# Gaussian Mixture Model
Student ID: F74082125
Student Name: 張議隆
See full output and data in [hackmd website](https://hackmd.io/@bebo1010/Gaussian_Mixture), full output was too long to be shown here.

## Some Test Results
:::warning
When the sample data grows, sample time grows significantly.
:::
| mu_prior_params_$\mu$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ |
|----------------------------------- |----------- |------------ |----------- |--------- |
| mu_prior_params_$\sigma$ | $4.0$ | $4.0$ | $4.0$ | $4.0$ |
| sigma_prior_param_a | $0.5$ | $0.5$ | $0.5$ | $0.5$ |
| sigma_prior_param_b | $2.0$ | $2.0$ | $2.0$ | $2.0$ |
| repeat num | $2000000$ | $20000000$ | $2000000$ | $20000$ |
| dataset num | $10$ | $10$ | $100$ | $10$ |
| Model1 sampling correct selection | $8/10$ | $7/10$ | $91/100$ | $8/10$ |
| Model2 sampling correct selection | $10/10$ | $8/10$ | $50/100$ | $5/10$ |
| Model1 summing correct selection | $10/10$ | $8/10$ | $83/100$ | $8/10$ |
| Model2 summing correct selection | $9/10$ | $7/10$ | $57/100$ | $6/10$ |
:::spoiler Full output of first test(default setting)

./Gaussian_poolOrNot 
Using default random seed
Starting computation for 10 datasets each. ...

Data generated with one component
generating data with: (μ,σ) =  (0.54,3.54)
Integrals by sampling= (1.86361e-50,2.24389e-50)  by summing: (3.24022e-50,3.12089e-50)

generating data with: (μ,σ) =  (-3.86,0.29)
Integrals by sampling= (3.55084e-10,6.97491e-11)  by summing: (9.98852e-15,2.12547e-15)

generating data with: (μ,σ) =  (1.84,9.89)
Integrals by sampling= (1.01493e-67,1.05214e-67)  by summing: (2.5593e-68,1.55591e-68)

generating data with: (μ,σ) =  (4.99,0.64)
Integrals by sampling= (9.99233e-21,2.24743e-21)  by summing: (2.21573e-22,1.10553e-22)

generating data with: (μ,σ) =  (1.40,2.07)
Integrals by sampling= (2.27837e-40,1.41155e-40)  by summing: (2.19598e-40,1.47314e-40)

generating data with: (μ,σ) =  (3.63,1.00)
Integrals by sampling= (2.63974e-29,9.05853e-30)  by summing: (1.45186e-29,6.20038e-30)

generating data with: (μ,σ) =  (-0.13,0.68)
Integrals by sampling= (2.3021e-20,6.34571e-21)  by summing: (7.31168e-21,2.6248e-21)

generating data with: (μ,σ) =  (0.97,0.83)
Integrals by sampling= (3.66915e-26,1.59224e-26)  by summing: (3.08058e-26,1.5787e-26)

generating data with: (μ,σ) =  (4.19,3.16)
Integrals by sampling= (1.25761e-49,9.92477e-50)  by summing: (1.55613e-49,1.09263e-49)

generating data with: (μ,σ) =  (4.70,0.60)
Integrals by sampling= (1.75858e-20,3.91985e-21)  by summing: (1.11411e-22,8.70714e-23)


Data generated with two components
generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.477; (8.83,1.30); (-1.44,2.71)
Integrals by sampling= (5.2043e-58,2.21519e-56)  by summing: (6.01777e-59,1.58436e-58)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.106; (7.99,1.39); (-3.64,1.45)
Integrals by sampling= (2.10854e-47,7.24461e-40)  by summing: (1.17881e-47,2.62187e-40)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.641; (-7.44,1.43); (2.93,0.58)
Integrals by sampling= (3.90553e-55,3.76945e-43)  by summing: (9.99491e-56,2.01073e-43)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.391; (-4.62,0.51); (-2.25,13.62)
Integrals by sampling= (3.92372e-62,3.67783e-58)  by summing: (1.08494e-61,9.23667e-58)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.058; (-0.49,3.60); (-2.17,3.72)
Integrals by sampling= (1.40194e-50,1.58673e-50)  by summing: (2.48966e-50,2.31388e-50)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.105; (-3.26,1.38); (-4.02,0.49)
Integrals by sampling= (6.30961e-19,1.93739e-18)  by summing: (1.12456e-20,2.66021e-20)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.869; (-5.26,0.79); (5.20,2.04)
Integrals by sampling= (6.72205e-35,8.91506e-26)  by summing: (5.14684e-35,1.97583e-25)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.569; (-3.51,1.13); (-9.10,0.97)
Integrals by sampling= (8.91131e-46,7.60151e-42)  by summing: (7.03125e-46,3.15685e-44)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.844; (-10.74,0.59); (5.00,0.70)
Integrals by sampling= (1.85561e-60,4.36834e-30)  by summing: (1.04399e-60,2.74986e-57)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.660; (-1.08,4.77); (-9.87,5.11)
Integrals by sampling= (3.20295e-55,3.46717e-55)  by summing: (7.82967e-56,1.1744e-55)

By sampling: Model1 data, correct selection 8/10
             Model2 data, correct selection 10/10
By summing:  Model1 data, correct selection 10/10
             Model2 data, correct selection 9/10

:::

:::spoiler Full output of second test(repeat num set to 20M)

Using default random seed
Starting computation for 10 datasets each. ...

Data generated with one component
generating data with: (μ,σ) =  (0.54,3.54)
Integrals by sampling= (1.85698e-50,2.24245e-50)  by summing: (3.24022e-50,3.12089e-50)

generating data with: (μ,σ) =  (-5.63,5.36)
Integrals by sampling= (2.3779e-52,2.60066e-52)  by summing: (5.12329e-52,3.33261e-52)

generating data with: (μ,σ) =  (3.99,3.03)
Integrals by sampling= (1.32279e-45,8.0314e-46)  by summing: (1.37755e-45,8.39752e-46)

generating data with: (μ,σ) =  (1.58,1.71)
Integrals by sampling= (6.76693e-37,3.67069e-37)  by summing: (6.96528e-37,3.87481e-37)

generating data with: (μ,σ) =  (-0.13,0.66)
Integrals by sampling= (9.64615e-22,2.85357e-22)  by summing: (7.29402e-22,2.24573e-22)

generating data with: (μ,σ) =  (4.48,0.96)
Integrals by sampling= (5.15729e-27,1.56724e-27)  by summing: (4.41028e-27,1.34462e-27)

generating data with: (μ,σ) =  (-2.68,29.69)
Integrals by sampling= (1.62014e-87,8.1677e-88)  by summing: (9.15151e-189,6.5147e-165)

generating data with: (μ,σ) =  (1.50,5.98)
Integrals by sampling= (3.40986e-59,5.3005e-59)  by summing: (1.5266e-59,6.46514e-59)

generating data with: (μ,σ) =  (6.45,2.07)
Integrals by sampling= (1.40161e-41,4.64802e-42)  by summing: (1.37014e-41,3.47168e-42)

generating data with: (μ,σ) =  (0.89,0.63)
Integrals by sampling= (7.08145e-18,1.77869e-18)  by summing: (5.07446e-18,1.2248e-18)


