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monitoring the change of the mixing model-related components
L M ( q )ofthe
variational bound
L M ( q )
between two successive iterations is lower than the system parameter Δ s L M ( q ),
then the algorithm assumes convergence and returns.
The parameters are initialised in Lines 2 to 4 of TrainMixing .The D V
L
( q ), as given by (7.95). If the magnitude of change of
K
mixing matrix V holds the vector v k that corresponds to classifier k in its
k th column. As by (7.13) the prior on each element of v k is given by a zero-
mean Gaussian with variance β k , each element of V is initialised by sampling
from N (0 ,b β /a β ) where the value of the random variable β k is approximated by
its prior expectation. The distribution parameters for q β ( β k ) are initialised by
setting them to the prior parameters.
An iteration starts by calling TrainMixWeights in Line 8 to get the parame-
ters of the variational posterior q V ( V ). These are subsequently used in Line 9 to
update the parameters of q β ( β k )foreach k by calling TrainMixPriors .Lines10
to 14 determine the magnitude of change of
×
L M ( q ) when compared to the last
iteration. This is achieved by computing the N
K mixing matrix G =( g k ( x n ))
by calling Mixing . Based on G , the responsibility matrix R =( r nk )isevalua-
ted by calling Responsibilities in Line 11. This allows for the evaluation of
L M ( q ) in Line 13 by calling VarMixBound , and determines the magnitude of
change Δ
×
L M ( q ) in the next Line, which is subsequently used to determine if
the parameter updates converged. In the performed experiments, the function
usually converged after 5-6 iterations.
Next, the Functions TrainMixWeights , TrainMixPriors , Mixing and
Responsibilities will be introduced, as they are all used by TrainMixing
to train the mixing model. VarMixBound is described in the later Sect. 8.1.4.
Function. Mixing( M , Φ , V )
Input : matching matrix M , mixing feature matrix Φ , mixing weight matrix V
Output : N × K mixing matrix G
get K from shape of V
1
G ΦV
2
limit all elements of G such that exp min ≤ g nk ln max ln K
3
G exp( G ) M
4
G G RowSum( G )
5
FixNaN( G , 1 /K )
6
return G
7
Starting with Mixing , this function is used to compute the mixing matrix G
that contains the values for g k ( x n ) for each classifier/input combination. It takes
the matching matrix M , the mixing features Φ , and the mixing weight matrix
V as arguments, and returns G .
The mixing matrix G is evaluated by computing (7.10) in several steps: firstly,
in Line 2, v k φ ( x n ) is computed for each combination of n and k .Beforethe
exponential of these values is taken, it needs to be ensured that this does not
cause any overflow/underflow. This is done by limiting the values in G in Line 3
 
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