1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
|
Mixture distribution
--------------------
.. math::
\mathbf{x}
&\sim
\mathrm{Mix}_{\mathcal{D}}
\left(
\lambda,
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\right)
.. math::
\lambda \in \{1, \ldots, N\},
\quad \mathcal{D} \text{ is an exp.fam. distribution},
\quad \mathbf{\Theta}^{(n)}_k \text{ are parameters of } \mathcal{D}
.. math::
\log\mathrm{Mix}_{\mathcal{D}}
\left(
\mathbf{x}
\left| \lambda,
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\right.
\right)
&=
\sum^N_{n=1} [\lambda=n]
\mathbf{u}_{\mathcal{D}}(\mathbf{x})^{\mathrm{T}}
\boldsymbol{\phi}_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K
\right)
\\
& \quad +
\sum^N_{n=1} [\lambda=n]
g_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K
\right)
+ f_{\mathcal{D}} (\mathbf{x})
.. math::
\mathbf{u} (\mathbf{x})
&=
\mathbf{u}_{\mathcal{D}} (\mathbf{x})
\\
\boldsymbol{\phi}
\left(
\lambda,
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\right)
&=
\sum^N_{n=1} [\lambda=n]
\boldsymbol{\phi}_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K
\right)
%
\\
%
\boldsymbol{\phi}_{\lambda}
\left(
\mathbf{x},
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\right)
&=
\left[\begin{matrix}
\mathbf{u}_{\mathcal{D}} (\mathbf{x})^{\mathrm{T}}
\boldsymbol{\phi}_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(1)}_1, \ldots, \mathbf{\Theta}^{(1)}_K
\right)
+ g_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(1)}_1, \ldots, \mathbf{\Theta}^{(1)}_K
\right)
\\
\vdots
\\
\mathbf{u}_{\mathcal{D}} (\mathbf{x})^{\mathrm{T}}
\boldsymbol{\phi}_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(N)}_1, \ldots, \mathbf{\Theta}^{(N)}_K
\right)
+ g_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(N)}_1, \ldots, \mathbf{\Theta}^{(N)}_K
\right)
\end{matrix}\right]
%
\\
%
\boldsymbol{\phi}_{\mathbf{\Theta}^{(m)}_l}
\left(
\mathbf{x},
\lambda,
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\setminus \left\{ \mathbf{\Theta}^{(m)}_l \right\}
\right)
&=
[\lambda=m] \boldsymbol{\phi}_{\mathcal{D}\rightarrow\mathbf{\Theta}_l}
\left(
\mathbf{x},
\left\{ \mathbf{\Theta}^{(m)}_k \right\}_{k\neq l}
\right)
%
\\
%
g
\left(
\lambda,
\left\{ \mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K \right\}^N_{n=1}
\right)
&=
\sum^N_{n=1} [\lambda=n]
g_{\mathcal{D}}
\left(
\mathbf{\Theta}^{(n)}_1, \ldots, \mathbf{\Theta}^{(n)}_K
\right)
\\
g (\boldsymbol{\phi})
&=
g_{\mathcal{D}} (\boldsymbol{\phi})
\\
f(\mathbf{x})
&=
f_{\mathcal{D}} (\mathbf{x})
\\
\overline{\mathbf{u}} (\boldsymbol{\phi})
&=
\overline{\mathbf{u}}_{\mathcal{D}} (\boldsymbol{\phi})
|
