Материал: искусственный интеллект

Step 4. Define the projectors. We can define the matrix for the pro-

which picks out the coordinates in ψ that are associated with the answer ai to attribute A. We can define the matrix for the projectors for the events

picks out the coordinates in ψ that are associated with a rating bj given to attribute B.

The 6 × 6 matrix representation of the projector P (C = cj ) is defined as follows. The columns of U provide the matrix representations of the basis vectors |C = l for the C basis. First we need construct 6 indicator matrices, denoted Cl, l = 1, . . . , 6, using the joint events from the AB basis as follows: Cl = P (A = ai) · P (B = bj), where l = 3 · (i − 1) + j, and so each Cl is just a diagonal matrix with zeros in all rows except for row j which contains the value 1. Then we rotate the basis from the AB basis to the C basis using a unitary matrix: U · Cj · U†, which simply picks the j-th column of U.

Although there are 6 basis vectors in the C basis, we need to map these into the observed 4 confidence rating values. So, we need to form 4 projectors from the 6 matrices, Cl, to represent the 4 possible confidence ratings. This choice can a ect the results. Here we define the projector for the first confidence level as P (C = c1) = U(C1 + C2)U† ; we define the projector for the second confidence level as P (C = c2) = UC3U†; we define the projector for the third confidence level as P (C = c3) = UC4U†; and finally, we define the projector for the final confidence level as P (C = c4) = U(C5 + C6)U†. In practice, this assignment can be made by fitting di erent combinations of coarse measurements and choosing the best fit.

Step 4 requires defining the unitary matrix U that rotates to the new C basis. We define a unitary matrix UA that operates on the two dimensional A subspace, and another unitary matrix UB that operates on the three dimensional B subspace. The complete unitary matrix is formed by the tensor product: U = UA UB which forms the 6 × 6 unitary matrix that operates on the full six dimensional Hilbert space.

The 2×2 matrix representation, UA was determined from Equation (7) by selecting a 2 × 2 Hermitian matrix, HA; the 3 × 3 matrix representation, UB was determined from Equation (7) by selecting a 3×3 Hermitian matrix. The

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parameters of each of these Hermitian matrices were estimated from the data. In general, the 2 × 2 Hermitian matrix has four coe cients, two real diagonal values and one complex o diagonal. However, one diagonal entry can be arbitrarily fixed, and so only 4 − 1 = 3 parameters are required to produce a total of 6 parameters. The 3 × 3 Hermitian matrix has 9 coe cients, three real diagonal values and three complex o diagonals. However, one diagonal entry can be arbitrarily fixed, and so only 9 − 1 = 8 parameters are required to fit the data. In many applications, the Hermitian matrix can be restricted to real values, in which case HA has only 1 + 1 = 2 parameters, and HB has only 2 + 3 = 5 parameters.

The unitary parameters determine the rotation from the basis of one variable to the basis of another variable. Psychologically, they tell us the relationship between the variables or attributes being examined, and can reveal the similarity between these variables, independent of the initial state (i.e., step 3) of the person. In addition, based on the unitary matrices, we can compute the probabilities the transition probabilities between basis vectors.

Step 5. Compute choice probabilities for each response sequence. The choice probabilities for each sequence were computed by the product of projectors corresponding to the sequence. For example, the probability of

Step 6. Estimate parameters by minimizing the KL divergence between model and observed probabilities. We fit the HSM model under the constraint that the initial state was real and the Hermitian matrices for the unitary transformation were real. This model required fitting 5 + 2 + 5 = 12 parameters. The 5 model parameters representing ψ, along with the 7 parameters representing UA and UB, are presented in the Appendix. The HSM model has a total of 12 free parameters, which is far fewer than the 23 required by the 3 − way joint probability model. Nevertheless, the HSM model almost perfectly fits all the relative frequencies in Table 1. The KL divergence for the quantum model fit is D < 10−8. Recall that the data were artificially generated for illustration, and so the results are not to be taken seriously. However, they help to show the application of a HSM model.

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Table 3: Transition matrices between basis vector for pairs of incompatible attributes

The unitary matrices, U, can be used to describe the transitions between incompatible measurements. The squared magnitudes of the entries in the unitary matrix describes the probability of transiting from a basis vector representing a row attribute (e.g., |aibj to a basis vector representing a column attribute (e.g., |cl . Table 3 presents the transition probabilities for the two incompatible pairs, AB, and C. For example, the probability of transiting from |a2b1 to |c2 . C equals 0.4458, which is one of the highest transition rates.

