Posterior Inference and MCMC Algorithms

Day 3

Recap: The Two-Parameter Normal Model and A Gibbs Sampler

The Normal Model and A Joint Prior

• The sampling model for \(Y_1, \cdots, Y_n\): \[\begin{equation} Y_1, \cdots, Y_n \mid \mu, \sigma \overset{i.i.d.}{\sim} \textrm{Normal}(\mu, \sigma^2) \end{equation}\]

• A joint prior distribution as: \[\begin{equation} f(\mu, \sigma^2) = f(\mu) f(\sigma^2) \end{equation}\]

• And let priors be: \[\begin{eqnarray} \mu &\sim& \textrm{Normal}(\theta, \tau^2)\\ 1/\sigma^2 &\sim& \textrm{Gamma}(\alpha, \beta) \end{eqnarray}\]

Full Conditional Posterior Distributions

\[\begin{eqnarray} \mu \mid y_1, \cdots, y_n,{\color{red}\phi} &\sim& \textrm{Normal}\left(\frac{\phi_0 \theta + n\phi\bar{y}}{\phi_0 + n \phi}, \frac{1}{\phi_0 + n \phi}\right) \\ \nonumber 1/\sigma^2 = \phi \mid y_1, \cdots, y_n,{\color{red}\mu} &\sim& \textrm{Gamma} \left(\alpha + \frac{n}{2}, \beta + \frac{1}{2}\sum_{i=1}^{n}(y_i - \mu)^2 \right) \nonumber \\ \end{eqnarray}\]

• We can use a Gibbs sampler for posterior sampling

R Function for A Gibbs Sampler

gibbs_normal <- function(input, S, seed){
  set.seed(seed)
  ybar <- mean(input$y)
  n <- length(input$y)
  para <- matrix(0, S, 2)
  phi <- input$phi_init
  for(s in 1:S){
    theta1 <- (input$theta/input$tau^2 + n*phi*ybar)/
    (1/input$tau^2 + n*phi)
    tau1 <- sqrt(1/(1/input$tau^2 + n*phi))
    mu <- rnorm(1, mean = theta1, sd = tau1)
    alpha1 <- input$alpha + n/2
    beta1 <- input$beta + sum((input$y - mu)^2)/2 
    phi <- rgamma(1, shape = alpha1, rate = beta1)
    para[s, ] <- c(mu, phi)
  }
  para 
}

A Two-Parameter Normal Model for The Concussion Example

Recall The Concussion Example

• Continued example from bayesrulesbook.com

• Let \((Y_1,Y_2,\ldots,Y_{25})\) denote the hippocampal volumes for the \(n = 25\) study subjects who played collegiate American football and who have been diagnosed with concussions:

# Load the data
library(bayesrules)
library(tidyverse)
data(football)

# Filter the data
football <- football %>%
  filter(group == "fb_concuss") 

Choosing Priors

\[\begin{eqnarray} \mu &\sim& \textrm{Normal}(\theta, \tau^2)\\ 1/\sigma^2 &\sim& \textrm{Gamma}(\alpha, \beta) \end{eqnarray}\]

• Previously, we selected the prior \(\mu \sim \textrm{Normal}(6.5, 0.5^2)\)

• For \(1/\sigma^2\) (or \(\phi = 1/\sigma^2)\), let’s try \(1/\sigma^2 \sim \textrm{Gamma}(1, 1)\)

Input for The Gibbs Sampler

Discussion question

What should be included in input?

gibbs_normal <- function(input, S, seed){
  set.seed(seed)
  ybar <- mean(input$y)
  n <- length(input$y)
  para <- matrix(0, S, 2)
  phi <- input$phi_init
  for(s in 1:S){
    theta1 <- (input$theta/input$tau^2 + n*phi*ybar)/
    (1/input$tau^2 + n*phi)
    tau1 <- sqrt(1/(1/input$tau^2 + n*phi))
    mu <- rnorm(1, mean = theta1, sd = tau1)
    alpha1 <- input$alpha + n/2
    beta1 <- input$beta + sum((input$y - mu)^2)/2 
    phi <- rgamma(1, shape = alpha1, rate = beta1)
    para[s, ] <- c(mu, phi)
  }
  para 
}

Running the Gibbs Sampler

• We create input to include: the data y, prior parameters of theta, tau, alpha, beta, and the initial value of phi

• We then run the gibbs_normal() function with input, S, seed

• We save the output of running the gibbs_normal() function in output

input <- list(y = football$volume, theta = 6.5, tau = 0.5,
              alpha = 1, beta = 1, phi_init = 1)
output <- gibbs_normal(input, S = 10000, seed = 123)

Discussion question

What do we expect to see in output?

Summarizing MCMC Output and Posterior Analysis

MCMC Output

print(output)

