Материал: искусственный интеллект

a 2 dimensional space, when the choice of a binary measure was an arbitrary choice by the researcher. Given that the number of values of the variable is an arbitrary choice of the researcher, there is no reason to think that this measurement will be a complete measurement, i.e., a measurement that is represented as a set of mutually exclusive and exhaustive one dimensional projections [24]. Of course, the simplest assumption is that a projector is one dimensional, but this assumption often fails, and we shall see that it does in this example.

Continuing with the case of a single variable represented by the Hilbert space HN1 , we can express each vector |φ in terms of its coordinates with respect to the eigenvectors of P (Y1 = yi) by using Equation (4). Using this basis, the coordinate representation of each projector, say P (Y1 = yi) is simply an N1 × N1 diagonal matrix, M1(i) with ones located in the rows corresponding to basis vectors that have an eigenvalue of one associated with the projector P (Y1 = yi), and zeros otherwise. The coordinate representation of |ψ with respect to this basis is a N1 × 1 column matrix ψ with coordinate ψi in row i, which satisfies ψ†ψ = 1. Then the probability distribution over the values of Y1 for i = 1, · · · , n1 is given by

There is little di erence between classical and quantum probability at this point.

Next suppose we measure two variables, Y1 with n1 values and Y2 with n2 values, with n1 ≥ n2. If these two variables are compatible, then the joint event (Y1 = yi ∩ Y2 = yj) is well defined for all pairs of values. Therefore the Hilbert space is partitioned into n1 · n2 orthogonal subspaces. Each subspace corresponds to a projector P (Y2 = yj )P (Y1 = yi) = P (Y1 = yi)P (Y2 = yj ) = P (Y1 = yi ∩ Y2 = yj ). These projectors are pairwise orthogonal and complete, and every pair of projectors is commutative. Each projector shares (N1 · N2) ≥ (n1 · n2) eigenvectors, but with di erent eigenvalues, to span a Hilbert space HN1·N2 . Using this basis, the projector P (Y1 = yi) is represented by the Kronecker product M1(i) IN2 , where IN2 is an N2 × N2 identity matrix. The projector P (Y2 = yj) is represented by the matrix Kronecker

product IN1 M2(j). Then P (Y2 = yj )P (Y1 = yi) = P (Y1 = yi ∩ Y2 = yj) is represented by the product (M1(i) IN2 ) · (IN1 M2(j)) = M1(i) M2(j), which is simply a diagonal matrix with ones located in the rows corresponding

to (Y1 = yi ∩ Y2 = yj ) and zeros otherwise. The coordinate representation of

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|ψ with respect to this basis is a (N1 · N2) × 1 column matrix, ψ, ψ†ψ = 1 , with coordinate ψij in row n2 · (i − 1) + j. Then the joint probability for a pair of values equals

P (Y2 = yj )P (Y1 = yi) |ψ 2 = M1(i) M2(j) · ψ 2 = |ψij |2 . (9)

There is still little di erence between the classical and quantum theories at this point. Adding variables increases the dimensionality of the space, just like it does with a Bayesian model.

Now suppose that variables Y1 (with n1 values) and Y2 (with n2 ≤ n1 values) are incompatible. In this case, we cannot define the joint occurrence of two events (Y1 = yi ∩Y2 = yj ), and we can only represent a sequence of two single events, e.g., (Y1 = yi) and then (Y2 = yj ) by the sequence of projectors P (Y2 = yj)P (Y1 = yi). As before, we define P (Y1 = yi) as the projector for the event (Y1 = yi) , and likewise, we define P (Y2 = yj) as projector for the event (Y2 = yj) . Both projectors are represented with a Hilbert space, HN1 , of dimension N1 ≥ n1. We can choose to express each vector |φ in terms of the coordinates with respect to the eigenvectors of P (Y1 = yi) by using Equation (4). Using this basis, the coordinate representation of projector P (Y1 = yi) is simply an N1 × N1 diagonal matrix, M1(i) with ones located in the rows corresponding to basis vectors that have an eigenvalue of one associated with this projector, and zeros otherwise. Using Equation (5), the projector P (Y2 = yj) can be expressed in terms of the P (Y1 = yi) basis by a unitary matrix, U. Then the matrix representation of P (Y2 = yj) isU · M1(j) · U† . Finally, the coordinate representation of the state vector |ψ with respect to the Y1 basis is a N1 ×1 column matrix ψ. The probability of the sequence of events (Y1 = yi) and then (Y2 = yj ) equals

This is where a key di erence between the classical and quantum theories occurs. Note that, unlike a Bayesian model, adding variable Y2 does not increase the dimensionality of the space.