Data generated with two components
generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.775; (-5.76,2.24); (-2.93,0.60)
Integrals by sampling= (1.57502e-42,5.91829e-42)  by summing: (1.90225e-42,5.12806e-42)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.953; (5.05,0.60); (-0.37,0.58)
Integrals by sampling= (8.66192e-32,2.47471e-23)  by summing: (4.40935e-32,7.47666e-23)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.347; (-1.32,2.33); (-5.01,0.55)
Integrals by sampling= (4.906e-39,8.81428e-34)  by summing: (5.09359e-39,3.41093e-35)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.985; (-5.74,4.68); (-3.49,4.75)
Integrals by sampling= (2.61691e-54,1.63871e-54)  by summing: (1.74732e-54,1.65003e-54)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.973; (-3.16,2.49); (1.17,1.29)
Integrals by sampling= (8.19082e-41,4.9056e-41)  by summing: (7.59859e-41,4.92856e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.721; (1.18,27.47); (1.08,1.49)
Integrals by sampling= (1.11161e-81,2.17462e-72)  by summing: (8.07206e-124,3.11886e-114)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.015; (4.61,0.84); (-2.48,1.01)
Integrals by sampling= (1.31573e-33,1.04553e-31)  by summing: (1.51188e-33,1.14911e-31)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.991; (0.49,0.59); (-7.24,0.44)
Integrals by sampling= (4.00608e-31,7.93797e-20)  by summing: (4.50384e-31,2.68623e-20)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.044; (-12.05,0.78); (-0.52,1.18)
Integrals by sampling= (1.13843e-40,1.8499e-36)  by summing: (1.0545e-40,1.37561e-36)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.815; (1.09,0.51); (0.66,0.67)
Integrals by sampling= (1.43809e-16,3.34187e-17)  by summing: (2.58918e-17,6.366e-18)

By sampling: Model1 data, correct selection 7/10
             Model2 data, correct selection 7/10
By summing:  Model1 data, correct selection 8/10
             Model2 data, correct selection 7/10

:::

:::spoiler Full output of third test(dataset num set to 100)

Using default random seed
Starting computation for 100 datasets each. ...

Data generated with one component
generating data with: (μ,σ) =  (0.54,3.54)
Integrals by sampling= (1.86361e-50,2.24389e-50)  by summing: (3.24022e-50,3.12089e-50)

generating data with: (μ,σ) =  (-3.86,0.29)
Integrals by sampling= (3.55084e-10,6.97491e-11)  by summing: (9.98852e-15,2.12547e-15)

generating data with: (μ,σ) =  (1.84,9.89)
Integrals by sampling= (1.01493e-67,1.05214e-67)  by summing: (2.5593e-68,1.55591e-68)

generating data with: (μ,σ) =  (4.99,0.64)
Integrals by sampling= (9.99233e-21,2.24743e-21)  by summing: (2.21573e-22,1.10553e-22)

generating data with: (μ,σ) =  (1.40,2.07)
Integrals by sampling= (2.27837e-40,1.41155e-40)  by summing: (2.19598e-40,1.47314e-40)

generating data with: (μ,σ) =  (3.63,1.00)
Integrals by sampling= (2.63974e-29,9.05853e-30)  by summing: (1.45186e-29,6.20038e-30)

generating data with: (μ,σ) =  (-0.13,0.68)
Integrals by sampling= (2.3021e-20,6.34571e-21)  by summing: (7.31168e-21,2.6248e-21)

generating data with: (μ,σ) =  (0.97,0.83)
Integrals by sampling= (3.66915e-26,1.59224e-26)  by summing: (3.08058e-26,1.5787e-26)

generating data with: (μ,σ) =  (4.19,3.16)
Integrals by sampling= (1.25761e-49,9.92477e-50)  by summing: (1.55613e-49,1.09263e-49)

generating data with: (μ,σ) =  (4.70,0.60)
Integrals by sampling= (1.75858e-20,3.91985e-21)  by summing: (1.11411e-22,8.70714e-23)

generating data with: (μ,σ) =  (10.13,3.21)
Integrals by sampling= (5.52487e-47,1.59976e-47)  by summing: (8.46157e-55,1.83779e-55)

generating data with: (μ,σ) =  (-4.06,0.72)
Integrals by sampling= (2.52372e-22,6.82752e-23)  by summing: (1.06443e-23,8.81803e-24)

generating data with: (μ,σ) =  (-1.19,1.05)
Integrals by sampling= (3.21125e-24,1.01069e-24)  by summing: (4.00104e-24,1.23492e-24)

generating data with: (μ,σ) =  (-5.11,0.52)
Integrals by sampling= (4.12602e-16,8.34698e-17)  by summing: (2.86363e-16,6.01008e-17)

generating data with: (μ,σ) =  (-0.51,1.16)
Integrals by sampling= (4.40101e-30,1.99733e-30)  by summing: (4.57234e-30,2.07032e-30)

generating data with: (μ,σ) =  (0.08,1.36)
Integrals by sampling= (9.07484e-37,5.75665e-37)  by summing: (9.16344e-37,5.99852e-37)

generating data with: (μ,σ) =  (-0.79,90.95)
Integrals by sampling= (4.2221e-104,1.5798e-104)  by summing: (0,0)

generating data with: (μ,σ) =  (-1.77,0.60)
Integrals by sampling= (3.94078e-18,9.58847e-19)  by summing: (5.29692e-18,1.25007e-18)

generating data with: (μ,σ) =  (-0.31,0.96)
Integrals by sampling= (2.279e-26,8.37148e-27)  by summing: (2.87275e-26,1.00536e-26)

generating data with: (μ,σ) =  (4.11,0.79)
Integrals by sampling= (1.31701e-22,3.8958e-23)  by summing: (8.80929e-24,5.50182e-24)

generating data with: (μ,σ) =  (4.21,3.40)
Integrals by sampling= (7.96805e-47,8.25143e-47)  by summing: (5.1421e-47,7.29445e-47)

generating data with: (μ,σ) =  (3.94,4.09)
Integrals by sampling= (4.93855e-55,3.84647e-55)  by summing: (1.50774e-55,2.56603e-55)

generating data with: (μ,σ) =  (-4.42,0.59)
Integrals by sampling= (2.0697e-18,4.88147e-19)  by summing: (6.65554e-18,1.51753e-18)

generating data with: (μ,σ) =  (0.36,0.98)
Integrals by sampling= (1.12039e-24,4.03207e-25)  by summing: (1.56251e-24,5.0636e-25)

generating data with: (μ,σ) =  (4.17,0.57)
Integrals by sampling= (4.73114e-18,9.54177e-19)  by summing: (1.4455e-17,3.10861e-18)

generating data with: (μ,σ) =  (-1.83,6.43)
Integrals by sampling= (8.12601e-58,5.39874e-58)  by summing: (8.31819e-59,1.26632e-58)

generating data with: (μ,σ) =  (-0.80,1.22)
Integrals by sampling= (1.92896e-31,1.03882e-31)  by summing: (1.97191e-31,1.11348e-31)

generating data with: (μ,σ) =  (-3.34,0.96)
Integrals by sampling= (3.53464e-25,1.10457e-25)  by summing: (3.32281e-25,1.04594e-25)

generating data with: (μ,σ) =  (0.13,0.80)
Integrals by sampling= (5.67417e-23,2.00984e-23)  by summing: (3.66025e-23,1.50132e-23)

generating data with: (μ,σ) =  (0.19,1.34)
Integrals by sampling= (1.40951e-34,8.0726e-35)  by summing: (1.5216e-34,8.6895e-35)

generating data with: (μ,σ) =  (4.45,6.11)
Integrals by sampling= (1.43795e-56,8.43117e-57)  by summing: (1.0714e-57,1.53114e-57)

generating data with: (μ,σ) =  (-2.33,1.47)
Integrals by sampling= (3.60073e-34,1.84675e-34)  by summing: (3.5749e-34,2.00613e-34)

generating data with: (μ,σ) =  (8.62,4.61)
Integrals by sampling= (2.38157e-51,1.21594e-51)  by summing: (1.89623e-51,5.30181e-52)

generating data with: (μ,σ) =  (1.81,1.16)
Integrals by sampling= (9.93755e-30,4.93098e-30)  by summing: (1.06581e-29,5.37003e-30)

