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developed a Markov model for the explicit task of associative learning (object
association with the correct button). It ignores the temporal context between
objects and gives the chance to an analysis of the associative learning process.
The remaining parts of the paper are organised as follows. Section 2 introduces
the behavioural Markov model along the line of the underlying experiment. In
Sect. 3 we derive the significance of two parameters by an analysis of the Markov
chain, representing the associative learning task. Finally, in Sect. 4, we present
the results of a simulation with the model and discuss the relevance of the
findings.
2 Behavioural Markov Model
The Markov property of a random process says, that the next state in the process
depends
only
on the current state. A Markov model is a particular type of
Markov process in which the process consists of a finite number of states.
We developed a Markov model for the given conditional learning task. The
effects of distinguishing objects, memorising previous actions, and memorising
success for an object are represented by separate parameters. The history of the
learning process (subject's actions) is accumulated in the current state. The basic
assumption is, that subjects follow a rational policy in choosing the action on a
certain object. This is: (i) once the successful button was found, this choice is
repeated in later trials and (ii) the choices which so far have not been successful
are not taken again, and the remaining options are chosen with equal probability.
Following these policy, we define the Markov model for learning the motor
response for an object by trial and error. The model consists of 16 states (
S
=
{
). Each represents one possible situation a subject could face
during learning. At the beginning the subject has no clue about any of the four
possible buttons (
s
1
). For each possible decision we define four states (
s
2
,...,s
5
).
After the first choice the subject has three possibilities left, which lead to the
next states and so on. Table 1 shows all possible situations/states whereas 'x'
implies 'chosen and memorised to be wrong' and '.' implies 'not chosen yet'.
The last state
s
16
represents success. In this state the subject found the correct
button and memorised it.
Knowing the state space of the learning process, we have to define the way
through this space during learning. This is described by transition probabilities,
summarised in the state transition matrix
s
1
,s
2
,...,s
16
}
T
. The general structure of the matrix
Table 1.
Possible states during associative learning of an object by trial and error
state
s
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
left
.
x
.
.
.
x
x
x
.
.
.
x
x
x
.
success
right
.
.
x
.
.
x
.
.
x
x
.
x
x
.
x
success
up
.
.
.
x
.
.
x
.
x
.
x
x
.
x
x
success
down
.
.
.
.
x
.
.
x
.
x
x
.
x
x
x
success
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