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A quantum framework for likelihood ratios

RACHAEL BOND

December 12th, 2015

University of Sussex

The annual scientiﬁc meeting of the

Mathematical, Statistical, & Computing Psychology Section

of the British Psychological Society

r.l.bond@sussex.ac.uk www.rachaelbond.com

@rachael_bond rlb.me/pdf1215

Contents

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Pseudodiagnosticity

Is probability subjective?

Describing an objective reality

Deconstructing the contingency table

Quantum mechanics 101

Describing the wave function

Solving the “c” functions

The objective covariate probability

The implications for psychology

The relational information seeker

Conclusions

References

1. Pseudodiagnosticity

Pseudo-

diagnosticity

Doherty,

Mynatt,

Tweney, &

Schiavo [1]

“An undersea explorer has found a

pot with a square base that has

been made from smooth clay.

Using the information below, you

must decide from which of two

nearby islands it came. You may

select one more piece of

information to help you make your

decision.”

Pseudo-

diagnosticity

Pseudo-

diagnosticity

Doherty,

Mynatt,

Tweney, &

Schiavo [1]

ShellIs. CoralIs.

#Finds 10 10

%Smooth 80 ?

%Sq.base ? ?

Pseudo-

diagnosticity

Pseudo-

diagnosticity

Doherty,

Mynatt,

Tweney, &

Schiavo [1]

ShellIs. CoralIs.

#Finds 10 10

%Smooth 80

%Sq.base

Doherty et al. expected their

participants to select the paired

datum to the given “anchor

information” in order to calculate a

Bayes' ratio. The majority didn't.

Pseudo-

diagnosticity

“Pseudodiagnosticity is clearly disfunctional.”

~ Doherty, Mynatt, Tweney, & Schiavo (1979) , p. 121

[1]

What if all the data are known?

ShellIs. CoralIs.

#Finds(Baserate)10 10

#Smoothclay 8 7

#Squarebase 6 5

What if all the data are known?

Base 10 10

8 7

6 5

To calculate the value

using Bayes' theorem, this

expression must be solved

However, the measures of

covariate intersection, ie.

, are unknowns.

What if all the data are known?

Base 10 10

8 7

6 5

Doherty et al. suggest that the

data should be treated as

conditionally independent. This

allows for a simple estimation of

from the multiplication of

marginal probabilities

What if all the data are known?

However, it would also be reasonable to note that the

covariate intersections form ranges:

ie.,

What if all the data are known?

Base 10 10

8 7

6 5

This means that it is also possible

to calculate a probability from the

mean value of these ranges:

What if all the data are known?

Base 10 10

8 7

6 5

Or, to take the mean value of the

minimum→maximum probability

range:

What if all the data are known?

Base 10 10

8 7

6 5

Other possible approaches

include regression analysis, which

would assume a low level of co-

linearity, or using an expectation-

maximisation algorithm (eg., see

Dempster, Laird, & Rubin, 1977)[2]

2. Is probability subjective?

Is probability subjective?

Given the variety of probability values which may be

reasonably calculated, one may conclude that there is

no objectively correct likelihood ratio.

Is probability subjective?

Given the variety of probability values which may be

reasonably calculated, one may conclude that there is

no objectively correct likelihood ratio.

The subjective nature of probability has moved to the

centre of statistical research since Bruno de Finetti

claimed that “probability does not exist”.

(de Finetti, 1974)[3]

3. Describing an objective reality

Describing an objective reality

(384-322 BCE) argued that “ ” is

described by the unity of form and substance:

“substance” being what something is made from,

and “form” being its innate characteristics.

In the contingency table, the “substances” (ie., the

differentiating characteristics), and their “forms” (ie.,

their values), are known. Yet an objective probability

value cannot be calculated from this description of the

table's reality.

Aristotlereality

4. Deconstructing the contingency table

Deconstructing the contingency table

Assuming, for the moment, the case of even base rates,

the contingency table may be deconstructed into 4

sub-contingency tables ...

8 7

6 5

8 6 7 5

Deconstructing the contingency table

... each of which provides two pieces of “pure”

information generated from the facts of and .

These are not logically separable.

8 7

6 5

8 6 7 5

Deconstructing the contingency table

While the relationships between and are

known (they are mutually exclusive), the relationships

between and cannot be stated.

{

{

8 6 7 5

Deconstructing the contingency table

What is needed is a mathematical approach which

allows the covariate intersections to be directly

mapped to and .

Deconstructing the contingency table

What is needed is a mathematical approach which

allows the covariate intersections to be directly

mapped to and .

In other words, the contingency table's internal

relationships must be rewritten in a way that includes

the covariate intersections, but does not make any

structural changes. This can only be achieved by using

the mathematics of quantum mechanics.

5. Quantum mechanics 101

Quantum mechanics 101

Instead of the used in

classical statistics, quantum mechanics works in

.

The vectors are normalised which are

orthogonal to each other in n-dimensions.

In psychology these vectors could, for instance,

represent attitudes, beliefs, or intent etc.

joint probability spaces

vector

spaces

wave functions

Quantum mechanics 101

Using the Dirac (1939) “ ” notation, the wave

functions are described by horizontal matrices known

as “kets”, written as

Their “ ” form vertical

matrix “bras”, written as

Any ket multiplied by its own bra is “ ”,

meaning that

[5] bra-ket

complex conjugate transposes

orthonormal

6. Describing the wave function

Describing the wave function

8 7

6 5

The bra can only collapse into the ket

if the inner product contains

both and . As a consequence,

the inner product is a measure of covariate overlap.

