/* 

  True skill in Picat.

  Example inspired by
  Johannes Borgstrom, Andrew D. Gordon, Michael Greenberg, James Margetson, and Jurgen Van Gael:
  "Measure Transformer Semantics for Bayesian Machine Learning"
  https://www.microsoft.com/en-us/research/publication/measure-transformer-semantics-for-bayesian-machine-learning-2011/?from=http%3A%2F%2Fresearch.microsoft.com%2Fpubs%2F135344%2Fmsr-tr-2011-18.pdf

  In this setup the three persons a, b and c has some skill, and they have a
  performance (of some unidentified task/action). We only observe performance, 
  but not knowing the skills.
  Here we observe that:
    - a performs better than both b and c
    - b berforms better than c.


  This is a port of my Gamble model gamble_true_skill.rkt.
  
  Cf:
  - ppl_true_skill_simple.pi, but this current model is quite simpler than that model.

  This program was created by Hakan Kjellerstrand, hakank@gmail.com
  See also my Picat page: http://www.hakank.org/picat/

*/

import ppl_distributions,ppl_utils.

main => go.

/*
  var : skills(1)
  Probabilities (truncated):
  110.356094944947088: 0.00059952038369305 (1 / 1668)
  109.993527643642452: 0.00059952038369305 (1 / 1668)
  109.641657089712837: 0.00059952038369305 (1 / 1668)
  109.224664900945072: 0.00059952038369305 (1 / 1668)
  .........
  93.084629954279521: 0.00059952038369305 (1 / 1668)
  92.722418466345346: 0.00059952038369305 (1 / 1668)
  92.649811284188402: 0.00059952038369305 (1 / 1668)
  91.814344865557558: 0.00059952038369305 (1 / 1668)
  mean = 101.633

  var : skills(2)
  Probabilities (truncated):
  109.381190329004596: 0.00059952038369305 (1 / 1668)
  109.079093397621847: 0.00059952038369305 (1 / 1668)
  108.922380070720692: 0.00059952038369305 (1 / 1668)
  108.783588875059095: 0.00059952038369305 (1 / 1668)
  .........
  92.626708449667831: 0.00059952038369305 (1 / 1668)
  92.60299547137663: 0.00059952038369305 (1 / 1668)
  92.578781025840669: 0.00059952038369305 (1 / 1668)
  89.365945066034357: 0.00059952038369305 (1 / 1668)
  mean = 100.015

  var : skills(3)
  Probabilities (truncated):
  106.791474649404591: 0.00059952038369305 (1 / 1668)
  106.299618876289259: 0.00059952038369305 (1 / 1668)
  106.043259638697933: 0.00059952038369305 (1 / 1668)
  105.990883269359031: 0.00059952038369305 (1 / 1668)
  .........
  89.826945166157216: 0.00059952038369305 (1 / 1668)
  89.604280468223024: 0.00059952038369305 (1 / 1668)
  89.202816851513319: 0.00059952038369305 (1 / 1668)
  88.753559232947239: 0.00059952038369305 (1 / 1668)
  mean = 98.2731

  var : performance(1)
  Probabilities (truncated):
  118.918097564094154: 0.00059952038369305 (1 / 1668)
  116.980542653897558: 0.00059952038369305 (1 / 1668)
  116.669790286945485: 0.00059952038369305 (1 / 1668)
  116.597275205553785: 0.00059952038369305 (1 / 1668)
  .........
  94.591587425659952: 0.00059952038369305 (1 / 1668)
  94.497938522414927: 0.00059952038369305 (1 / 1668)
  94.20561363407603: 0.00059952038369305 (1 / 1668)
  93.311249208775578: 0.00059952038369305 (1 / 1668)
  mean = 104.113

  var : performance(2)
  Probabilities (truncated):
  111.450364215319553: 0.00059952038369305 (1 / 1668)
  110.778342365320626: 0.00059952038369305 (1 / 1668)
  109.853839073761691: 0.00059952038369305 (1 / 1668)
  109.766116879680965: 0.00059952038369305 (1 / 1668)
  .........
  89.274429594997031: 0.00059952038369305 (1 / 1668)
  88.226959817387652: 0.00059952038369305 (1 / 1668)
  87.943630537949133: 0.00059952038369305 (1 / 1668)
  87.467412370224707: 0.00059952038369305 (1 / 1668)
  mean = 99.9461

  var : performance(3)
  Probabilities (truncated):
  108.556598800640032: 0.00059952038369305 (1 / 1668)
  107.48309012714347: 0.00059952038369305 (1 / 1668)
  106.977050608345323: 0.00059952038369305 (1 / 1668)
  106.879205278283905: 0.00059952038369305 (1 / 1668)
  .........
  84.41816728577669: 0.00059952038369305 (1 / 1668)
  83.417570249296134: 0.00059952038369305 (1 / 1668)
  83.279453692764235: 0.00059952038369305 (1 / 1668)
  81.680254397514631: 0.00059952038369305 (1 / 1668)
  mean = 95.7115

*/
go ?=>
  reset_store(),
  run_model(10_000,$model,[show_probs_rat_trunc,mean,
                                 presentation=["skills(1)","skills(2)","skills(3)",
                                               "performance(1)","performance(2)","performance(3)"]
                                ]),
  % fail,
  nl.
go => true.


model() =>
  % There are three people, a, b, and c
  % Each person has an unknown Skill and a
  % known performance, where the skill is
  % reflected in the performance (with uncertainties).
   
  A = 1, B = 2, C = 3,  
  NumPlayers = 3,
  
  % Each player has some fixed skill
  Skills = normal_dist_n(100,sqrt(10),NumPlayers),

  % Performances per player
  % The performance this time is - however - depending on the skill
  % but might vary.
  Performance = [ normal_dist(Skills[P],sqrt(15)) : P in 1..NumPlayers],

  % Now we see that a is better than b and c, and b is better than c.
  
  observe(Performance[A] > Performance[B]),
  observe(Performance[A] > Performance[C]),
  observe(Performance[B] > Performance[C]),
  
  if observed_ok() then
    add("skills(1)",Skills[1]),
    add("skills(2)",Skills[2]),
    add("skills(3)",Skills[3]),
    % Peformance for each game  
    add("performance(1)",Performance[1]),
    add("performance(2)",Performance[2]),
    add("performance(3)",Performance[3])
  end.