How Engines Learn to Generate Recommendations: Adaptive Learning Algorithms - Realtime Data Mining - page 73

Thus, we have completed the conversion into the conditional probabilities j p a
which we again update with respect to the accepted and rejected recommendations
according to Algorithm 4.1 to obtain the updated conditional probabilities jþ 1 p a .
Using the inverse mappings F 1 j and G 1

S a , we reconvert them into our internal
single probabilities jþ 1 p [ a ] . We then carry the unconditional probabilities of the
recommendations jþ 1

Π a

Π a over unchanged to the next update step.

Algorithm 5.3: Update of the internal from conditional probabilities for
multiple recommendations

Input: vector of i nte rnal probabilities j p ½ and fixed probabilities j

Π a , delivered

recommendations a ¼ a 1 , ð ... , a k Þ , index of product t ra nsition l , step size α j

Output: updated vector of internal probabili tie s jþ 1 p ½ and jþ 1 Π a

1: proce d ure UPDA T E_P_DP_MULTI( j p ½ , j

Π a , a , l , α j )

j p fg ¼ G S a j p ½

2: ⊳ conversion into intermediate

probabilities

j p a j p fg

3: ¼ F j ⊳ conversion into conditional probabilities

Π a

jþ 1 p a : ¼ U PD ATE_P_
SIN GL E( j p a , l ,

4:

α j ) ⊳ update of conditional probabilities

⊳

jþ 1 p fg ¼ F 1
j

jþ 1 p a

5: conversion into intermediate probabilities

Π a

jþ 1 p ½ ¼ G 1
S a

jþ 1 p fg

6: ⊳ conversion into internal probabilities

jþ 1 j

7: Π a :¼ Π a ⊳ unchanged take-over of the fixed

component

return ( jþ 1 p ½ ,

jþ 1

8: Π a )

9: end procedure

A closer look at Algorithm 5.3 reveals that it may be arranged in a different way by
updating the conditional recommendation probabilities in a bundle by means of
Algorithm 4.2 and updating the unconditional

(non-fixed) recommendations

separately.
Indeed, since the unconditional probabilities of the recommended products s 0 ∈ S a

Π a , also their sum X
s 0 ∈

jþ 1 j

are kept fix, i.e., Π a ¼ p ss 0 does not change, and due to

S a

X

p ss 0 ¼ 1 X
s 0 ∈S a

p ss 0 also the sum of all unconditional probabilities of the

s 0 2S a

non-recommended products is constant. Thus, if one of the recommendations is
accepted, all unconditional probabilities remain unchanged. Only if no recommen-
dation is accepted, the unconditional probabilities of the non-recommended products
will change (but not their sum).
In order to formulate the algorithm, let us denote all parts of vectors
corresponding to the recommended products by index c and t othe
non-recommended products by index u . Especially, we denote p ½ p ½

ss 0

¼

c

s 0 ∈S a