16 September 2018   6:31:53.110 PM
 
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 =   0.500000    
  ALPHA2 =   0.500000    
 
      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:           0     0
  ALPHA1/ALPHA2:      0.500000      0.500000    
  THETA1/THETA2:      0.500000      0.500000    
 
  Observation data:
 
  Tosses:                5
  Heads/Tails:           3     2
 
  Post observation data:
 
  Tosses:                5
  Heads/Tails:           3     2
  ALPHA1/ALPHA2:       3.50000       2.50000    
  THETA1/THETA2:      0.583333      0.416667    
 
BAYES_BETA - After observation      2
 
  Prior observation data:
 
  Tosses:                5
  Heads/Tails:           3     2
  ALPHA1/ALPHA2:       3.50000       2.50000    
  THETA1/THETA2:      0.583333      0.416667    
 
  Observation data:
 
  Tosses:                5
  Heads/Tails:           5     0
 
  Post observation data:
 
  Tosses:               10
  Heads/Tails:           8     2
  ALPHA1/ALPHA2:       8.50000       2.50000    
  THETA1/THETA2:      0.772727      0.227273    
 
BAYES_BETA - After observation      3
 
  Prior observation data:
 
  Tosses:               10
  Heads/Tails:           8     2
  ALPHA1/ALPHA2:       8.50000       2.50000    
  THETA1/THETA2:      0.772727      0.227273    
 
  Observation data:
 
  Tosses:                5
  Heads/Tails:           3     2
 
  Post observation data:
 
  Tosses:               15
  Heads/Tails:          11     4
  ALPHA1/ALPHA2:       11.5000       4.50000    
  THETA1/THETA2:      0.718750      0.281250    
 
BAYES_BETA - After observation      4
 
  Prior observation data:
 
  Tosses:               15
  Heads/Tails:          11     4
  ALPHA1/ALPHA2:       11.5000       4.50000    
  THETA1/THETA2:      0.718750      0.281250    
 
  Observation data:
 
  Tosses:                5
  Heads/Tails:           4     1
 
  Post observation data:
 
  Tosses:               20
  Heads/Tails:          15     5
  ALPHA1/ALPHA2:       15.5000       5.50000    
  THETA1/THETA2:      0.738095      0.261905    
 
Plot of Beta distribution,
 
  ALPHA1 =    15.5000    
  ALPHA2 =    5.50000    
 
      T            BETA(T,ALPHA1,ALPHA2)
 
   0.00000       0.00000    
  0.500000E-01  0.150402E-13
  0.100000      0.273227E-09
  0.150000      0.755337E-07
  0.200000      0.372625E-05
  0.250000      0.708497E-04
  0.300000      0.730519E-03
  0.350000      0.489248E-02
  0.400000      0.236583E-01
  0.450000      0.882360E-01
  0.500000      0.264766    
  0.550000      0.656368    
  0.600000       1.36425    
  0.650000       2.38771    
  0.700000       3.49462    
  0.750000       4.18361    
  0.800000       3.90726    
  0.850000       2.57880    
  0.900000      0.952684    
  0.950000      0.922125E-01
   1.00000       0.00000    
 
BAYES_BETA:
  Normal end of execution.
 
16 September 2018   6:31:53.111 PM