27.11.2013

Boncukta tutum aramak: The Beads Task


A
B

Observation

Induction

No assumption, no hypothesis

Empirical stance / method


Repeated observation of presence or absence, occurence or nonoccurence of the event is key, i.e. prediction is an inference  with a probability of error

Power of a series of observations is dependent on (a) the degree to which the event is observable, as well as (b) the number of observations

Relevant to Bottom-Up information processing Frequentist, Bayesian


Reasoning, calculation

Deduction

An absolute numerical probability is at hand

Rationalistic stance / method


Prediction is is based on a mathematical operation, ie. an extension of logic



Prediction is binomial i.e. absolute and either true or false.


 Relevant to Top-Down information processing Probabilistic


In the A condition, probability of error is dependent on the frequency distribution in the set of repeated observations, which varies with every “bit” of new information / observation—Yes or No, True or False, Occurence or Nonoccurence. The prediction that is based on the initial bit of information carries the highest probability of error, decreasing as observations accumulate.

In the B condition, i.e. when the probability that an event takes place or that a proposition is true is a given, predictions are based on, and only on, this constant value of probability.
Inference based on observation cannot be justified in the B condition, for the probability is constant, rendering observational evidence redundant.

The Beads Task is designed as a measure of the tendency on the part of the observer-subject to jump to conclusions. I find it hard to understand—literally: Although the subjects are informed at the outset regarding the constant figure of probability for True or False, they are offered a series of repeated observations until they feel confident to make a guess, a forced choice between True and False.
  
M: Number of beads that are in the majority
m: Number of beads that are in the minority
B: Total number of beads in each of the two jars = M+m
D: Draw
n: Number of draws (observations)
Pr: Probability of a correct guess regarding the jar of origin for the bead presented = Ratio of the number of beads in the Majority to the Total number of beads in each jar
Pw: Probability of an incorrect guess regarding the jar of origin for the bead presented = Ratio of the number of beads in the Minority to the Total number of beads in each jar
The initial observation is the only basis for the forced choice, and this is independent from the ratio of M/n.
Induction, based on probability of error (Pw) is redundant, because:
Pw= (m/B)ⁿ for any number of draws (observations)
Pr= M/B at the initial draw (observation)
M, m, n and B are positive integers.
If m is smaller than M, then M/m is bigger than 1.
Pw will decrease at every draw, to reach 0 when the draw number is ∞.
For no value of n will Pw be bigger than M/B, i.e. the Pr at the initial draw.  

Hence my question:
What is the question of the Beads Task?
How quickly does the subject jump to the conclusion? (1), or
How long does it take the subject to figure out the redundancy of observation following the initial draw? (2)
If (1), is it that the task takes advantage of a trick that involves misguiding the subject? (3)
If (3), how is this really relevant to psychotic thought disorder?  

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