             [,1]      [,2]
    [1,] 5.736095 2.3315252
    [2,] 5.569675 3.2078545
    [3,] 5.759087 2.4041373
    [4,] 5.624355 1.9789330
    [5,] 6.022990 2.3750755
    [6,] 6.100500 2.1822315
    [7,] 5.818015 1.7869819
    [8,] 5.868858 1.2615033
    [9,] 5.689173 1.9975885
   [10,] 5.761680 1.8270417
   [11,] 5.784111 3.2622525
   [12,] 5.754666 1.7469124
   [13,] 5.633982 3.3471072
   [14,] 5.815064 2.2534904
   [15,] 5.768681 3.1285974
   [16,] 5.755142 2.7178405
   [17,] 5.396668 1.5581388
   [18,] 5.747824 2.0307184
   [19,] 5.898625 2.2323282
   [20,] 5.941987 1.6078101
   [21,] 5.864590 1.8394275
   [22,] 5.784058 2.6501774
   [23,] 5.774773 2.4488782
   [24,] 5.815719 1.7885238
   [25,] 5.575549 2.7416488
   [26,] 5.791347 2.6188816
   [27,] 5.824217 2.1127057
   [28,] 6.073705 1.2958483
   [29,] 5.869031 2.6903951
   [30,] 5.783843 3.1466298
   [31,] 5.997116 1.8547816
   [32,] 5.668460 1.9857261
   [33,] 5.697736 3.2352716
   [34,] 5.739760 1.7167153
   [35,] 5.685582 1.9680388
   [36,] 5.844970 2.1752562
   [37,] 5.686466 2.0632553
   [38,] 5.936805 2.5479515
   [39,] 5.740211 3.3433896
   [40,] 5.875714 2.7496267
   [41,] 5.804680 2.0493568
   [42,] 5.963915 1.9348633
   [43,] 6.004839 1.9816625
   [44,] 5.816884 3.8048994
   [45,] 5.769272 3.3559546
   [46,] 5.709857 1.8753841
   [47,] 5.684734 1.4613551
   [48,] 5.773965 3.5323808
   [49,] 5.951332 2.5478784
   [50,] 5.747527 2.5924999
   [51,] 5.945042 2.1383753
   [52,] 5.703359 1.9315630
   [53,] 5.911238 2.5917456
   [54,] 5.820091 2.8662635
   [55,] 5.990220 2.5263144
   [56,] 5.686696 2.0668647
   [57,] 5.660682 2.4074001
   [58,] 5.930202 3.4388493
   [59,] 5.788975 2.2087142
   [60,] 5.933317 1.4455110
   [61,] 5.910092 2.2633610
   [62,] 5.875297 2.2057473
   [63,] 5.752038 1.9942647
   [64,] 5.719104 1.5805350
   [65,] 5.941777 1.5359362
   [66,] 5.927789 2.7970414
   [67,] 5.814651 1.8129036
   [68,] 5.793922 2.4067479
   [69,] 5.735818 3.2075886
   [70,] 5.730141 3.2670584
   [71,] 5.656985 1.6702252
   [72,] 5.533408 3.2138680
   [73,] 5.709458 3.1975472
   [74,] 5.769040 2.7363781
   [75,] 6.013245 1.4686406
   [76,] 5.804455 4.1250445
   [77,] 5.691554 1.7811632
   [78,] 5.656735 4.1732565
   [79,] 5.743197 3.2055992
   [80,] 5.676407 2.1367145
   [81,] 5.673699 2.3069953
   [82,] 5.732695 2.4106041
   [83,] 5.876302 2.0564629
   [84,] 5.885092 2.8664346
   [85,] 5.932081 1.9263790
   [86,] 6.069837 2.3009858
   [87,] 5.624838 2.0256139
   [88,] 6.111992 1.3354720
   [89,] 5.773126 2.8665665
   [90,] 5.727430 2.1697997
   [91,] 5.786491 1.9111467
   [92,] 5.786220 2.5545014
   [93,] 5.809296 3.3659789
   [94,] 5.714366 1.8483148
   [95,] 5.818190 4.6299575
   [96,] 5.786329 2.4497625
   [97,] 5.710466 2.9284255
   [98,] 5.900559 2.8145752
   [99,] 5.733600 2.5292171
  [100,] 5.694206 2.0097118
  [101,] 5.881565 1.6703663
  [102,] 5.788155 2.6200822
  [103,] 5.690064 2.0973308
  [104,] 5.911736 2.7803814
  [105,] 5.759978 4.9512310
  [106,] 5.890908 1.6675124
  [107,] 5.945370 1.4243606
  [108,] 5.595827 1.9599860
  [109,] 5.901679 2.1165669
  [110,] 5.742806 2.5502485
  [111,] 5.973982 2.1344976
  [112,] 5.753037 1.8755983
  [113,] 5.579880 2.0252726
  [114,] 5.549540 2.6055208
  [115,] 5.544241 1.9733914
  [116,] 5.773771 1.4182317
  [117,] 5.963629 1.8584145
  [118,] 5.930773 2.7309898
  [119,] 5.612839 3.0021426
  [120,] 5.723104 2.6090756
  [121,] 5.787786 2.7732644
  [122,] 5.779232 1.4633224
  [123,] 5.678533 2.7475491
  [124,] 5.880689 3.2211085
  [125,] 5.914046 2.1878798
  [126,] 5.758168 2.6016843
  [127,] 5.919558 3.0211443
  [128,] 5.901520 1.9737739
  [129,] 5.820596 1.5075534
  [130,] 5.598814 2.3966197
  [131,] 5.938843 1.8356104
  [132,] 5.729155 3.1061547
  [133,] 5.874546 2.1697264
  [134,] 5.523240 2.3817198
  [135,] 5.937921 3.7241026
  [136,] 5.898174 2.8546835
  [137,] 5.576323 1.9719249
  [138,] 5.721971 3.2358564
  [139,] 5.837051 1.9349225
  [140,] 5.918994 2.4546951
  [141,] 5.932099 1.5090188
  [142,] 5.595920 2.5743928
  [143,] 5.946966 2.3520819
  [144,] 5.918234 1.4008867
  [145,] 5.975951 1.5231398
  [146,] 5.585395 1.7488717
