] -1.1763626203 -1.2575793661
#> [282,] -0.9831725022 0.0263462257
#> [283,] 0.1215149652 0.0767490491
#> [284,] -1.5005243044 -2.3234399038
#> [285,] 0.6394252938 -0.6769324231
#> [286,] 0.4401706044 0.2597026936
#> [287,] -0.7328901751 0.9938274964
#> [288,] -0.4301601026 0.4285820574
#> [289,] 0.2260091415 0.0470028952
#> [290,] -0.3599338724 -0.5728610205
#> [291,] 0.6728396747 -1.1226581338
#> [292,] 0.4792842726 -0.1507534419
#> [293,] 0.0757014771 0.0171556594
#> [294,] 1.2117489191 2.8168976913
#> [295,] -1.5903269163 3.0223632985
#> [296,] -0.1029681625 -0.0207935820
#> [297,] -0.7171842573 -1.0570486693
#> [298,] 1.3391923380 0.2585501184
#> [299,] -0.2706262255 0.0837781722
#> [300,] -0.7360494135 0.6579410840
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 1.00037870 -0.01898342
#> [2,] -0.01898342 0.95161054
#>