## LogLikehood Values, for the Alarm Data

The log likelihood values, over all 100 queries:
 beta normal -93,682.04 -93,772.99
which corresponds to an average difference of
0.9095 / instance
-- which means the beta distribution claims each instance is more likely than does the normal distribution, by an average factor of e0.9095 = 2.483.

Unfortunately, it is not clear whether this is significant. Moreover, the log-likelihood measure unnaturately favours the normal model, especially when the response is near 0 or near 1. That is, the log-likelihood for Normal(μ ,σ2) model is a constant plus a sum of terms

-(1/2) × [ (x-μ)/σ ]2
while the log-likelihood for Beta(a,b) is a constant plus a sum of terms
(a-1) × log(x) + (b-1) × log(1-x)

Notice that the Beta log likelihood blows up (large negative) for x near 0 and 1, and so may be strongly affected by values near the boundary of the unit interval. The normal log likelihood does not have this behaviour.

Attached are the results for the 100 queries. The first column is the query number for alarm. The second (resp., third) column is the likelihood of each samples from the true query response under the beta (resp., normal) model. The fourth column is the difference between the beta log-likelihood and the normal log-likelihood (the more positive this value, the better the beta model is doing).

```         beta    normal   beta-normal

1   -93680.66 -93693.20   12.54693086
2   -93813.69 -93843.52   29.82991899
3   -93735.34 -93886.43  151.09273046
4   -93621.21 -93618.14   -3.07003414
5   -93902.45 -94100.64  198.19226720
6   -93579.83 -93585.04    5.21622705
7   -93614.47 -93639.66   25.19767966
8   -93489.66 -93535.02   45.36258028
9   -93495.37 -93549.24   53.86471157
10  -93810.75 -93940.84  130.08770791
11  -95207.35 -97092.94 1885.58283343
12  -93871.11 -93990.78  119.67039099
13  -93712.19 -93729.87   17.67731863
14  -93951.44 -93989.56   38.12030886
15  -93718.27 -93881.93  163.66144350
16  -93424.57 -93604.27  179.69169866
17  -93660.94 -93686.93   25.99113056
18  -93455.76 -93502.17   46.41421602
19  -93527.38 -93560.92   33.54121419
20  -93605.55 -93609.26    3.71140569
21  -93662.64 -93701.11   38.47538781
22  -93647.82 -93792.35  144.53072512
23  -93550.32 -93581.21   30.89018696
24  -93681.84 -93781.83   99.98897815
25  -93600.43 -93633.15   32.71664293
26  -93656.72 -93687.27   30.55002949
27  -93779.49 -93832.06   52.56887538
28  -93396.78 -93486.63   89.84242364
29  -93637.89 -93635.89   -2.00108213
30  -93640.71 -93645.28    4.56844677
31  -93611.78 -93636.23   24.45117443
32  -93802.22 -93986.68  184.46132607
33  -93804.50 -93936.52  132.01854575
34  -93569.56 -93615.15   45.59418104
35  -93643.50 -93692.48   48.98107255
36  -93624.53 -93631.49    6.95954538
37  -93671.27 -93869.33  198.06054834
38  -93682.59 -93718.37   35.78171507
39  -93626.98 -93647.89   20.91556097
40  -94040.89 -93765.51 -275.38117640
41  -93643.42 -93769.01  125.58137485
42  -93717.54 -93727.47    9.92921296
43  -93601.52 -93622.89   21.36924436
44  -93652.89 -93694.27   41.37366320
45  -93911.86 -94202.37  290.51242713
46  -93630.62 -93649.40   18.77737471
47  -93240.63 -93450.60  209.96998732
48  -93640.29 -93653.73   13.44605600
49  -93875.94 -94021.58  145.64150913
50  -93918.84 -94160.29  241.45435442
51  -93623.88 -93627.31    3.42724436
52  -93920.22 -94283.91  363.68840460
53  -93773.61 -93804.58   30.96696501
54  -93634.34 -93642.49    8.14930149
55  -94438.28 -95125.59  687.31082419
56  -93489.92 -93515.08   25.16536792
57  -93603.33 -93631.94   28.61148456
58  -93619.18 -93643.12   23.94492465
59  -93595.81 -93595.84    0.03568137
60  -93861.69 -93973.14  111.45193747
61  -93576.59 -93607.10   30.50761081
62  -93636.15 -93640.25    4.10046664
63  -92855.54 -92992.00  136.45960901
64  -93718.56 -93823.28  104.72187232
65  -93608.71 -93627.76   19.04720410
66  -93429.46 -93501.63   72.16872676
67  -93513.09 -93606.85   93.76171759
68  -93595.30 -93649.15   53.84893200
69  -93690.24 -93718.56   28.31818668
70  -93643.75 -93666.90   23.15575144
71  -93644.25 -93653.95    9.70330292
72  -93286.02 -93493.18  207.15656899
73  -93504.40 -93540.47   36.06817969
74  -93842.88 -93897.66   54.77882638
75  -93646.48 -93653.43    6.95398203
76  -93696.33 -93694.05   -2.27431226
77  -93741.43 -93751.58   10.14264471
78  -93626.72 -93622.56   -4.15412339
79  -93626.98 -93630.66    3.67938843
80  -93625.93 -93652.10   26.17707411
81  -93597.73 -93603.66    5.93325532
82  -93757.88 -93898.93  141.05470754
83  -93687.55 -93717.15   29.60031635
84  -93578.13 -93692.84  114.70822963
85  -93764.42 -93849.16   84.73474654
86  -93655.38 -93660.09    4.71075119
87  -93654.67 -93660.67    6.00048920
88  -93615.10 -93612.16   -2.94002128
89  -93720.97 -93783.95   62.98162882
90  -93641.04 -93667.11   26.07484414
91  -93711.03 -93809.07   98.03387606
92  -93561.01 -93569.47    8.45929247
93  -93696.08 -93703.32    7.23687481
94  -93650.67 -93652.33    1.65961749
95  -93702.61 -93802.97  100.35923198
96  -93629.00 -93648.44   19.43582846
97  -93621.24 -93630.38    9.13812577
98  -94132.08 -94574.57  442.48970793
99  -93730.95 -93756.51   25.55462917
100 -93983.69 -94471.66  487.96092989
```