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Recursive computation of the normalization constant of a multivariate
Gaussian distribution truncated on a simplex
Nicolas Dobigeon and Jean-Yves Tourneret
E-mail: [email protected]
TECHNICAL REPORT – 2008, January
University of Michigan, Department of EECS, Ann Arbor, MI 48109-2122, USA
University of Toulouse, IRIT/INP-ENSEEIHT, 31071 Toulouse cedex 7, France
I. P ROBLEM STATEMENT
Let S denote the following simplex:
(
)
R−1
X
S = ααr ≥ 0, ∀r = 1, . . . , R − 1,
αr ≤ 1 ,
(1)
r=1
Let NS (A, B) denote the truncated multivariate normal distribution defined on the simplex
S with mean vector A and covariance matrix B. The probability density function (pdf) of this
truncated multivariate normal distribution denoted as φS (·|A, B) satisfies the following relation:
φS (α|A, B) ∝ φ(α|A, B)1S (α),
(2)
where
•
φ (·|A, B) is the standard Gaussian pdf with mean vector A and covariance matrix Σ,
•
1S (·) is the indicator function defined on S,
•
∝ stands for “proportional to”.
This report proposes to evaluate the normalization constant of the multivariate truncated normal
distribution NS (u, σ02 IR−1 ), IR−1 is the (R − 1) × (R − 1) identity matrix. This normalization
constant, denoted KS (u, σ02 ), can be derived directly from the definition of φS (α|A, B):
"
#
kα − uk2
1
exp −
φS (α|A, B) =
.
KS (u, σ02 )
2σ02
(3)
Consequently, it can be written:
KS u, σ02
Z
=
fu,σ02 (α) dα,
S
(4)
2
with

 α = [α1 , . . . , αR−1 ]T ,
i
h PR−1
2
 fu,σ2 (α) = exp r=1 (α2r −ur ) .
(5)
2σ0
0
II. C ASE R = 2
For R = 2, the pdf of the Gaussian distribution truncated on the simplex S = {α1 |0 ≤ α1 ≤ 1}
reduces to the two-sided truncated normal distribution. As an example, the pdf of such distribution
with σ02 = 0.2 and u = 0 is depicted in Figure 1.
Fig. 1.
Pdf of the normal distribution for σ02 = 0.2 truncated on the simplex S (R = 2).
Consequently, the case R = 2 consists in computing the integral of the function fu,σ02 (α1 ) =
h
i
−u1 )2
exp − (α12σ
on the set S = {α1 |0 ≤ α1 ≤ 1}:
2
0
"
#
Z 1
(α1 − u1 )2
2
KS u, σ0 =
exp −
dα1 .
(6)
2σ02
0
Let t = α√1 −u21 ,
2σ0
KS u, σ02
=
q
2σ02
Z
√1
2σ0
exp −t2 dt.
u1
2σ0
(7)
−√
Finally,
2
KS u, σ0
p
2πσ02
1 − u1
u1
=
erf √
+ erf √
.
2
2σ0
2σ0
(8)
3
III. G ENERAL CASE
The problem consists in computing the following quantity:
Z 1 Z 1−α1 Z 1−α1 −α2
2
...
KS u, σ0 =
0
0
0
"
#
Z 1−PR−2
2
2
r=1 αr
(α1 − u1 ) + . . . + (αR−1 − uR−1 )
exp −
...
dαR−1 dαR−2 . . . dα1 ,
2σ02
0
(9)
that can be rewritten as:
"
#Z
"
#Z
Z 1
1−α1
1−α1 −α2
(α1 − u1 )2
(α2 − u2 )2
2
exp −
exp −
...
KS u, σ0 =
2σ02
2σ02
0
0
0
"
# Z PR−2
"
#
1− r=1 αr
(αR−2 − uR−2 )2
(αR−1 − uR−1 )2
. . . exp −
exp −
dαR−1 dαR−2 . . . dα1 .
2σ02
2σ02
0
(10)
Inspired by [1], we introduce ∀x ∈ R, ∀y ∈ RR−1 , ∀s2 ∈ R+ :
"
#
Z x
2
(t
−
y
)
R−1
gR−1 x, y, s2 =
exp −
dt
2
2s
0
√
2πs2
x − yR−1
yR−1
√
=
erf
+ erf √
.
2
2s
2s
(11)
Then the following sequence of functions is defined as follows, r = 1, . . . , R − 2:
gr
x, y, s2 =
Z
0
x
#
(t − yr )2
2
g
x
−
t,
y,
s
exp −
dt.
r+1
2s2
"
(12)
Therefore, the normalization constant for the multivariate truncated Gaussian distribution
NS (u, σ02 IR−1 ) is:
KS u, σ02 = g1 (1, u, σ02 ) .
(13)
R EFERENCES
[1] A. Genz and P. Joyce, “Computation of the normalization constant for exponentially weighted dirichlet distribution integrals,”
Computing Science and Statistics, vol. 35, pp. 557–563, 2003.