generating data with: (μ,σ) =  (1.56,0.47)
Integrals by sampling= (5.28923e-13,1.08735e-13)  by summing: (1.05682e-14,2.33668e-15)

generating data with: (μ,σ) =  (3.36,1.35)
Integrals by sampling= (8.61372e-30,7.72021e-30)  by summing: (1.09935e-29,6.89218e-30)

generating data with: (μ,σ) =  (3.32,1.58)
Integrals by sampling= (2.79755e-37,1.5961e-37)  by summing: (3.09618e-37,1.70756e-37)

generating data with: (μ,σ) =  (-0.90,0.74)
Integrals by sampling= (1.59432e-20,4.72666e-21)  by summing: (8.56421e-21,2.75721e-21)

generating data with: (μ,σ) =  (5.63,1.41)
Integrals by sampling= (8.33582e-33,3.03231e-33)  by summing: (1.77549e-32,5.32546e-33)

generating data with: (μ,σ) =  (3.64,48.70)
Integrals by sampling= (7.3976e-94,2.95075e-94)  by summing: (0,2.08973e-286)

generating data with: (μ,σ) =  (1.81,1.03)
Integrals by sampling= (4.43663e-29,1.73735e-29)  by summing: (5.19201e-29,2.00416e-29)

generating data with: (μ,σ) =  (-5.69,0.83)
Integrals by sampling= (1.81054e-22,4.90338e-23)  by summing: (8.70627e-24,5.50597e-24)

generating data with: (μ,σ) =  (-0.75,27.33)
Integrals by sampling= (1.624e-80,2.38709e-80)  by summing: (8.30189e-114,1.08759e-105)

generating data with: (μ,σ) =  (-7.01,1.43)
Integrals by sampling= (1.23636e-35,3.43713e-36)  by summing: (3.17894e-35,7.78293e-36)

generating data with: (μ,σ) =  (3.74,1.99)
Integrals by sampling= (1.73714e-40,9.1341e-41)  by summing: (1.70811e-40,9.40016e-41)

generating data with: (μ,σ) =  (1.63,1.51)
Integrals by sampling= (4.83593e-33,2.43289e-33)  by summing: (4.61303e-33,2.49947e-33)

generating data with: (μ,σ) =  (-3.41,13.54)
Integrals by sampling= (3.59976e-67,2.09438e-67)  by summing: (1.79729e-67,9.49724e-68)

generating data with: (μ,σ) =  (-0.00,18.33)
Integrals by sampling= (3.94892e-75,3.22751e-75)  by summing: (3.17277e-87,7.70653e-84)

generating data with: (μ,σ) =  (2.53,0.76)
Integrals by sampling= (7.70376e-24,2.27672e-24)  by summing: (2.46537e-24,1.0276e-24)

generating data with: (μ,σ) =  (-1.79,2.06)
Integrals by sampling= (1.51336e-38,9.50126e-39)  by summing: (1.70421e-38,1.02688e-38)

generating data with: (μ,σ) =  (-5.09,1.14)
Integrals by sampling= (1.86889e-25,4.71131e-26)  by summing: (1.72112e-25,4.64293e-26)

generating data with: (μ,σ) =  (5.23,1.36)
Integrals by sampling= (3.48882e-34,1.35478e-34)  by summing: (3.83183e-34,1.61435e-34)

generating data with: (μ,σ) =  (-4.21,1.32)
Integrals by sampling= (9.37548e-29,2.80962e-29)  by summing: (7.50359e-29,2.64819e-29)

generating data with: (μ,σ) =  (-1.95,2.40)
Integrals by sampling= (1.89293e-43,5.42375e-43)  by summing: (2.43016e-43,4.38626e-43)

generating data with: (μ,σ) =  (-2.65,1.09)
Integrals by sampling= (2.65659e-26,1.00034e-26)  by summing: (2.31467e-26,9.48197e-27)

generating data with: (μ,σ) =  (10.09,0.64)
Integrals by sampling= (6.42022e-21,1.29566e-21)  by summing: (7.78978e-50,1.59462e-50)

generating data with: (μ,σ) =  (-1.15,0.68)
Integrals by sampling= (9.19956e-21,3.59667e-21)  by summing: (1.22175e-20,5.36672e-21)

generating data with: (μ,σ) =  (0.30,2.11)
Integrals by sampling= (1.00387e-37,6.12411e-38)  by summing: (1.08888e-37,6.65203e-38)

generating data with: (μ,σ) =  (1.25,3.08)
Integrals by sampling= (3.72461e-46,2.90043e-46)  by summing: (2.92492e-46,2.62647e-46)

generating data with: (μ,σ) =  (-2.11,1.03)
Integrals by sampling= (1.54667e-25,5.01818e-26)  by summing: (1.09234e-25,4.10651e-26)

generating data with: (μ,σ) =  (3.88,0.49)
Integrals by sampling= (3.58091e-19,8.18674e-20)  by summing: (2.00051e-20,6.82317e-21)

generating data with: (μ,σ) =  (-1.91,1.30)
Integrals by sampling= (4.21968e-28,3.10167e-28)  by summing: (5.27087e-28,3.2524e-28)

generating data with: (μ,σ) =  (-2.17,1.45)
Integrals by sampling= (3.52526e-34,1.6784e-34)  by summing: (3.99347e-34,1.89927e-34)

generating data with: (μ,σ) =  (11.98,1.59)
Integrals by sampling= (2.99327e-36,6.6514e-37)  by summing: (1.78842e-58,3.76834e-59)

generating data with: (μ,σ) =  (-0.61,0.43)
Integrals by sampling= (8.19453e-12,1.60073e-12)  by summing: (2.3836e-14,5.28481e-15)

generating data with: (μ,σ) =  (-5.38,1.60)
Integrals by sampling= (2.7245e-35,3.3284e-35)  by summing: (2.67282e-35,4.19384e-35)

generating data with: (μ,σ) =  (2.27,0.88)
Integrals by sampling= (7.12807e-25,2.20414e-25)  by summing: (8.11517e-25,2.43054e-25)

generating data with: (μ,σ) =  (-1.90,0.63)
Integrals by sampling= (1.70649e-18,4.00402e-19)  by summing: (2.37522e-19,7.00215e-20)

generating data with: (μ,σ) =  (0.97,0.87)
Integrals by sampling= (9.07792e-26,3.80808e-26)  by summing: (7.74683e-26,3.49573e-26)

generating data with: (μ,σ) =  (-5.72,2.16)
Integrals by sampling= (2.54662e-43,1.09774e-43)  by summing: (1.91927e-43,8.94688e-44)

generating data with: (μ,σ) =  (-2.57,1.01)
Integrals by sampling= (7.41044e-26,2.91596e-26)  by summing: (3.96729e-26,1.92111e-26)

generating data with: (μ,σ) =  (-1.69,0.96)
Integrals by sampling= (7.99311e-28,3.43558e-28)  by summing: (6.22097e-28,2.9474e-28)

generating data with: (μ,σ) =  (-8.81,11.97)
Integrals by sampling= (9.71407e-68,6.04978e-68)  by summing: (2.2639e-68,6.98228e-69)

generating data with: (μ,σ) =  (-1.18,2.11)
Integrals by sampling= (4.42049e-37,5.46502e-37)  by summing: (4.55837e-37,5.61272e-37)

generating data with: (μ,σ) =  (-0.06,0.87)
Integrals by sampling= (1.51792e-26,5.54971e-27)  by summing: (1.93937e-26,6.89854e-27)

generating data with: (μ,σ) =  (0.40,0.83)
Integrals by sampling= (4.29643e-25,1.84892e-25)  by summing: (2.75691e-25,1.52253e-25)

generating data with: (μ,σ) =  (-7.15,15.67)
Integrals by sampling= (5.03477e-75,3.17387e-75)  by summing: (9.18262e-86,3.42676e-84)