The transition matrices produced by unitary matrices are always symmetric (see chapter 2 in [10] for a discussion). This is because each entry in the unitary matrix contains the inner product between vectors from di erent bases, and the squared magnitude is the same in both directions. This can provide a test of the quantum model, but only assuming the most restrictive assumption that the projectors are uni-dimensional (see, e.g., [6]). Unfortunately, what has been often overlooked is the fact that when the projectors are multi-dimensional, the conditional probabilities of events are not necessarily symmetric (see, e.g., [27]). Very often in applications, the projectors must be multi-dimensional, as they have to be in this case, and this is not an arbitrary assumption. If events are represented by multi-dimensional projectors, then the conditionals can be asymmetric. For example, the projector for the Confidence rating (C = c1) is 2-dimensional and the projector for the low judgment of Brilliance (B = b1) is 2 dimensional, and the quantum model quantum model predicts p(C = c1|B = b1) = p(B = b1, C = c1)/p(B = b1) = 0.3875, but p(B = b1|C = c1) = p(C = c1, B = b1)/p(C = c1) = 0.2485, which exhibits asymmetry.

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7. Application to real data

7.1. Test of joint probability model

Now we apply HSM theory to a real data set. A total of 184 participants (70% female students from the Ohio State University) observed pictures of female avatars and made binary (Yes, No) judgments about Attractiveness (A), Intelligence (I), Sociability (S), and Honesty (H) of each avatar. For each presentation of an avatar, the participant was asked to judge a pair of attributes (e.g., judge Attractiveness and Intelligence of this avatar) by choosing one of four pairs of answers (YY, YN, NY, NN, where for example YN indicates Yes to the first attribute and No to the second). The choice from the 4 answers was entered into a 2 × 2 contingency table. A total of 6 tables {AI, AS, AH, IS, IH, SH} were formed from all combinations of pairs generated from the 4 attributes. The avatars were sampled from two di erent types: Sexualized versus non-sexualized avatars. Altogether, each participant judged both types of avatars under all 6 contexts, and each avatar for each contex was replicated on 4 di erent presentations. The avatars and contexts were randomized organized across presentations with a di erent random ordering for each participant.

The aggregate results are presented in Table 4. The results are presented separately for each stimulus type. For example, when the non-Sexualized avatar was presented, the relative frequency of Y to Attractive and N to Intelligent was 0.0312, and the corresponding result for the sexualized avatar was 0.4226. Altogether, each 2 × 2 joint frequency table for each type of avatar (sexualized or not) and for each of the 6 contexts contains a total (184 · 4 = 736) observations.

We conducted a statistical chi-square test of a 4 − way joint probability model composed from four binary random variables, A, I, S, H used in the study. The test was based a comparison of the model predictions with the data in Table 4.6 The joint probability model states that the 6 rows of 2 × 2 tables for each stimulus type are produced by a joint distribution, π(A = w ∩ I = x∩S = y∩H = z) where w = 0, 1, x = 0, 1, y = 0, 1 and z = 0, 1, that has 16−1 = 15 free parameters per stimulus type or 30 parameters altogether. A

6This test should be evaluated with some caution because we are ignoring individual di erences for this analysis. In future work, we plan to use a hierarchical Bayesian model that introduces assumptions about the distribution of individual di erences and priors on these hyper parameters.

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Table 4: Observed results for 6 contexts and 2 types of stimuli. Each row is a 2 × 2 table containing the relative frequency of answers. Each column indicates a pair of answers. Table on the left is the non-sexualized type of avatar, and table on the right is the sexualized type.

completely unconstrained saturated model requires 3 parameters for each 2× 2 table, producing a total of 18 parameters per stimulus type or 36 parameters altogether. Using maximum likelihood estimation, we computed the G2sat and

G2joint for each table and computed the di erence G2Diff = G2sat − G2joint = 20.6521, which produced a statistically significant G2 di erence: based on 6

degrees using the p = .0021. This provides some evidence that the 4 − way joint probability model systematically deviates from the observed results for the two tables. It is noteworthy that violations of the joint probability model occur in this collection of contexts even though there are no changes in order of measurement. In this case, the violations must arise from either violations of the marginals or correlations between variables.

7.2. HSM model of real data

Each pair of attributes forms a measurement context, and so there were 6 di erent measurement contexts. Each event in the model represents an answer to a question about an attribute within a context. We applied a simple HSM model to the data as follows

First, we used a 4−dimensional space to represent the events and the state vector. After selecting a basis for an event, the state vector can be represented by 4 coordinates, where each coordinate contains the amplitude

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