Finally suppose that we measure three variables, Y1 with n1 values, Y2 with n2 values, and Y3 with n3 values. Suppose Y1 is compatible with Y2 and Y2 is compatible with Y3 but Y1 is incompatible with Y3. In this case, we can partition the Hilbert space using projectors P (Y1 = yi ∩ Y2 = yj ), i = 1, · · · n1, j = 1, · · · n2, which are pairwise orthogonal and complete, and every

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pair of these projectors is commutative. Using the eigenvectors of these projectors as the basis, the projector P (Y1 = yi) is represented by the Kronecker product M1(i) IN2 , and the projector P (Y2 = yj) is represented by the Kronecker product IN1 M2(j). Using a unitary transformation, U, the matrix representation of the projector P (Y3 = yk) is given U · M1(k) · U† IN2 . Then, the probability of the two compatible events (Y1 = yi) and (Y2 = yj) equals

the probability of the two compatible events (Y2 = yi) and (Y3 = yj) equals

and the probability of the sequence of two incompatible events (Y1 = yi) and then (Y3 = yk) equals

5.3. Constraints on HSM models

The methods described above generalize in a fairly straightforward manner for more variables. Note that when variables are compatible, quantum probability theory works like classical probability theory, and the Hilbert space dimensionality increases exponentially as the number of compatible variables increases. However, when variables are incompatible, it is unlike classical probability theory, and the Hilbert space dimensionality remains constant as the number of incompatible variables increases.

The use of incompatible variables comes with some costs. HSM theory is not completely general, and it must satisfy several of its own consistency constraints. For example, consider once again a collection of four 2 by 2 tables (AC, AD, BC, BD) generated from 4 binary variables (A, B, C, D). Although quantum probabilities do not have to satisfy the famous Bell inequalities, they do have to satisfy another inequality called the Cirelson inequality. See [12] for general methods for determining constraints on correlations by quantum models. Even more demands are placed on the transition probabilities generated by the unitary transformation used to form incompatible variables. The transition probability matrix is produced by squaring the magnitude of the entries of the unitary matrix, and this transition matrix must be doubly stochastic, and it must satisfy the law of reciprocity (see [24], Ch. 2).

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6.Application to the Artificial Data Set

An HSM is constructed from the following 6 steps.

•Step 1 detemines the compatibility and incompatibility relations among the variables.

•Step 2 determines the dimension of the Hilbert space based on assumed compatibility relations.

•Step 3 defines the initial state given the dimension of the Hilbert space.

•Step 4 defines the projectors for the variables using unitary transformations to change the basis

•Step 5 computes the choice probabilities given the initial state and the projectors

•Step 6 interprets the model parameters to provide an deeper understanding of the complex tables

We apply the above principles to the artificial data set using six steps:5 Step 1. Determine compatibility of variables. Psychologically, this step

determines whether two variables can be measured simultaneously (compatible) or they have to be measured sequentially (incompatible). Based on the order e ects observed in Table 1, we infer that the pair of variables A,C were incompatible, as well as the pair B,C. The design did not include manipulations of order to test compatibility between variables A,B. In this case, another way to empirically test compatibility is to compare model fits that make compatibility vs. incompatibility assumptions about these variables. Here for the purpose of illustration, we assumed that they were compatible.

Step 2. Define the Hilbert space. Assuming that A,B are compatible means that we can define all of the events obtained from all of the combination of values of these two variables: (A = ai ∩ B = bj ), for (ai = a1, a2) and (bj = b1, b2, b3). However, we cannot define combinations for more variables because of the incompatibilities with C. The simplest model is a model that

5The Matlab code used to perform these computations is available at http://mypage.iu.edu/ jbusemey/quantum/HilbertSpaceModelPrograms.htm

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assumes that each event (A = ai ∩B = bj) is represented by only 1 dimension, which produces a total of 6 dimensions. Therefore, the minimum size of the Hilbert space was set to 6 dimensions, and we started with this minimum.

Step 3. Define the initial state. We chose a basis that provided the most meaningful parameters for the initial state. For this application, we chose to use the basis defined by the combination of variables A and B. Using this basis, the matrix representation of the basis vector |A = ai ∩ B = bj is simply a 6 × 1 column matrix with zeros everywhere except for a 1 in row

i,j

The six coe cients in Equation (14) form a 6 × 1 column matrix

ψ11

ψ12

ψ= ψ13 .

ψ21

ψ22 ψ23

For example, |ψ13|2 equals the probability of yes to Adeptness and a high rating to Brilliance. The parameters in ψ are estimated from the data under the constraint that ψ†ψ = 1. In general, the 6 coe cients can be complex valued, and so each coe cient contributes a magnitude and a phase. However, the magnitudes must satisfy the unit length constraint that ψ†ψ = 1. Also, one phase for one coe cient can be set to an arbitrary value without changing the final choice probabilities. Therefore, only 6 × 2 − 2 = 10 free parameters are required for the initial state. In many applications, the initial state can be restricted to real values, in which case there are only 6 − 1 = 5 parameters.

The initial state parameters tell us what the initial state of the psychological system (e.g., initial belief or attitude towards attributes A and B) is before any measurement is taken on the system, and can be used to compute the probability of certain response to an attribute when it is measured alone. That is, we can estimate more “context free” responses from the respondents–free from influences from measurement e ects from the other attributes–even though we didn’t collect such actual empirical data.

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