  [147,] 5.501680 1.4477694
  [148,] 5.845558 1.8896760
  [149,] 5.881545 3.0366835
  [150,] 5.959836 2.2531877
  [151,] 5.778655 1.4185688
  [152,] 5.640284 2.5796939
  [153,] 5.621867 2.1731328
  [154,] 5.526166 1.9944611
  [155,] 5.748544 2.3646999
  [156,] 5.834893 2.4071613
  [157,] 5.447807 1.4263765
  [158,] 5.915201 2.4863170
  [159,] 5.906855 2.8796032
  [160,] 5.750823 2.7538684
  [161,] 5.665860 3.0794245
  [162,] 5.721173 4.4329231
  [163,] 5.607184 2.0965525
  [164,] 5.591319 3.5659758
  [165,] 5.697389 1.5751242
  [166,] 5.803007 1.3903116
  [167,] 5.698242 2.4494778
  [168,] 5.650910 2.7806138
  [169,] 5.937029 2.5614988
  [170,] 5.641926 2.1520961
  [171,] 5.902366 2.9587311
  [172,] 5.818854 2.3809245
  [173,] 5.710833 3.6288175
  [174,] 5.687380 3.0891216
  [175,] 5.632412 2.1943903
  [176,] 5.741923 1.4662617
  [177,] 5.805381 2.9369281
  [178,] 5.586561 2.1401505
  [179,] 5.644905 1.7271987
  [180,] 5.871154 2.5617913
  [181,] 5.858787 2.3183469
  [182,] 5.863391 2.9129595
  [183,] 5.841234 2.5455068
  [184,] 5.829096 1.6381362
  [185,] 5.845426 2.5022278
  [186,] 5.936551 1.8375108
  [187,] 5.881091 2.7541289
  [188,] 5.757762 1.7414263
  [189,] 5.771374 2.7469023
  [190,] 5.811925 1.8163423
  [191,] 5.813735 1.8448270
  [192,] 5.948910 1.0716911
  [193,] 5.894715 3.3521393
  [194,] 5.658215 3.6440488
  [195,] 5.712347 1.9871642
  [196,] 5.851529 3.0353553
  [197,] 5.543301 1.3671536
  [198,] 5.832105 2.5769498
  [199,] 5.782774 3.4473137
  [200,] 5.614259 1.4434936
  [201,] 5.734374 2.6494944
  [202,] 5.742971 3.1253316
  [203,] 5.733425 2.8584447
  [204,] 5.730198 1.7908535
  [205,] 5.678944 2.6394350
  [206,] 5.658827 3.2925098
  [207,] 5.556636 2.2969245
  [208,] 5.615642 2.2492735
  [209,] 5.719358 1.8377839
  [210,] 5.749751 2.7503439
  [211,] 5.749986 2.5005399
  [212,] 5.819138 3.4300534
  [213,] 5.781518 2.9882606
  [214,] 5.861291 3.0422450
  [215,] 5.708646 3.7309863
  [216,] 5.727448 2.4150739
  [217,] 5.822565 2.9381686
  [218,] 5.657937 2.3401723
  [219,] 5.806583 2.5709970
  [220,] 5.823509 2.8298881
  [221,] 5.705941 1.8457897
  [222,] 5.787824 2.1163190
  [223,] 5.775786 1.9003156
  [224,] 5.898686 2.5780959
  [225,] 5.708601 3.0777125
  [226,] 5.742754 2.1726388
  [227,] 5.706378 2.0028303
  [228,] 5.638225 2.5158335
  [229,] 5.798684 2.3746578
  [230,] 5.769313 2.0234204
  [231,] 5.489804 3.8959362
  [232,] 5.858269 2.4663363
  [233,] 5.707991 2.0277668
  [234,] 5.872433 2.0572293
  [235,] 5.865400 1.7169815
  [236,] 5.916764 3.1099411
  [237,] 5.811741 2.0563670
  [238,] 5.740056 1.8774448
  [239,] 5.893372 1.6318342
  [240,] 5.795500 1.5948993
  [241,] 5.900931 4.1711841
  [242,] 5.709326 3.1002236
  [243,] 5.685522 3.2942557
  [244,] 5.796937 3.7235860
  [245,] 5.618066 2.3753894
  [246,] 5.695919 3.6477507
  [247,] 5.784466 1.9138089
  [248,] 5.642607 3.5453982
  [249,] 5.644404 2.5092268
  [250,] 5.845726 2.8256697
  [251,] 5.740691 2.5427579
  [252,] 5.896852 1.4904905
  [253,] 5.771621 2.7101404
  [254,] 5.649153 2.6183407
  [255,] 5.946430 2.0477570
  [256,] 5.793583 2.2555175
  [257,] 5.695030 1.5777950
  [258,] 5.647445 2.2925237
  [259,] 5.960855 2.2911697
  [260,] 5.751286 2.7296424
  [261,] 5.682315 3.2900059
  [262,] 5.651321 3.0211873
  [263,] 5.799846 2.5582571
  [264,] 5.867755 1.8518770
  [265,] 6.057724 2.6277646
  [266,] 5.775253 2.4551738
  [267,] 5.657246 2.5253190
  [268,] 6.153322 1.7623283
  [269,] 5.926904 2.8558240
  [270,] 5.960730 2.2508066
  [271,] 5.940665 1.2970394
  [272,] 5.996757 3.2166795
  [273,] 5.584143 2.7233619
  [274,] 5.719060 2.1774653
  [275,] 5.897718 2.4559746
  [276,] 5.673184 3.1511736
  [277,] 5.790334 1.7947879
  [278,] 5.698306 2.6164522
  [279,] 5.813787 1.7881332
  [280,] 5.841916 3.4783156
  [281,] 5.785735 2.5673343
  [282,] 5.817494 2.2653864
  [283,] 5.819904 2.5531410
  [284,] 5.856198 2.1463048
  [285,] 5.796379 3.2185944
  [286,] 5.714826 2.9337263
  [287,] 5.793964 2.4432867
  [288,] 5.857104 1.5324482
  [289,] 5.609608 2.1757863
  [290,] 5.857496 2.9405259
  [291,] 5.787809 3.0984416
  [292,] 5.925139 3.7075123
  [293,] 5.763377 1.1756540