Describing the wave function

8 7

6 5

The reverse, complex conjugate transposed,

inner product is also true.

Describing the wave function

8 7

6 5

Because both inner products are real, and consistent

with the conditional independence of and ,

it follows that they also equal to each other.

Describing the wave function

8 7

6 5

all other bra-kets

Thus, the complete quantum contingency table

consists of 4 orthonormal kets, and 2 inner products.

It exactly matches the classical description.

Describing the wave function

In doing so, it returns four base kets that give a full

system description and includes the inner products.

This allows the fully normalized system wave function

to be described.

Describing the wave function

The correct expression for may be

found through rearrangement.

Describing the wave function

This expression fully generalizes, and the individual

elements may be weighted to incorporate the prior

distributions.

7. Solving the functions

Solving the functions

There are known features of which may be used to

generate constraints. These include “data

dependence”: must be, in some way, dependent

upon the data in the table;

Solving the functions

a “valid probability range”: the values of must

fall between 0 and 1;

Solving the functions

“complementarity”: the law of total probability requires

that the sum of all probabilities=1;

Solving the functions

“symmetry”: the exchanging of rows in the contingency

table should not affect the calculated probability

value, and if the columns are exchanged then the

values should map;

Solving the functions

→

→

→

→

→

→

“known probabilities”: there are certain contingency

table structures which must return speciﬁc

probabilities.

Solving the functions

Using these principles and constraints demonstrates

that are anti-symmetric bivariate functional

equations, to which only one solution exists.

8. The objective covariate probability

The objective covariate probability

8 7

6 5

Substituting in the derived functional expressions

allows for a ﬁnal probability to be calculated.

9. The implications for psychology

The implications for psychology

“Calculating probabilities for predicting performance”

With only 10 data points in the “pot“ example, there is

not much difference between 0.5896 (QT) and 0.578

(classical Bayes' theorem) and is unlikely to affect

ordinal predictions. However, in modelling phenomena

based on thousands, or millions, of data points (eg., in

perception, memory, social learning etc.) this

difference will matter a lot more.

The implications for psychology

“Predicting new phenomena”

Bayesian learning lends itself to modelling systems

that develop linearly. However, humans often show

nonlinear, sometimes seemingly nondeterministic,

behaviours, such as sudden switches in strategy that

don't necessarily accord with the available data.

10. The relational information seeker

The relational information seeker

We conducted an experiment with a larger, 3x4,

contingency table, giving the participants (n=150) 5

degrees of freedom in their selections.

For the ﬁrst 4 selections, the choices made followed an

information gain model, based on Shannon's entropy,

with a signiﬁcance of for each choice (using

a Chi-squared test of predicted selection against

random).

The relational information seeker

However, the ﬁnal selection demonstrated a strategy

change towards “weak” information. This suggests that

the search process only follows information theory in-

so-far as it is required to identify the diagnostically

important relationships.

The relational information seeker

However, the ﬁnal selection demonstrated a strategy

change towards “weak” information. This suggests that

the search processonly follows information theory in-

so-far as it is required to identify the diagnostically

important relationships.

This is not the same as mental model building. Rather,

information search reﬁnes the mental representation

created by the question.

The relational information seeker

It is unclear as to whether these relationships are

classical, or quantum, in nature.

11. Conclusions

Conclusions

Any full description of objective reality may have to

include mathematical concepts that only exist in

quantum mechanics.

Conclusions

Any full description of objective reality may have to

include mathematical concepts that only exist in

quantum mechanics.

Quantum mechanics can describe models, and provide

solutions to them, which lie beyond the scope of

classical mathematics.

Conclusions

Any full description of objective reality may have to

include mathematical concepts that only exist in

quantum mechanics.

Quantum mechanics can describe models, and provide

solutions to them, which lie beyond the scope of

classical mathematics.

Bayes' theorem is a special case of a more general,

quantum mechanical expression.

Download this presentation from

http://rlb.me/pdf1215

RACHAEL BOND

University of Sussex

PROFESSOR TOM ORMEROD

University of Sussex

PROFESSOR YANG-HUI HE

City University; Nankai University;

Merton college, Oxford University

References

[1] Doherty, M.E., Mynatt, C.R., Tweney, R.D., & Schiavo, M.D. (1979).

Pseudodiagnosticity. Acta Psychologica, vol. 43(2), pp. 111-121. doi:

10.1016/0001-6918(79)90017-9

[2] Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood

from incomplete data via the EM algorithm. Journal of the Royal

Statistical Society. Series B (Statistical Methodology), vol. 39(1), pp. 1-38.

[3] de Finetti, B. (1974). Theory of probability: A critical introductory

treatment. New York, New York: Wiley.

[4] Caves, C.M., Fuchs, C.A., & Schack, R. (2002). Unknown quantum states:

The quantum de Finetti representation. Journal of Mathematical

Physics, vol. 43(9), pp. 4537-4559. doi: 10.1063/1.1494475

[5] Dirac, P.A.M. (1939). A new notation for quantum mechanics.

Mathematical Proceedings of the Cambridge Philosophical Society, vol.

35(03), pp. 416-418. doi: 10.1017/S0305004100021162

[6] Strang, G. (1980). Linear algebra and its applications (2nd ed.). New

York, New York: Academic Press.