November 15 2002   8:18:37.171 AM
  
 BAYES_BETA
   Simple Bayesian Statistics demonstrations.
  
 BAYES_BETA
   Simple Bayesian Statistics demonstrations.
  
   Suppose we're watching a "system" and trying
   to analyze its behavior.  Each time we observe
   the system, it flips a coin a certain number of
   times, and reports the number of heads and
   tails.  We want to estimate THETA1 and THETA2,
   the probabilities of heads and of tails.
  
   We treat the values of THETA1 and THETA2 as
   random variables themselves, controlled by a
   Beta probability density function, which has
   parameters ALPHA1 and ALPHA2.  We make an 
   arbitrary or informed guess for initial values
   of ALPHA1 and ALPHA2.  We observe the system,
   and adjust ALPHA1 and ALPHA2 using Bayes's
   Law.  We continue until we are satisfied
   that our estimates seem to have converged.
  
 Run parameters:
  
  THETA1/THETA2:      0.750000      0.250000    
   Number of observations to make =      4
   Number of tosses per observation =    5
  
   Initial data:
  
  ALPHA1/ALPHA2       0.500000      0.500000    
  THETA1/THETA2:      0.500000      0.500000    
  
 Plot of Beta distribution,
  
   ALPHA1 =  0.5
   ALPHA2 =  0.5
  
       T            BETA(T,ALPHA1,ALPHA2)
  
   0.00000       0.00000    
  0.500000E-01   1.46051    
  0.100000       1.06103    
  0.150000      0.891446    
  0.200000      0.795775    
  0.250000      0.735105    
  0.300000      0.694609    
  0.350000      0.667359    
  0.400000      0.649747    
  0.450000      0.639827    
  0.500000      0.636620    
  0.550000      0.639827    
  0.600000      0.649747    
  0.650000      0.667359    
  0.700000      0.694609    
  0.750000      0.735105    
  0.800000      0.795775    
  0.850000      0.891446    
  0.900000       1.06103    
  0.950000       1.46051    
   1.00000       0.00000    
  
 BAYES_BETA - After observation  1
  
   Prior observation data:
  
   Tosses:            0
   Heads/Tails:       2*0
  ALPHA1/ALPHA2:      0.500000      0.500000    
  THETA1/THETA2:      0.500000      0.500000    
  
   Observation data:
  
   Tosses:            5
   Heads/Tails:       5,  0
  
   Post observation data:
  
   Tosses:            5
   Heads/Tails:       5,  0
  ALPHA1/ALPHA2:       5.50000      0.500000    
  THETA1/THETA2:      0.916667      0.833333E-01
  
 BAYES_BETA - After observation  2
  
   Prior observation data:
  
   Tosses:            5
   Heads/Tails:       5,  0
  ALPHA1/ALPHA2:       5.50000      0.500000    
  THETA1/THETA2:      0.916667      0.833333E-01
  
   Observation data:
  
   Tosses:            5
   Heads/Tails:       4,  1
  
   Post observation data:
  
   Tosses:            10
   Heads/Tails:       9,  1
  ALPHA1/ALPHA2:       9.50000       1.50000    
  THETA1/THETA2:      0.863636      0.136364    
  
 BAYES_BETA - After observation  3
  
   Prior observation data:
  
   Tosses:            10
   Heads/Tails:       9,  1
  ALPHA1/ALPHA2:       9.50000       1.50000    
  THETA1/THETA2:      0.863636      0.136364    
  
   Observation data:
  
   Tosses:            5
   Heads/Tails:       4,  1
  
   Post observation data:
  
   Tosses:            15
   Heads/Tails:       13,  2
  ALPHA1/ALPHA2:       13.5000       2.50000    
  THETA1/THETA2:      0.843750      0.156250    
  
 BAYES_BETA - After observation  4
  
   Prior observation data:
  
   Tosses:            15
   Heads/Tails:       13,  2
  ALPHA1/ALPHA2:       13.5000       2.50000    
  THETA1/THETA2:      0.843750      0.156250    
  
   Observation data:
  
   Tosses:            5
   Heads/Tails:       3,  2
  
   Post observation data:
  
   Tosses:            20
   Heads/Tails:       16,  4
  ALPHA1/ALPHA2:       16.5000       4.50000    
  THETA1/THETA2:      0.785714      0.214286    
  
 Plot of Beta distribution,
  
   ALPHA1 =  16.5
   ALPHA2 =  4.5
  
       T            BETA(T,ALPHA1,ALPHA2)
  
   0.00000       0.00000    
  0.500000E-01  0.229817E-15
  0.100000      0.881377E-11
  0.150000      0.386986E-08
  0.200000      0.270454E-06
  0.250000      0.685645E-05
  0.300000      0.908944E-04
  0.350000      0.764833E-03
  0.400000      0.457904E-02
  0.450000      0.209593E-01
  0.500000      0.768679E-01
  0.550000      0.232906    
  0.600000      0.594113    
  0.650000       1.28739    
  0.700000       2.36733    
  0.750000       3.64381    
  0.800000       4.53747    
  0.850000       4.24256    
  0.900000       2.48928    
  0.950000      0.508657    
   1.00000       0.00000    
  
 BAYES_BETA:
   Normal end of execution.