  [294,] 5.745521 2.6972926
  [295,] 5.664737 2.9354728
  [296,] 5.784079 2.3989917
  [297,] 5.832196 2.0753678
  [298,] 5.822393 2.3454454
  [299,] 5.675621 2.0156698
  [300,] 5.655330 2.3436883
  [301,] 5.657515 1.7954550
  [302,] 5.668421 2.6630496
  [303,] 5.888549 3.5735417
  [304,] 5.796954 2.2167678
  [305,] 5.940711 1.7911972
  [306,] 5.693873 2.8061698
  [307,] 5.739591 3.1073587
  [308,] 5.762641 2.0149301
  [309,] 5.582000 1.9253232
  [310,] 6.018579 3.1310967
  [311,] 5.928214 1.6989722
  [312,] 5.833676 2.9612395
  [313,] 5.893957 2.2089394
  [314,] 5.619282 4.0601899
  [315,] 5.747784 2.1389082
  [316,] 5.749968 2.7504720
  [317,] 5.823956 1.7868565
  [318,] 5.796914 2.9568280
  [319,] 5.757501 2.0808053
  [320,] 5.864588 2.5184502
  [321,] 5.690764 3.2002522
  [322,] 5.960738 2.7915003
  [323,] 5.549822 3.2794022
  [324,] 5.764161 2.8572247
  [325,] 5.846809 2.3507561
  [326,] 6.200448 2.0960977
  [327,] 5.627047 3.0569170
  [328,] 5.587405 2.2335300
  [329,] 5.499988 4.1458537
  [330,] 5.716519 4.3927561
  [331,] 5.796615 3.6291840
  [332,] 5.755647 2.8502344
  [333,] 5.799946 2.3371855
  [334,] 6.000394 1.7017038
  [335,] 5.501697 2.0840713
  [336,] 5.493993 2.0837190
  [337,] 5.563252 2.5683750
  [338,] 5.896675 3.2428561
  [339,] 5.746198 2.2699463
  [340,] 5.777386 3.0194139
  [341,] 5.788028 2.0826652
  [342,] 5.826551 3.8739079
  [343,] 5.764554 2.2944769
  [344,] 5.711833 4.3523414
  [345,] 5.715695 1.9837359
  [346,] 5.630993 1.6197976
  [347,] 5.515226 2.0125939
  [348,] 5.535016 1.8781983
  [349,] 5.601236 1.6271318
  [350,] 5.853000 2.2945315
  [351,] 5.939009 2.4115242
  [352,] 5.603291 1.8175192
  [353,] 5.847200 1.9940362
  [354,] 5.561611 1.6478511
  [355,] 5.823775 2.3904444
  [356,] 5.748707 2.2200254
  [357,] 5.460928 1.2932936
  [358,] 5.991407 1.6514401
  [359,] 5.924282 2.1689854
  [360,] 5.949337 1.8543267
  [361,] 5.622211 2.9376079
  [362,] 5.764712 3.0560143
  [363,] 5.742824 2.0958423
  [364,] 5.749157 2.5237930
  [365,] 5.926890 2.8114670
  [366,] 5.568780 2.5758922
  [367,] 5.748748 2.2986209
  [368,] 5.820619 1.4014820
  [369,] 5.877516 2.6015453
  [370,] 5.710543 2.5523452
  [371,] 5.725965 1.3965847
  [372,] 5.917439 2.2870136
  [373,] 5.931627 2.9562285
  [374,] 5.818376 1.8823237
  [375,] 5.780930 3.2220634
  [376,] 5.645074 4.2651401
  [377,] 5.704894 1.5797074
  [378,] 6.048183 2.0318290
  [379,] 5.624637 2.2128849
  [380,] 5.605806 1.8526302
  [381,] 5.675356 2.3763438
  [382,] 5.924923 1.9669006
  [383,] 5.935526 3.5936898
  [384,] 5.503846 0.9334012
  [385,] 6.130401 2.9211710
  [386,] 5.865224 3.7684164
  [387,] 5.891547 2.2642454
  [388,] 5.516233 2.2266388
  [389,] 5.943979 2.6024125
  [390,] 5.833968 1.7810100
  [391,] 5.782499 2.0426312
  [392,] 5.656719 2.8494024
  [393,] 5.646583 3.8383395
  [394,] 5.811297 1.9995072
  [395,] 5.924539 2.0228800
  [396,] 5.823668 1.8030731
  [397,] 5.859729 1.4288028
  [398,] 5.983776 1.8071337
  [399,] 5.867274 2.3310225
  [400,] 5.816755 4.5094129
  [401,] 5.651123 2.5160062
  [402,] 5.562743 2.7193304
  [403,] 5.906384 3.3489145
  [404,] 5.563956 2.5996974
  [405,] 5.970806 1.9500873
  [406,] 5.897258 3.3360122
  [407,] 5.910591 2.2552784
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 [1885,] 6.062714 1.5987738
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 [1897,] 5.700136 1.3089709
 [1898,] 5.776854 2.7429030
 [1899,] 6.021357 2.5391674
 [1900,] 5.700921 3.3203643
 [1901,] 5.623030 1.7854146
 [1902,] 5.563014 2.3541571
 [1903,] 5.843909 3.9656096
 [1904,] 5.868167 2.6095366
 [1905,] 5.900964 1.9088041
 [1906,] 5.981477 2.1224260
 [1907,] 6.043285 2.1602825
 [1908,] 5.878067 2.5718343
 [1909,] 5.989996 1.8645042
 [1910,] 5.818091 3.2255558
 [1911,] 5.774961 2.3913724
 [1912,] 5.892561 2.2389188
 [1913,] 5.937761 2.5798768
 [1914,] 5.770108 3.7369234
 [1915,] 5.775574 2.7357454
 [1916,] 5.825562 2.6454213
 [1917,] 5.878405 1.9760818
 [1918,] 5.764153 3.0587655
 [1919,] 5.776932 2.3630163
 [1920,] 5.773135 1.9073222
 [1921,] 5.796542 5.2096703
 [1922,] 5.757148 4.9082854
 [1923,] 5.746869 4.4081970
 [1924,] 5.765942 2.8699008