generating data with: (μ,σ) =  (-3.36,10.44)
Integrals by sampling= (7.89052e-69,3.55131e-69)  by summing: (3.64778e-70,2.53372e-70)

generating data with: (μ,σ) =  (-0.57,2.50)
Integrals by sampling= (4.57174e-43,5.25742e-43)  by summing: (5.56647e-43,5.73008e-43)

generating data with: (μ,σ) =  (-0.32,0.40)
Integrals by sampling= (5.2049e-10,1.05726e-10)  by summing: (3.53989e-13,7.64696e-14)

generating data with: (μ,σ) =  (-5.77,0.90)
Integrals by sampling= (3.75832e-25,9.00712e-26)  by summing: (2.98979e-28,4.16567e-27)

generating data with: (μ,σ) =  (3.96,0.41)
Integrals by sampling= (1.36024e-11,3.23935e-12)  by summing: (5.44701e-16,1.2594e-16)

generating data with: (μ,σ) =  (-1.09,4.58)
Integrals by sampling= (5.60455e-53,5.15605e-53)  by summing: (7.17588e-53,5.03398e-53)

generating data with: (μ,σ) =  (-0.99,0.44)
Integrals by sampling= (7.08843e-14,1.53371e-14)  by summing: (1.42121e-15,3.28283e-16)

generating data with: (μ,σ) =  (-6.61,1.83)
Integrals by sampling= (2.93845e-41,1.09698e-41)  by summing: (4.95832e-41,2.02731e-41)

generating data with: (μ,σ) =  (1.12,0.81)
Integrals by sampling= (1.04669e-23,3.97622e-24)  by summing: (1.74163e-23,6.06284e-24)

generating data with: (μ,σ) =  (9.10,0.65)
Integrals by sampling= (4.45353e-20,8.13427e-21)  by summing: (1.48045e-43,3.05937e-44)

generating data with: (μ,σ) =  (6.60,0.68)
Integrals by sampling= (1.25216e-23,2.82344e-24)  by summing: (7.87345e-23,1.74881e-23)

generating data with: (μ,σ) =  (0.31,4.15)
Integrals by sampling= (5.81816e-55,4.71541e-55)  by summing: (1.4104e-55,1.90602e-55)

generating data with: (μ,σ) =  (-2.15,1.17)
Integrals by sampling= (1.57822e-31,1.09486e-31)  by summing: (1.87111e-31,1.10114e-31)

generating data with: (μ,σ) =  (1.08,2.23)
Integrals by sampling= (1.9408e-39,1.15366e-39)  by summing: (2.08734e-39,1.25105e-39)

generating data with: (μ,σ) =  (-1.52,2.04)
Integrals by sampling= (7.83294e-38,5.88241e-38)  by summing: (8.53094e-38,6.27351e-38)

generating data with: (μ,σ) =  (0.81,36.58)
Integrals by sampling= (4.43033e-89,1.39863e-89)  by summing: (2.44453e-215,5.79775e-185)

generating data with: (μ,σ) =  (-4.67,2.86)
Integrals by sampling= (6.46202e-45,2.83769e-45)  by summing: (8.10157e-45,3.27646e-45)

generating data with: (μ,σ) =  (-6.71,1.02)
Integrals by sampling= (1.22893e-25,2.94231e-26)  by summing: (4.97631e-25,1.34004e-25)

generating data with: (μ,σ) =  (3.86,1.57)
Integrals by sampling= (2.89439e-37,1.57628e-37)  by summing: (2.58522e-37,1.62055e-37)

generating data with: (μ,σ) =  (7.19,3.02)
Integrals by sampling= (8.11127e-48,3.6093e-48)  by summing: (7.24352e-48,2.65998e-48)

generating data with: (μ,σ) =  (2.72,0.76)
Integrals by sampling= (5.69612e-21,1.49948e-21)  by summing: (6.31589e-21,1.81199e-21)

generating data with: (μ,σ) =  (-5.63,3.23)
Integrals by sampling= (4.80731e-44,6.61579e-44)  by summing: (5.96283e-44,8.39159e-44)

generating data with: (μ,σ) =  (-0.51,0.82)
Integrals by sampling= (4.2557e-21,1.34603e-21)  by summing: (2.07281e-21,9.0956e-22)


