A Markov Model of Conditional Associative Learning in a Cognitive Behavioural Scenario - Foundations on Natural and Artificial Computation - page 18

which is by the definition of the eigenvalues

16

π (0) , e i

λ i e i .

π ( t )= (13)

| e i |

2

i =1

Since λ i is taken to the power of t , the contribution of the eigenvalues λ 2 ... 5 to
the sum in (13) is lowered with every step (

< 1 (10)). The smaller they
are the faster they disappear in the sum and the faster the sum converges to the
steady state. Figure 4(b) shows

| λ 2 ... 5 |

against the parameters p suc and p sr .The
impact of p suc is stronger than those of p sr .

| λ 2 ... 5 |

4 Results and Discussion

We fitted the Markov model on experimental data published in [4]. Further, a
similar experiment is presented in [5]. The subjects were tested on the random
and deterministic ordered sequence as described above. The experiment lasts for
28 cycles. The three model parameters were manually selected to fit the subjects'
learning behaviour (Fig. 5). Yet, we did not use an optimisation procedure to
gain better fits and/or other parameter combinations.
Compared to the reinforcement learning model in [5] and the neural network
model in [3], the Markov model ignores temporal context. Therefore, it makes
no predictions concerning the performance difference which occurred in the ex-
periment for different sequential orders of object presentation. Nevertheless, it
gives the chance to an analysis of the underlying learning task (object associa-
tion with the correct button) which was not possible with the existent models
in such explicit way.
The associative learning of a single object is described with 16 possible states.
The assumed rational behaviour gives the state transitions. The parameters
( p suc , p sr , p or ) model typical mistakes. The analysis of the proposed Markov
chain produced two main results.

1 1

0.8 0.8

0.6 0.6

P P

0.4 0.4

0.2 0.2

Markov model
subjects

Markov model

subjects

0 0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

cycle cycle

Fig. 5. Learning process of the basic Markov model on a deterministic (left) and a
random sequence (right). The parameters were p suc =0 . 6, p sr =1, p or =0 . 9875 (left)
and p suc =0 . 3, p sr =1, p or =0 . 9875 (right). The mean probability of success P is

averaged over all subjects.