 [1925,] 5.686874 3.3368816
 [1926,] 5.730374 2.8520739
 [1927,] 5.626356 2.8675694
 [1928,] 5.825050 3.4396996
 [1929,] 5.937618 0.9877464
 [1930,] 5.690328 2.3326634
 [1931,] 5.575763 2.3248810
 [1932,] 5.819633 2.7440294
 [1933,] 5.651337 2.0191238
 [1934,] 5.752565 1.7108935
 [1935,] 5.942351 2.3460408
 [1936,] 6.009945 2.1965218
 [1937,] 5.833905 2.7298448
 [1938,] 5.642868 2.5346515
 [1939,] 5.740997 2.6271509
 [1940,] 5.776630 2.2051690
 [1941,] 5.992265 2.9645894
 [1942,] 5.808302 2.6884543
 [1943,] 5.827542 3.0826201
 [1944,] 5.815230 2.8788297
 [1945,] 5.848076 2.5683714
 [1946,] 5.866719 2.4265713
 [1947,] 5.907706 2.0538802
 [1948,] 5.707713 2.8300148
 [1949,] 5.808511 2.0503905
 [1950,] 5.965573 1.6009003
 [1951,] 6.243028 1.4534879
 [1952,] 5.519972 3.2726884
 [1953,] 5.501645 1.7993440
 [1954,] 5.817091 1.3095522
 [1955,] 5.762906 2.1004215
 [1956,] 5.646664 1.6925576
 [1957,] 5.495823 1.6631694
 [1958,] 5.905985 1.8209093
 [1959,] 5.633651 1.8688266
 [1960,] 5.879228 3.5438846
 [1961,] 5.707311 1.9121292
 [1962,] 5.828134 2.7488943
 [1963,] 5.640372 2.3945156
 [1964,] 5.848912 1.6613573
 [1965,] 5.940786 1.5170890
 [1966,] 5.743122 2.8690945
 [1967,] 5.922840 2.2664048
 [1968,] 5.765957 3.2546402
 [1969,] 5.914627 3.4042250
 [1970,] 5.649072 2.5728415
 [1971,] 5.832968 1.5697614
 [1972,] 5.793180 2.3142417
 [1973,] 5.876103 2.0401798
 [1974,] 5.807473 2.0477533
 [1975,] 5.947647 2.3383521
 [1976,] 5.479945 1.7862837
 [1977,] 5.977839 2.9807266
 [1978,] 5.760935 2.9834161
 [1979,] 5.532422 2.5540508
 [1980,] 5.766455 2.9118483
 [1981,] 5.786877 3.4389591
 [1982,] 5.750637 3.0180047
 [1983,] 5.761248 2.9558527
 [1984,] 5.634397 2.8553779
 [1985,] 5.730342 2.7675844
 [1986,] 5.674116 2.8506453
 [1987,] 6.120097 2.7033164
 [1988,] 5.893245 1.9609758
 [1989,] 5.839615 2.6013369
 [1990,] 5.831796 2.1124240
 [1991,] 5.776600 2.1253138
 [1992,] 5.807108 1.9996117
 [1993,] 5.881154 2.4025286
 [1994,] 5.791852 3.3245487
 [1995,] 5.907062 1.9569123
 [1996,] 5.623590 1.7884581
 [1997,] 6.026095 2.0888099
 [1998,] 5.793563 3.1827367
 [1999,] 5.749724 1.9368637
 [2000,] 5.627587 2.0125041
 [2001,] 5.769702 1.5873405
 [2002,] 5.966976 3.3794696
 [2003,] 5.688323 2.6808041
 [2004,] 5.705493 1.8471627
 [2005,] 5.801305 2.6109187
 [2006,] 5.731455 2.2675271
 [2007,] 5.512186 1.5854909
 [2008,] 5.784221 3.1909282
 [2009,] 5.642310 3.2910057
 [2010,] 5.729259 2.9999332
 [2011,] 5.939169 2.8601507
 [2012,] 5.914897 3.0001912
 [2013,] 5.795459 2.9090086
 [2014,] 5.700666 3.0029746
 [2015,] 5.685981 2.5785777
 [2016,] 5.867037 1.7030178
 [2017,] 5.852313 4.3576033
 [2018,] 5.695017 3.4040592
 [2019,] 5.638516 2.8009851
 [2020,] 5.761835 1.7215226
 [2021,] 5.613441 2.0949915
 [2022,] 5.752166 3.3864354
 [2023,] 5.767404 2.1445856
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 [2027,] 5.838323 2.4058502
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 [2033,] 6.107962 1.5363691
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 [2356,] 5.835366 2.2450385
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 [9670,] 5.742744 2.0774217
 [9671,] 5.791910 2.0732023
 [9672,] 5.841566 2.0977785
 [9673,] 5.648360 3.6052521
 [9674,] 5.755964 1.7493997
 [9675,] 5.980507 1.0940921
 [9676,] 6.076861 2.5232083
 [9677,] 5.840948 2.6893447
 [9678,] 5.801013 2.6663616
 [9679,] 5.512252 3.4288350
 [9680,] 5.809512 1.7031935
 [9681,] 5.802474 1.6317085
 [9682,] 5.856568 2.6668458
 [9683,] 5.825368 2.9862520
 [9684,] 5.838238 2.3923514
 [9685,] 5.813782 2.0084844
 [9686,] 5.731902 1.4384151
 [9687,] 5.912561 2.2504639
 [9688,] 5.902482 3.1655752
 [9689,] 5.832199 2.7609012
 [9690,] 5.773978 1.9157084
 [9691,] 5.783116 1.7214262
 [9692,] 5.604425 1.5463112
 [9693,] 5.811754 1.8676554
 [9694,] 5.918602 1.9010396
 [9695,] 5.730527 2.7671515
 [9696,] 5.708834 3.0530077
 [9697,] 5.696799 1.8459525
 [9698,] 5.528313 2.2398143
 [9699,] 5.654859 2.9879425
 [9700,] 5.584347 2.4225850
 [9701,] 5.740135 3.5970962
 [9702,] 5.766431 3.8813433
 [9703,] 5.723844 3.2387614
 [9704,] 5.664483 3.1221229
 [9705,] 5.692118 2.5451500
 [9706,] 5.805500 2.4630998