Data generated with two components
generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.225; (-6.65,2.28); (9.05,6.93)
Integrals by sampling= (6.9886e-67,1.94642e-66)  by summing: (5.91359e-67,1.71339e-66)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.590; (-4.20,1.49); (0.18,1.33)
Integrals by sampling= (8.53697e-46,6.89045e-45)  by summing: (7.96456e-46,5.49727e-45)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.014; (2.96,1.36); (6.54,4.07)
Integrals by sampling= (9.10337e-50,6.52108e-50)  by summing: (1.07688e-49,7.36541e-50)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.400; (1.73,1.04); (-5.27,1.90)
Integrals by sampling= (1.07245e-51,2.31832e-48)  by summing: (2.01695e-51,1.81006e-48)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.147; (4.25,3.47); (5.26,2.89)
Integrals by sampling= (2.73099e-44,1.62642e-44)  by summing: (4.10592e-44,2.06621e-44)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.072; (7.46,2.15); (0.34,0.82)
Integrals by sampling= (4.19459e-28,6.40084e-28)  by summing: (4.05784e-28,7.5489e-28)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.001; (0.31,7.13); (1.40,1.94)
Integrals by sampling= (1.92484e-37,1.02117e-37)  by summing: (2.06207e-37,1.11356e-37)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  1.000; (-2.49,0.39); (1.24,3.27)
Integrals by sampling= (1.16521e-13,2.38351e-14)  by summing: (7.33872e-16,1.66887e-16)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.301; (-3.84,1.29); (-4.89,2.65)
Integrals by sampling= (9.35803e-35,3.79357e-35)  by summing: (6.79313e-35,3.20264e-35)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.233; (-1.72,0.58); (-5.03,2.24)
Integrals by sampling= (3.77045e-45,1.18613e-43)  by summing: (4.37568e-45,1.16243e-43)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.082; (3.77,1.13); (-8.46,0.88)
Integrals by sampling= (1.46836e-51,2.68872e-34)  by summing: (2.53347e-51,9.71294e-47)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.007; (1.75,1.02); (-3.12,0.45)
Integrals by sampling= (4.81613e-15,1.05791e-15)  by summing: (9.4265e-17,2.20145e-17)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.033; (-6.54,0.83); (3.03,1.90)
Integrals by sampling= (1.44919e-47,1.51983e-45)  by summing: (8.18366e-48,1.1503e-45)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.978; (2.89,41.07); (-1.85,0.73)
Integrals by sampling= (1.619e-94,5.98279e-95)  by summing: (0,9.01864e-305)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.114; (1.91,1.06); (-1.28,2.05)
Integrals by sampling= (2.22824e-38,1.77952e-38)  by summing: (2.4967e-38,1.92395e-38)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.078; (-0.02,4.19); (-6.47,2.65)
Integrals by sampling= (7.65173e-46,4.03349e-46)  by summing: (5.1033e-46,3.86549e-46)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.600; (8.37,2.25); (4.68,4.06)
Integrals by sampling= (6.44808e-46,3.52222e-46)  by summing: (1.0796e-45,4.63695e-46)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.115; (0.41,0.63); (4.95,66.35)
Integrals by sampling= (2.21465e-93,8.05696e-92)  by summing: (1.02596e-316,3.45152e-275)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.870; (4.45,7.34); (2.63,1.03)
Integrals by sampling= (8.08245e-61,7.21813e-61)  by summing: (1.06261e-60,1.1329e-60)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.508; (0.20,0.63); (-7.53,0.92)
Integrals by sampling= (5.49803e-51,6.60914e-36)  by summing: (1.02466e-50,3.56112e-37)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.984; (-7.55,1.00); (-7.27,0.61)
Integrals by sampling= (2.53692e-25,5.20028e-26)  by summing: (2.59499e-29,5.96315e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.027; (1.37,3.09); (6.89,13.95)
Integrals by sampling= (3.36478e-74,1.71176e-74)  by summing: (7.51049e-83,3.70124e-83)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.838; (0.15,0.69); (1.17,2.49)
Integrals by sampling= (1.45728e-35,6.0038e-28)  by summing: (1.49917e-35,8.57344e-28)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.962; (2.61,1.12); (-11.63,0.70)
Integrals by sampling= (2.33722e-42,4.43335e-30)  by summing: (2.42172e-42,2.68406e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.121; (-0.39,0.91); (-1.93,2.89)
Integrals by sampling= (5.15692e-44,4.88014e-44)  by summing: (6.81165e-44,5.48025e-44)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.225; (1.60,1.74); (3.04,2.81)
Integrals by sampling= (4.50071e-46,4.16883e-46)  by summing: (3.65238e-46,4.13755e-46)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.000; (-1.25,3.64); (-0.42,4.82)
Integrals by sampling= (6.90285e-53,5.8044e-53)  by summing: (9.2695e-53,7.11103e-53)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.854; (2.82,1.73); (2.02,1.38)
Integrals by sampling= (4.8126e-36,2.38908e-36)  by summing: (5.28419e-36,2.6128e-36)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.052; (0.73,6.90); (1.77,17.29)
Integrals by sampling= (6.84094e-78,4.45838e-78)  by summing: (3.13569e-99,9.48552e-92)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.312; (10.28,4.52); (-0.81,2.65)
Integrals by sampling= (3.15199e-60,1.64742e-58)  by summing: (3.00417e-60,1.42683e-58)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.068; (0.03,0.76); (-4.66,1.60)
Integrals by sampling= (1.0618e-40,6.07796e-41)  by summing: (9.93292e-41,6.41939e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  1.000; (0.57,20.49); (-0.17,1.56)
Integrals by sampling= (1.45903e-79,6.41797e-80)  by summing: (9.66557e-109,8.03223e-99)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.469; (-2.06,0.69); (-0.19,0.74)
Integrals by sampling= (3.22618e-30,2.04349e-30)  by summing: (3.84356e-30,2.11352e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.874; (-4.82,3.24); (0.88,2.16)
Integrals by sampling= (4.35565e-51,2.87191e-51)  by summing: (8.16511e-51,4.46601e-51)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.511; (1.59,5.82); (-3.33,2.81)
Integrals by sampling= (3.34433e-54,2.10448e-53)  by summing: (1.81818e-54,9.14204e-54)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.463; (9.28,2.64); (3.70,1.74)
Integrals by sampling= (1.72002e-51,2.2061e-51)  by summing: (3.45739e-51,2.73751e-51)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.115; (-4.46,0.85); (1.52,0.70)
Integrals by sampling= (4.07095e-42,1.37609e-31)  by summing: (4.11119e-42,2.49942e-31)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.451; (-1.76,1.34); (-1.89,0.79)
Integrals by sampling= (1.95832e-30,9.01611e-31)  by summing: (2.40419e-30,1.06024e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.975; (1.90,12.39); (5.69,5.93)
Integrals by sampling= (6.72913e-71,1.05213e-70)  by summing: (1.49731e-74,5.11014e-74)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.015; (-2.00,8.61); (1.89,1.01)
Integrals by sampling= (1.43225e-28,5.76368e-29)  by summing: (1.47005e-28,5.94687e-29)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.416; (6.87,0.90); (-0.56,2.94)
Integrals by sampling= (5.62402e-50,2.082e-47)  by summing: (8.44576e-50,3.21657e-48)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.211; (5.11,1.19); (-3.50,2.18)
Integrals by sampling= (2.61983e-50,6.57035e-49)  by summing: (4.34456e-50,7.54546e-49)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.000; (-2.35,0.96); (10.22,9.40)
Integrals by sampling= (1.19945e-64,4.32609e-65)  by summing: (7.62919e-68,1.9176e-68)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.034; (-1.08,0.47); (4.24,22.73)
Integrals by sampling= (6.26787e-84,2.2938e-84)  by summing: (3.04823e-144,3.39191e-126)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.874; (-3.73,1.02); (4.65,1.26)
Integrals by sampling= (1.91323e-47,1.41608e-37)  by summing: (1.07631e-47,1.33669e-37)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  1.000; (2.80,12.55); (-8.59,6.20)
Integrals by sampling= (1.72849e-68,9.97312e-69)  by summing: (1.24013e-69,8.22976e-70)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.391; (-1.29,0.94); (6.40,2.44)
Integrals by sampling= (4.02195e-51,6.43048e-50)  by summing: (7.50949e-51,5.60608e-50)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.035; (-0.73,16.78); (-6.26,4.36)
Integrals by sampling= (1.62538e-57,1.01662e-57)  by summing: (9.33424e-59,4.45046e-58)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.321; (-0.16,3.38); (3.40,0.78)
Integrals by sampling= (2.1389e-47,1.06622e-34)  by summing: (1.20634e-47,2.09876e-34)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.178; (-1.97,9.74); (-1.25,2.27)
