How Engines Learn to Generate Recommendations: Adaptive Learning Algorithms - Realtime Data Mining - page 71

Nonlinear Approach

We remind the reader, though, that according to ( 4.5 ), p ss 0

can only be applied to

s 0 S a , i .e., to the successor states corresponding to the recommendations. Yet we

need p ss 0 for all admissible subsequent states s 0 .
To this end, we extend Assumption 5.2 to all of the k issued recommendations
and generalize ( 5.3 )to

∈

p ss 0 ¼ cs s 0 2 S a :

ð p ss 0 , ; ð 5 : 16 Þ

While being able t o calculate the transition probabilities to each of the issued
recommended states p ss 0 , s 0 ∈

S a according to ( 4.4 ), we use ( 5.16 ) for the remaining
successor states. The scaling factor cs ðÞ is determined similarly to c ( s , a )as ;

described in Sect. 5.2 :

X

ð X
s 0 2S a

p ss 0 þ cs ; p ss 0 ¼ 1

s 0 ∈ S a

together with X
s 0

p ss 0 ¼ 1 yields

1 X
s 0 ∈S a

p ss 0

1 X
s 0 ∈

cs ðÞ¼ ;

p ss 0 :

ð 5 : 17 Þ

S a

We are now in a positi on to generalize Assumption 5.2 to the case of a given
multi ple recommendation a and to compute its corresponding transition probabil-
ities p ss 0 from the single recommendation probabilities p ðÞ

ss 0 .
Thus, we have gathered all of the tools necessary to estimate the transition
probabilities from multiple recommendations.
Using the transformation F s 0 from ( 4.5 ), we may calculate the joint probabilities
p ss 0 , s 0 S a from the single probabilities of the issued recommendations p a s 0

ss 0 , s 0

∈ ∈ S a

(equivalent notation: p a l

ss a l ,

l ¼ 1, ... , k ) and thereupon define the intermediate

probabilities p fg
ss 0

:

(

p ss 0 , s 0 2S a

p ss 0 , s 0 2S a
p ss 0 , s 0 ∈

, s 0 ∈

p fg

ss 0 ¼

¼ , ð 5 : 18 Þ

F s 0 p a 1 p a k

ss a k

S a

ss a 1 ; ...;

S a

which enables to introduce the vector function G S a in terms of the vectors p ½

s 0 s 0 :

p ½
ss 0

p fg
ss 0

and p fg ¼ ¼