 [9707,] 5.801802 2.3680808
 [9708,] 6.014322 2.4351299
 [9709,] 5.731801 2.7080395
 [9710,] 5.840131 2.0765483
 [9711,] 5.677473 1.1022351
 [9712,] 5.706697 1.7955278
 [9713,] 5.720417 1.8859058
 [9714,] 5.562118 3.1008133
 [9715,] 5.661795 1.3340754
 [9716,] 5.898725 2.0887728
 [9717,] 5.689892 3.1608795
 [9718,] 5.773675 1.9586052
 [9719,] 5.872985 2.8556076
 [9720,] 5.723400 2.3341261
 [9721,] 5.964389 2.5357526
 [9722,] 5.774763 2.1756741
 [9723,] 5.829962 1.6621280
 [9724,] 5.815884 2.1386496
 [9725,] 5.931481 1.3372211
 [9726,] 5.684559 1.6678178
 [9727,] 5.921223 1.3896460
 [9728,] 5.621282 2.1803554
 [9729,] 5.903041 2.2155738
 [9730,] 5.793163 2.4217924
 [9731,] 5.757216 3.3413830
 [9732,] 5.783114 2.3672309
 [9733,] 5.869671 2.5031744
 [9734,] 5.649826 4.3106082
 [9735,] 5.707483 3.8103501
 [9736,] 5.763962 2.6461268
 [9737,] 5.625685 2.3185301
 [9738,] 5.794023 3.1761140
 [9739,] 5.669086 2.1948345
 [9740,] 5.781459 1.3620288
 [9741,] 5.738970 1.9782121
 [9742,] 5.845054 2.5661156
 [9743,] 5.750455 1.1769934
 [9744,] 5.632196 3.1133504
 [9745,] 5.699879 1.6489411
 [9746,] 5.735602 1.8056385
 [9747,] 5.726212 2.8668687
 [9748,] 5.797469 1.7043476
 [9749,] 5.799277 2.4659411
 [9750,] 5.697500 3.1013810
 [9751,] 5.812107 1.9781778
 [9752,] 5.740730 3.2676480
 [9753,] 5.852204 1.8888995
 [9754,] 5.878404 2.7580395
 [9755,] 5.814251 1.9080137
 [9756,] 5.682087 2.6593440
 [9757,] 5.699643 2.7481292
 [9758,] 5.726811 4.2833732
 [9759,] 5.763223 2.1718347
 [9760,] 5.746854 2.0763894
 [9761,] 5.543788 1.6848797
 [9762,] 5.982221 2.5786306
 [9763,] 5.758638 2.5391416
 [9764,] 5.808097 2.6243982
 [9765,] 5.977876 2.0842945
 [9766,] 5.701671 2.6681321
 [9767,] 5.898347 2.4870703
 [9768,] 5.873286 3.0861041
 [9769,] 5.978485 2.3114189
 [9770,] 5.756073 3.2712899
 [9771,] 5.773791 1.9139412
 [9772,] 5.948227 2.6094588
 [9773,] 5.733232 3.2669193
 [9774,] 5.723014 3.4779566
 [9775,] 5.829249 1.8573562
 [9776,] 5.665043 1.9441195
 [9777,] 5.698254 4.1860242
 [9778,] 5.699787 2.1807633
 [9779,] 5.634573 2.7315672
 [9780,] 5.882610 2.1977081
 [9781,] 5.720439 3.0093948
 [9782,] 5.629220 2.5602790
 [9783,] 6.048252 3.3384865
 [9784,] 5.954731 2.0955646
 [9785,] 5.925128 2.5496842
 [9786,] 5.853060 3.1211910
 [9787,] 5.831157 2.8076444
 [9788,] 5.807180 3.0811380
 [9789,] 5.756515 1.5460044
 [9790,] 5.727848 2.7835292
 [9791,] 5.924243 3.6310941
 [9792,] 5.687285 0.7962476
 [9793,] 5.452423 1.8023602
 [9794,] 6.125319 1.6270801
 [9795,] 5.670287 2.2121361
 [9796,] 5.588983 1.4883274
 [9797,] 5.459886 1.8747131
 [9798,] 5.744157 3.4952437
 [9799,] 5.879648 1.8700018
 [9800,] 5.530603 3.3591069
 [9801,] 5.874582 2.6370351
 [9802,] 5.686983 1.8104768
 [9803,] 5.889030 2.2211591
 [9804,] 5.830364 3.0166040
 [9805,] 5.654328 1.6381516
 [9806,] 5.555719 2.3111943
 [9807,] 5.898168 2.9285831
 [9808,] 5.814148 1.7945343
 [9809,] 5.797048 2.6290129
 [9810,] 5.703252 3.0479314
 [9811,] 5.717828 3.4011257
 [9812,] 5.652219 1.7099144
 [9813,] 5.556107 2.3366104
 [9814,] 5.734583 1.4485366
 [9815,] 5.644844 2.0083205
 [9816,] 5.695788 2.5124394
 [9817,] 5.944414 2.1989031
 [9818,] 5.671104 1.5872971
 [9819,] 5.807067 2.9539250
 [9820,] 5.993852 1.9299398
 [9821,] 5.936739 2.6732347
 [9822,] 5.970429 2.2767410
 [9823,] 5.811742 2.3641406
 [9824,] 5.753874 1.6617340
 [9825,] 5.911843 2.0897476
 [9826,] 5.876654 2.7247818
 [9827,] 5.829729 1.9176289
 [9828,] 5.603127 2.2645575
 [9829,] 5.579399 2.2887045
 [9830,] 5.833492 1.9867320
 [9831,] 5.691687 2.6463574
 [9832,] 5.578692 2.1447803
 [9833,] 5.964134 3.2057570
 [9834,] 5.547887 2.1321705
 [9835,] 5.899660 3.4279411
 [9836,] 5.764353 2.8666569
 [9837,] 5.809048 2.1471841
 [9838,] 5.678450 2.2770121
 [9839,] 5.782060 1.8739364
 [9840,] 5.824619 2.8671909
 [9841,] 5.993764 2.4221908
 [9842,] 5.848178 2.7191125
 [9843,] 5.899563 1.7256111
 [9844,] 5.988229 2.7491697
 [9845,] 5.868372 2.4661210
 [9846,] 5.969052 1.6719175
 [9847,] 5.695468 3.1463493
 [9848,] 5.885623 4.5097663
 [9849,] 5.599999 3.3944271
 [9850,] 5.810975 2.6265604
 [9851,] 5.746375 2.7796683
 [9852,] 5.735024 3.2931171
 [9853,] 5.736131 0.7941465