Integrals by sampling= (2.76409e-55,1.66831e-50)  by summing: (4.64004e-56,6.00394e-51)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.985; (7.07,2.87); (1.00,0.99)
Integrals by sampling= (1.50953e-48,8.11978e-49)  by summing: (1.17247e-48,6.60845e-49)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.709; (0.24,0.86); (-1.22,1.86)
Integrals by sampling= (6.82643e-37,2.50752e-36)  by summing: (7.04036e-37,2.80103e-36)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.092; (2.56,0.71); (-1.20,2.08)
Integrals by sampling= (6.8285e-42,7.84006e-42)  by summing: (6.60921e-42,8.09543e-42)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.046; (-1.11,16.07); (-0.35,0.73)
Integrals by sampling= (3.32837e-56,1.06185e-33)  by summing: (1.83746e-57,2.28669e-34)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.001; (-3.00,16.23); (-5.79,1.16)
Integrals by sampling= (4.45667e-30,1.27712e-30)  by summing: (8.00356e-31,5.24089e-31)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.581; (-5.98,6.72); (-1.32,0.82)
Integrals by sampling= (2.79019e-60,8.11804e-59)  by summing: (2.73825e-60,2.267e-58)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.308; (0.51,4.92); (-0.78,0.57)
Integrals by sampling= (5.7145e-35,2.25451e-29)  by summing: (5.93643e-35,1.09179e-29)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.005; (-0.39,27.78); (4.62,1.11)
Integrals by sampling= (1.65244e-27,5.10262e-28)  by summing: (4.01484e-28,1.73768e-28)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.849; (-1.83,1.22); (-3.39,0.81)
Integrals by sampling= (1.66363e-31,8.60459e-32)  by summing: (1.6731e-31,8.82482e-32)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.182; (-0.07,2.34); (-1.59,5.55)
Integrals by sampling= (9.02809e-56,2.71479e-55)  by summing: (8.33164e-57,2.08378e-55)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.466; (-5.02,0.41); (-4.99,0.68)
Integrals by sampling= (2.69165e-17,2.78821e-17)  by summing: (2.74945e-18,1.09107e-18)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.958; (-2.83,0.25); (-4.34,39.96)
Integrals by sampling= (7.43106e-05,9.60011e-06)  by summing: (1.80425e-11,3.75446e-12)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.005; (1.71,1.43); (-0.08,1.41)
Integrals by sampling= (1.20797e-36,6.71589e-37)  by summing: (1.21716e-36,7.11103e-37)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.026; (1.02,1.12); (-2.12,2.07)
Integrals by sampling= (2.51188e-42,1.43717e-42)  by summing: (2.61143e-42,1.52489e-42)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.956; (2.03,3.94); (-0.43,3.58)
Integrals by sampling= (3.14455e-53,3.119e-53)  by summing: (3.64937e-53,3.1189e-53)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.205; (3.05,9.02); (-4.71,0.94)
Integrals by sampling= (1.0861e-50,1.22906e-34)  by summing: (1.91886e-50,1.26023e-35)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.431; (-0.93,3.10); (-4.86,5.44)
Integrals by sampling= (5.97058e-56,4.74672e-56)  by summing: (5.08697e-57,1.70508e-56)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  1.000; (-2.76,10.53); (-1.29,4.24)
Integrals by sampling= (4.52488e-69,2.64764e-69)  by summing: (1.45656e-70,1.21931e-70)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.032; (7.75,9.69); (-0.09,1.30)
Integrals by sampling= (5.86554e-30,4.14164e-30)  by summing: (6.49065e-30,4.73054e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.751; (2.99,0.91); (-8.29,5.71)
Integrals by sampling= (7.05127e-61,3.86904e-47)  by summing: (1.15123e-60,3.81898e-47)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.408; (1.91,5.30); (-4.94,2.63)
Integrals by sampling= (1.92493e-53,1.65671e-52)  by summing: (1.98051e-53,1.42328e-52)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.915; (-0.08,2.62); (5.91,2.59)
Integrals by sampling= (6.25462e-48,1.05552e-47)  by summing: (3.95522e-48,1.02408e-47)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.997; (-6.68,1.76); (-4.25,1.34)
Integrals by sampling= (7.14683e-41,2.61615e-41)  by summing: (1.5869e-40,4.65822e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.764; (-4.76,1.26); (-0.20,1.21)
Integrals by sampling= (3.39077e-43,1.56656e-41)  by summing: (4.07817e-43,1.96865e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.310; (-1.05,25.36); (-4.61,3.91)
Integrals by sampling= (1.30834e-71,1.47093e-66)  by summing: (4.0026e-76,4.40218e-75)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.698; (-2.66,2.19); (0.88,4.17)
Integrals by sampling= (3.97097e-44,8.95455e-44)  by summing: (5.2247e-44,1.00262e-43)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.426; (-1.17,1.05); (1.17,1.06)
Integrals by sampling= (2.23278e-37,1.52454e-37)  by summing: (2.35385e-37,1.63439e-37)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.007; (3.12,1.09); (6.00,2.92)
Integrals by sampling= (4.1216e-44,2.91153e-44)  by summing: (3.3947e-44,1.87325e-44)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.955; (4.11,3.15); (4.28,0.66)
Integrals by sampling= (1.94467e-45,1.08044e-45)  by summing: (2.1819e-45,1.20854e-45)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.873; (-6.00,1.86); (-1.04,3.29)
Integrals by sampling= (8.56567e-45,2.26657e-40)  by summing: (1.11785e-44,5.86396e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.486; (8.00,2.59); (-0.42,16.77)
Integrals by sampling= (8.23012e-75,5.10955e-73)  by summing: (6.17846e-85,2.22753e-83)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.505; (3.19,0.46); (0.02,1.28)
Integrals by sampling= (3.57533e-37,2.52297e-34)  by summing: (3.73854e-37,2.87421e-35)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.924; (-1.23,1.90); (-4.64,6.99)
Integrals by sampling= (2.17639e-42,1.36962e-40)  by summing: (2.31306e-42,1.17742e-40)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.762; (4.64,9.75); (1.44,4.26)
Integrals by sampling= (7.368e-65,3.82023e-65)  by summing: (2.23824e-64,1.04369e-64)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.186; (-4.25,0.47); (1.47,0.75)
Integrals by sampling= (1.65458e-44,9.09983e-33)  by summing: (2.12562e-44,4.46399e-33)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.999; (4.52,10.13); (2.57,72.13)
Integrals by sampling= (2.38872e-70,1.17008e-70)  by summing: (4.51462e-73,4.54174e-73)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.018; (2.03,0.66); (-8.60,0.67)
Integrals by sampling= (1.71952e-17,3.08841e-18)  by summing: (9.40204e-39,1.96134e-39)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.287; (0.30,1.97); (-3.94,1.28)
Integrals by sampling= (1.46793e-42,5.22259e-40)  by summing: (1.58056e-42,6.39447e-40)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.675; (-0.06,9.10); (2.00,1.73)
Integrals by sampling= (8.49164e-63,1.40206e-62)  by summing: (2.97543e-62,3.73297e-62)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.011; (-6.75,1.26); (5.21,0.59)
Integrals by sampling= (9.05223e-17,2.01731e-17)  by summing: (1.68472e-16,3.54909e-17)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.149; (0.41,17.49); (0.38,0.41)
Integrals by sampling= (9.05427e-53,3.64919e-29)  by summing: (1.27901e-52,7.3921e-30)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.011; (2.18,1.04); (6.04,8.30)
Integrals by sampling= (3.36323e-64,1.71289e-64)  by summing: (1.16149e-63,4.70201e-64)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.035; (-3.60,0.75); (4.19,1.01)
Integrals by sampling= (1.98531e-37,1.61482e-32)  by summing: (1.79639e-37,2.39334e-32)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.561; (-1.79,2.97); (-0.03,0.52)
Integrals by sampling= (1.39121e-41,5.76642e-41)  by summing: (1.29384e-41,6.66843e-41)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.375; (3.69,1.01); (-12.53,1.91)
Integrals by sampling= (7.41686e-63,2.10176e-50)  by summing: (1.89566e-62,1.12264e-60)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.955; (-4.12,3.87); (-4.47,3.01)
Integrals by sampling= (4.96146e-52,3.45614e-52)  by summing: (9.32484e-52,4.94718e-52)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.257; (-3.25,0.54); (-1.46,0.78)
Integrals by sampling= (2.2067e-30,8.82448e-31)  by summing: (1.87626e-30,8.59649e-31)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.236; (-2.15,1.52); (1.88,0.86)
Integrals by sampling= (6.36468e-41,7.43948e-40)  by summing: (5.69475e-41,6.68421e-40)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.076; (-0.62,7.16); (-2.12,2.56)
Integrals by sampling= (2.26876e-47,1.14416e-46)  by summing: (1.2752e-47,1.06695e-46)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.146; (-9.32,0.68); (1.96,2.49)
Integrals by sampling= (3.28025e-58,2.70433e-47)  by summing: (5.66921e-59,3.87375e-53)