 [9854,] 5.893434 2.2586525
 [9855,] 5.758779 2.7042637
 [9856,] 5.815032 2.1967403
 [9857,] 5.847520 1.8172746
 [9858,] 5.686676 3.2470211
 [9859,] 5.633348 1.3529054
 [9860,] 5.697358 2.9793718
 [9861,] 5.647762 2.6230558
 [9862,] 5.815598 3.4522804
 [9863,] 5.945802 2.0892703
 [9864,] 6.049016 2.4168164
 [9865,] 5.810147 2.0757631
 [9866,] 5.598076 1.4711788
 [9867,] 5.724806 1.3120708
 [9868,] 5.950428 2.4959426
 [9869,] 5.776134 3.4886072
 [9870,] 5.793669 3.2480870
 [9871,] 5.900074 1.6897751
 [9872,] 5.761096 2.9706528
 [9873,] 5.683393 2.5292064
 [9874,] 5.658096 3.0989372
 [9875,] 5.842491 2.1666504
 [9876,] 5.574125 1.4253952
 [9877,] 5.762363 3.2982306
 [9878,] 5.769123 2.7479808
 [9879,] 5.626071 1.1604485
 [9880,] 5.613247 3.5013794
 [9881,] 5.684824 1.8864646
 [9882,] 5.869896 2.7635101
 [9883,] 5.768891 2.7016869
 [9884,] 5.643133 2.4378071
 [9885,] 5.862719 3.7451445
 [9886,] 5.812204 3.9483057
 [9887,] 5.653330 1.6502319
 [9888,] 5.858584 1.3125300
 [9889,] 5.863768 2.4579562
 [9890,] 5.809879 4.1321943
 [9891,] 5.711286 1.9577392
 [9892,] 5.714500 3.0318215
 [9893,] 5.724185 2.5993818
 [9894,] 5.594502 3.0044915
 [9895,] 5.473172 2.9233011
 [9896,] 5.811773 3.0026837
 [9897,] 5.884972 2.3233361
 [9898,] 6.087976 2.3854859
 [9899,] 5.847546 2.1661993
 [9900,] 5.780705 3.5330077
 [9901,] 5.771533 2.4575451
 [9902,] 5.993310 2.9202351
 [9903,] 5.736442 3.5625094
 [9904,] 5.927945 3.0529462
 [9905,] 5.986328 2.8667861
 [9906,] 5.599567 3.9591753
 [9907,] 5.769017 2.3906459
 [9908,] 6.006654 2.9235848
 [9909,] 5.600738 2.4399692
 [9910,] 5.779180 1.5533593
 [9911,] 5.883974 2.3863886
 [9912,] 5.893526 2.3721037
 [9913,] 5.731600 2.2113261
 [9914,] 5.913002 1.6709855
 [9915,] 5.637433 3.1374786
 [9916,] 5.924556 2.0704586
 [9917,] 5.748032 2.6523149
 [9918,] 5.925762 2.4192167
 [9919,] 5.820936 2.9694714
 [9920,] 5.859371 1.8468723
 [9921,] 5.867656 1.6255854
 [9922,] 5.963321 2.2759088
 [9923,] 5.546988 2.8087900
 [9924,] 5.785868 1.4937113
 [9925,] 5.867965 1.3835456
 [9926,] 5.865096 2.3185527
 [9927,] 5.694750 2.3735035
 [9928,] 5.975749 2.4247892
 [9929,] 5.527786 4.1249718
 [9930,] 5.542674 2.8597320
 [9931,] 5.826651 1.9756850
 [9932,] 5.780099 1.5052593
 [9933,] 5.624587 2.7001754
 [9934,] 5.873539 1.9260352
 [9935,] 5.859354 2.1154248
 [9936,] 6.064803 1.5935581
 [9937,] 5.898899 3.0909386
 [9938,] 5.640023 1.7630437
 [9939,] 5.770042 2.7192926
 [9940,] 5.930841 3.2063961
 [9941,] 5.657136 3.0065886
 [9942,] 5.885723 1.9017350
 [9943,] 5.819337 2.4694020
 [9944,] 5.802969 1.7080684
 [9945,] 5.751010 1.6085910
 [9946,] 5.769631 1.9222588
 [9947,] 5.513795 2.0103786
 [9948,] 5.715988 3.3778497
 [9949,] 5.760825 2.8659403
 [9950,] 5.948758 1.8223763
 [9951,] 5.748122 2.5984029
 [9952,] 5.796983 3.3406644
 [9953,] 5.773686 2.5516716
 [9954,] 5.774564 3.6902799
 [9955,] 5.698304 3.1187455
 [9956,] 5.882032 3.0822445
 [9957,] 5.697092 3.2965852
 [9958,] 5.765230 2.4651919
 [9959,] 5.667306 3.0384372
 [9960,] 5.627987 2.4931606
 [9961,] 5.674607 2.1506565
 [9962,] 5.817254 2.0659492
 [9963,] 5.722835 2.4345043
 [9964,] 5.687280 3.0651536
 [9965,] 5.591030 2.1316865
 [9966,] 5.902549 1.5826685
 [9967,] 5.763541 1.9915671
 [9968,] 5.776277 2.1451519
 [9969,] 5.810741 3.8045128
 [9970,] 5.912253 2.8627182
 [9971,] 5.907763 1.3795670
 [9972,] 6.105498 3.8589867
 [9973,] 5.796253 1.8126088
 [9974,] 5.917175 2.9892265
 [9975,] 5.537116 2.3294949
 [9976,] 5.640143 3.3585776
 [9977,] 5.839356 2.0531132
 [9978,] 5.576636 2.3919953
 [9979,] 6.022224 2.1852914
 [9980,] 5.655229 2.0716758
 [9981,] 5.735662 2.3432880
 [9982,] 5.862520 2.2145522
 [9983,] 5.888416 2.8619261
 [9984,] 5.665813 2.1726705
 [9985,] 5.800523 3.8581925
 [9986,] 5.576234 1.3286935
 [9987,] 6.078717 1.7130570
 [9988,] 6.085538 1.9588669
 [9989,] 5.780599 1.7493429
 [9990,] 5.619011 3.3234588
 [9991,] 5.763451 2.3068303
 [9992,] 6.020020 2.5494954
 [9993,] 5.851437 2.8964771
 [9994,] 5.825988 4.4392982
 [9995,] 5.803194 1.9979412
 [9996,] 5.868015 1.6313122
 [9997,] 5.962179 2.7460072
 [9998,] 5.774155 2.0582675
 [9999,] 5.550044 1.3517533
[10000,] 5.917369 1.5651517