By sampling: Model1 data, correct selection 91/100
             Model2 data, correct selection 50/100
By summing:  Model1 data, correct selection 83/100
             Model2 data, correct selection 57/100

:::

:::spoiler Full output of fourth test(repeat num set to 20K)

Using default random seed
Starting computation for 10 datasets each. ...

Data generated with one component
generating data with: (μ,σ) =  (0.54,3.54)
Integrals by sampling= (2.01732e-50,2.14882e-50)  by summing: (3.24022e-50,3.12089e-50)

generating data with: (μ,σ) =  (-1.87,2.16)
Integrals by sampling= (2.81648e-39,1.58237e-39)  by summing: (3.20225e-39,1.81856e-39)

generating data with: (μ,σ) =  (-7.01,1.14)
Integrals by sampling= (2.00585e-32,9.26428e-33)  by summing: (2.90153e-32,7.75657e-33)

generating data with: (μ,σ) =  (-2.32,0.62)
Integrals by sampling= (1.08742e-20,3.58438e-21)  by summing: (1.95745e-20,4.9171e-21)

generating data with: (μ,σ) =  (5.88,1.12)
Integrals by sampling= (7.31889e-28,1.67512e-28)  by summing: (2.01319e-29,2.2994e-29)

generating data with: (μ,σ) =  (-0.87,0.53)
Integrals by sampling= (1.71382e-16,3.41412e-17)  by summing: (1.51446e-17,4.01038e-18)

generating data with: (μ,σ) =  (-0.92,5.78)
Integrals by sampling= (9.65839e-61,2.10993e-60)  by summing: (1.43676e-60,3.70144e-60)

generating data with: (μ,σ) =  (3.45,0.81)
Integrals by sampling= (5.16593e-26,1.84302e-26)  by summing: (1.74837e-26,9.52794e-27)

generating data with: (μ,σ) =  (-0.03,1.02)
Integrals by sampling= (2.87918e-28,1.26763e-28)  by summing: (2.62173e-28,1.13217e-28)

generating data with: (μ,σ) =  (-1.03,1.37)
Integrals by sampling= (5.90994e-31,3.04709e-31)  by summing: (5.59816e-31,3.07399e-31)


Data generated with two components
generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.719; (-0.25,18.36); (0.10,2.16)
Integrals by sampling= (5.97186e-72,4.22775e-72)  by summing: (5.39971e-77,1.70271e-75)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.198; (-6.44,9.76); (2.33,7.11)
Integrals by sampling= (1.47295e-61,1.78842e-61)  by summing: (3.36043e-61,3.50655e-61)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.136; (-4.90,1.07); (-5.18,0.68)
Integrals by sampling= (7.56107e-21,1.9025e-21)  by summing: (1.00649e-20,4.07395e-21)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.477; (-1.34,0.95); (-4.75,3.55)
Integrals by sampling= (2.10694e-46,7.57591e-46)  by summing: (1.60217e-46,8.60866e-46)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.955; (-2.96,0.96); (-1.75,0.70)
Integrals by sampling= (1.31154e-24,3.53804e-25)  by summing: (2.05715e-24,5.65351e-25)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.691; (2.90,0.98); (-0.59,11.09)
Integrals by sampling= (4.92001e-57,2.39173e-50)  by summing: (2.49793e-58,5.24274e-50)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.597; (0.93,0.61); (1.28,1.75)
Integrals by sampling= (4.08634e-25,1.66084e-25)  by summing: (2.52347e-25,1.29223e-25)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.427; (-3.75,1.82); (-2.56,2.36)
Integrals by sampling= (8.08189e-43,4.84749e-43)  by summing: (9.26013e-43,5.48336e-43)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.029; (-0.08,3.35); (5.86,2.31)
Integrals by sampling= (3.80772e-43,8.76871e-42)  by summing: (4.76926e-43,5.44976e-42)

generating data with:  m; (μ1,σ1); (μ2,σ2) =  0.878; (0.53,1.10); (-14.79,0.86)
Integrals by sampling= (1.86986e-49,5.30699e-39)  by summing: (2.33995e-49,8.0067e-39)

By sampling: Model1 data, correct selection 8/10
             Model2 data, correct selection 5/10
By summing:  Model1 data, correct selection 8/10
             Model2 data, correct selection 6/10

:::

After doing some test, I am not even certain about what I am doing.
All the data I have seen does not really make any sense.

## Reviewing assignment requirement
:::info
Compile and run the program `Gaussian_poolOrNot.c`. Look at the source
code and figure out how it works. Edit the program to explore some different
parameter settings. How does the ability to discriminate between 1 and 2-
component increase with sample size. How about with the parameter setting
on the priors?
The priors are: `Gauss_params` `mu_prior_params` for the mean of the
data generating Gaussian distribution and `sigma_prior_param_a` and `sigma_prior_param_b` for the Gamma distribution used for the prior on
the precision $\dfrac{1}{\sigma^2}$. The mixture coefficient is selected according to a $\text{Beta}(\dfrac{1}{2}, \dfrac{1}{2})$, but that could also be explored.
Also evaluate the accuracy of our estimation of the integrals. We fixed data, when the number of samples is increased does the estimate seem to be converging? Instrument the code so you can graph how the estimate of the integral changes over time on a long run, i.e. with `sampleRepeatNum` set to a large number.
:::

From the above test results, it seems that I could answer only one questions:
* How does the ability to discriminate between 1 and 2-component increase with sample size
* From the test I have done, 1-component model seems to do better than 2-component model on larger dataset. But not really better with more repeat number.

But I don't really have much clue for anything else.
1. How about with the parameter setting on the priors?
* I am not even certain about what this means. Should I attempt to change parameters and see the result? But what kind of result I should be expecting?
2. Also evaluate the accuracy of our estimation of the integrals.
* From my observation, only thing I see in the codes are generating data points by sampling. If so, how do I calculate the accuracy? Or more precisely, what should be calculated against the data I have?
3. Instrument the code so you can graph how the estimate of the integral changes over time on a long run, i.e. with `sampleRepeatNum` set to a large number.
* I could see that I should set `sampleRepeatNum` to a large number, but what does **the integral changes over time** precisely mean? I attempted to record and plot the data generated by the code(`prob_data1_bySampling` and `prob_data2_bySampling`), But it does not really fit the answer it is asking for.
* Two graphs below are plotted with `sampleRepeatNum` set to 2M and `datasets_n` set to 100, X label shows `prob_data1_bySampling` and Y label shows `prob_data2_bySampling`. I will put the original data in a spoiler block under the plot so that you could review.

:::info
By the way, from my knowledge it is not possible to show any graph on terminal with `gnuplot`.
(I am using SSH to connect to my working computer)
I did attempt to use `X11 forwarding` so that the graph could maybe show in my application on my notebook, but I have zero clue how to porperly connect the `gnuplot` to my application.
At last, I find out I just need to store the plot into an image and see the graph.
:::

:::spoiler `gnuplot` command to plot the data

set title "One Component" font ",20"
set key left box

set term png
set out "one_component.png"
set xlabel 'prob data1 bySampling'
set ylabel 'prob data2 bySampling'
set grid
set style data lines
plot "one_component.txt"
set out

set title "Two Component" font ",20"
set key left box

set term png
set out "two_component.png"
set xlabel 'prob data1 bySampling'
set ylabel 'prob data2 bySampling'
set grid
set style data lines
plot "two_component.txt"
set out
exit

:::

:::spoiler 1-component model data

# prob_data1_bySampling	prob_data2_bySampling
1.863612E-50	2.243892E-50
3.550838E-10	6.974909E-11
1.014932E-67	1.052138E-67
9.992330E-21	2.247433E-21
2.278372E-40	1.411546E-40
2.639745E-29	9.058528E-30
2.302095E-20	6.345712E-21
3.669148E-26	1.592240E-26
1.257609E-49	9.924768E-50
1.758580E-20	3.919850E-21
5.524871E-47	1.599759E-47
2.523724E-22	6.827519E-23
3.211253E-24	1.010686E-24
4.126015E-16	8.346980E-17
4.401013E-30	1.997327E-30
9.074840E-37	5.756654E-37
4.222100E-104	1.579801E-104
3.940782E-18	9.588468E-19
2.279003E-26	8.371477E-27
1.317013E-22	3.895804E-23
7.968051E-47	8.251429E-47
4.938550E-55	3.846471E-55
2.069697E-18	4.881474E-19
1.120395E-24	4.032068E-25
4.731135E-18	9.541774E-19
8.126012E-58	5.398741E-58
1.928956E-31	1.038821E-31
3.534643E-25	1.104567E-25
5.674168E-23	2.009843E-23
1.409509E-34	8.072600E-35
1.437952E-56	8.431173E-57
3.600729E-34	1.846751E-34
2.381574E-51	1.215935E-51
9.937549E-30	4.930977E-30
5.289233E-13	1.087351E-13
8.613722E-30	7.720207E-30
2.797546E-37	1.596098E-37
1.594322E-20	4.726658E-21
8.335825E-33	3.032305E-33
7.397598E-94	2.950747E-94
4.436629E-29	1.737351E-29
1.810542E-22	4.903384E-23
1.624000E-80	2.387094E-80
1.236361E-35	3.437127E-36
1.737138E-40	9.134102E-41
4.835935E-33	2.432893E-33
3.599758E-67	2.094382E-67
3.948924E-75	3.227509E-75
7.703756E-24	2.276723E-24
1.513364E-38	9.501256E-39
1.868888E-25	4.711309E-26
3.488823E-34	1.354783E-34
9.375480E-29	2.809623E-29
1.892932E-43	5.423751E-43
2.656589E-26	1.000345E-26
6.420224E-21	1.295656E-21
9.199559E-21	3.596670E-21
1.003870E-37	6.124106E-38
3.724611E-46	2.900428E-46
1.546669E-25	5.018185E-26
3.580910E-19	8.186740E-20
4.219677E-28	3.101669E-28
3.525256E-34	1.678405E-34
2.993273E-36	6.651400E-37
8.194535E-12	1.600731E-12
2.724504E-35	3.328398E-35
7.128068E-25	2.204144E-25
1.706488E-18	4.004017E-19
9.077922E-26	3.808083E-26
2.546624E-43	1.097736E-43
7.410441E-26	2.915960E-26
7.993112E-28	3.435583E-28
9.714071E-68	6.049778E-68
4.420488E-37	5.465021E-37
1.517915E-26	5.549711E-27
4.296425E-25	1.848916E-25
5.034772E-75	3.173873E-75
7.890520E-69	3.551313E-69
4.571737E-43	5.257416E-43
5.204905E-10	1.057258E-10
3.758323E-25	9.007118E-26
1.360236E-11	3.239349E-12
5.604550E-53	5.156051E-53
7.088427E-14	1.533714E-14
2.938452E-41	1.096976E-41
1.046689E-23	3.976222E-24
4.453527E-20	8.134273E-21
1.252163E-23	2.823439E-24
5.818161E-55	4.715407E-55
1.578221E-31	1.094864E-31
1.940800E-39	1.153656E-39
7.832943E-38	5.882412E-38
4.430326E-89	1.398626E-89
6.462020E-45	2.837689E-45
1.228932E-25	2.942309E-26
2.894392E-37	1.576277E-37
8.111273E-48	3.609295E-48
5.696121E-21	1.499477E-21
4.807308E-44	6.615787E-44
4.255704E-21	1.346032E-21

:::

:::spoiler 2-component model data

# prob_data1_bySampling	prob_data2_bySampling
6.988604E-67	1.946423E-66
8.536968E-46	6.890449E-45
9.103367E-50	6.521084E-50
1.072446E-51	2.318322E-48
2.730994E-44	1.626420E-44
4.194590E-28	6.400841E-28
1.924841E-37	1.021169E-37
1.165211E-13	2.383513E-14
9.358032E-35	3.793573E-35
3.770449E-45	1.186127E-43
1.468364E-51	2.688723E-34
4.816131E-15	1.057913E-15
1.449194E-47	1.519833E-45
1.618997E-94	5.982794E-95
2.228242E-38	1.779519E-38
7.651732E-46	4.033493E-46
6.448084E-46	3.522220E-46
2.214646E-93	8.056958E-92
8.082453E-61	7.218131E-61
5.498033E-51	6.609137E-36
2.536923E-25	5.200279E-26
3.364779E-74	1.711762E-74
1.457279E-35	6.003804E-28
2.337216E-42	4.433347E-30
5.156920E-44	4.880140E-44
4.500710E-46	4.168829E-46
6.902854E-53	5.804401E-53
4.812597E-36	2.389081E-36
6.840942E-78	4.458376E-78
3.151986E-60	1.647423E-58
1.061801E-40	6.077963E-41
1.459027E-79	6.417969E-80
3.226176E-30	2.043493E-30
4.355647E-51	2.871912E-51
3.344326E-54	2.104482E-53
1.720018E-51	2.206101E-51
4.070950E-42	1.376092E-31
1.958320E-30	9.016108E-31
6.729126E-71	1.052132E-70
1.432249E-28	5.763681E-29
5.624018E-50	2.082000E-47
2.619834E-50	6.570346E-49
1.199453E-64	4.326086E-65
6.267872E-84	2.293804E-84
1.913234E-47	1.416080E-37
1.728488E-68	9.973116E-69
4.021949E-51	6.430477E-50
1.625380E-57	1.016621E-57
2.138903E-47	1.066217E-34
2.764086E-55	1.668310E-50
1.509532E-48	8.119782E-49
6.826428E-37	2.507524E-36
6.828501E-42	7.840058E-42
3.328374E-56	1.061854E-33
4.456673E-30	1.277117E-30
2.790192E-60	8.118040E-59
5.714496E-35	2.254508E-29
1.652436E-27	5.102622E-28
1.663625E-31	8.604589E-32
9.028087E-56	2.714791E-55
2.691648E-17	2.788207E-17
7.431059E-05	9.600112E-06
1.207966E-36	6.715894E-37
2.511877E-42	1.437174E-42
3.144554E-53	3.119000E-53
1.086102E-50	1.229063E-34
5.970583E-56	4.746724E-56
4.524877E-69	2.647642E-69
5.865537E-30	4.141639E-30
7.051272E-61	3.869040E-47
1.924928E-53	1.656707E-52
6.254621E-48	1.055523E-47
7.146834E-41	2.616149E-41
3.390769E-43	1.566555E-41
1.308340E-71	1.470928E-66
3.970974E-44	8.954555E-44
2.232777E-37	1.524541E-37
4.121601E-44	2.911527E-44
1.944670E-45	1.080444E-45
8.565667E-45	2.266565E-40
8.230123E-75	5.109549E-73
3.575332E-37	2.522974E-34
2.176391E-42	1.369625E-40
7.368000E-65	3.820227E-65
1.654579E-44	9.099834E-33
2.388725E-70	1.170084E-70
1.719516E-17	3.088413E-18
1.467928E-42	5.222592E-40
8.491637E-63	1.402062E-62
9.052233E-17	2.017306E-17
9.054267E-53	3.649194E-29
3.363233E-64	1.712895E-64
1.985312E-37	1.614824E-32
1.391205E-41	5.766421E-41
7.416863E-63	2.101756E-50
4.961462E-52	3.456143E-52
2.206700E-30	8.824484E-31
6.364682E-41	7.439479E-40
2.268756E-47	1.144157E-46
3.280247E-58	2.704331E-47

:::

---
* For the third question I have no clue about, I did attempt to wrire another code to try to see the result.
With the remind of my classmate(杜孟聰), he mentioned that I should plot how the data variate with more `sampleRepeatNum`. So I modified my code and tried to plot this(I did overwrite my code to see this, so the above code to record data was gone.)
The plot I made failed to show the convergence I wish to see, yet I am not certain about what went wrong.

:::spoiler `command` to gnuplot

set title "Converge" font ",20"
set key left box

set term png
set out "converge.png"
set xlabel 'sample repeat num'
set ylabel 'data'
plot "one-component.txt", "two-component.txt"
set out
exit

:::
:::spoiler one-component.txt

0       3.987700E-288
1       2.440956E-62
2       2.440956E-62
3       2.440956E-62
4       2.440956E-62
5       2.440956E-62
6       2.440956E-62
7       2.440956E-62
8       2.440956E-62
9       2.440956E-62
...

:::
:::spoiler two-component.txt

0       1.291333E-82
1       1.397360E-82
2       1.603648E-74
3       1.603648E-74
4       1.603648E-74
5       1.603648E-74
6       3.805955E-62
7       3.805955E-62
8       4.572622E-51
9       4.673324E-51
...

:::
###### tags: `1112_courses` `probability models`