Discussion question

How can we extract posterior draws of mu and phi from output?

Extracting MCMC Output for Posterior Analysis

• Code
• Plot 1
• Plot 2

para_post <- as.data.frame(output)
names(para_post) <- c("mu", "phi")

para_post %>%
  summarize(quants = quantile(mu, probs = c(0.025, 0.975)))

    quants
1 5.537106
2 6.039795

para_post %>%
  summarize(quants = quantile(phi, probs = c(0.025, 0.975)))

    quants
1 1.332576
2 3.995338

Posterior Draws and MCMC Diagnostics

• In our demo, we use every parameter draw for posterior analysis

• However, posterior draws in an MCMC could be highly dependent

• We will need to assess the convergence of MCMC through the use of MCMC diagnostics

• Only when are eliminate any convergence issues, can we use independent posterior parameter draws for posterior analysis

• We will introduce MCMC diagnostics later with using the brms package

MCMC Algorithms

Commonly Used MCMC Algorithms

• Gibbs sampler

• Metropolis and Metropolis-Hastings

• Hamiltonian Monte Carlo

Gibbs Sampler

• Require derivation of full conditional posterior distributions

• Every draw is “accepted”

• Not necessarily feasible for any model and prior choices

• Demo animation

Metropolis and Metropolis-Hastings

• No not require derivation of full conditional posterior distributions

• Random walk: a new draw is proposed from a jumping/proposal distribution given at the current draw

• Every draw is “accepted” with a calculated ratio

• Demo animation

Hamiltonian Monte Carlo (HMC)

• Some fancy algorithm involved

• Compared to a random walk jumping/proposal distribution in Metropolis-Hastings, “HMC reduces the correlation between successive sampled states by proposing moves to distant states which maintain a high probability of acceptance…”(Wiki)

• More efficient

• The main MCMC algorithm behind Stan and brms

• Demo animation