Core: states, measurement, fitting and comparison

Everything else in the library is built from the definitions on this page.

Notation

Symbol	Meaning
\(\psi \in \mathbb{R}^d\) or \(\mathbb{C}^d\), \(\lVert\psi\rVert = 1\)	belief state (pure)
\(\rho = \lvert\psi\rangle\langle\psi\rvert\), \(\rho \succeq 0\), \(\operatorname{tr}\rho = 1\)	density matrix (pure or mixed)
\(P = P^\dagger = P^2\)	orthogonal projector of a “yes” answer; \(\bar P = I - P\) is “no”
\(\theta\)	vector of model parameters
\(\mathcal{D} = \{c: n_c\}\)	data: outcome counts \(n_{c,o}\) per condition \(c\)

Born rule and Lüders rule

Answering a yes/no question with projector \(P\) gives

\[ p(\text{yes}) = \lVert P\psi\rVert^2 = \langle\psi\rvert P\lvert\psi\rangle, \qquad \psi \;\mapsto\; \psi' = \frac{P\psi}{\lVert P\psi\rVert}. \]

For a sequence of questions \(q_1, \dots, q_m\) with answers \(a_k \in \{0, 1\}\) and projectors \(P_k^{(1)} = P_k\), \(P_k^{(0)} = I - P_k\), applying the rule repeatedly gives

\[ p(a_1, \dots, a_m) = \bigl\lVert P_m^{(a_m)} \cdots P_2^{(a_2)} P_1^{(a_1)}\, \psi \bigr\rVert^2 . \]

If two projectors do not commute, \(P_B P_A \neq P_A P_B\), the joint probability depends on the order: this single fact produces question-order effects, conjunction fallacies and interference. (core.linalg.luders, core.linalg.sequence_probabilities)

The projector onto the span of vectors \(v_1, \dots, v_r\) is \(P = QQ^\dagger\), where the columns of \(Q\) are an orthonormal basis of the span (QR decomposition). (core.linalg.projector)

Parameterising states and subspaces

A real unit vector in \(\mathbb{R}^d\) is written with hyperspherical angles,

\[ v_i = \cos\theta_i \prod_{j<i}\sin\theta_j \quad (i < d), \qquad v_d = \prod_{j<d}\sin\theta_j , \]

and an orthonormal frame with \(d(d-1)/2\) Givens rotations \(G_{ij}(\phi_{ij})\) (rotation by \(\phi_{ij}\) in the \((i, j)\) plane), \(F = \prod_{i<j} G_{ij}(\phi_{ij})\). (angles_to_unit, givens_frame)

Mixed states, dephasing and open-system dynamics

Partial dephasing in a basis of projectors \(\{P_k\}\) with strength \(s \in [0, 1]\):

\[ \rho \mapsto (1 - s)\,\rho + s \sum_k P_k \rho P_k . \]

The Lindblad (GKSL) equation with Hamiltonian \(H\), jump operators \(L_k\) and rates \(\gamma_k \ge 0\):

\[ \dot\rho = \mathcal{L}\rho = -i[H, \rho] + \sum_k \gamma_k \Bigl(L_k \rho L_k^\dagger - \tfrac12\{L_k^\dagger L_k, \rho\}\Bigr), \qquad \rho(t) = e^{t\mathcal{L}}\rho(0). \]

The library builds \(\mathcal{L}\) as a \(d^2 \times d^2\) matrix acting on the row-stacked \(\rho\) and uses the matrix exponential. (lindblad_superoperator, evolve_density)

Parameters and their transforms

Each parameter has a kind that maps it to an unconstrained value \(z\) used by the optimiser:

Kind	Range	Map from \(z\)
prob	\((0, 1)\)	\(p = 1/(1 + e^{-z})\)
positive	\((0, \infty)\)	\(x = e^{z}\)
bounded	\((\ell, h)\)	\(x = \ell + (h - \ell)/(1 + e^{-z})\)
angle, real	\(\mathbb{R}\)	\(x = z\)

Likelihood

For multinomial models (counts of outcomes), with predicted probabilities \(p_{c,o}(\theta)\),

\[ \log L(\theta) = \sum_c \sum_o n_{c,o}\log p_{c,o}(\theta) \]

(the multinomial coefficients are constant and omitted; probabilities are clipped at \(10^{-12}\)).

For judgement models (observed mean values \(y_c\) against predictions \(\hat y_c(\theta)\)) the loss is the sum of squared errors \(\mathrm{SSE} = \sum_c (y_c - \hat y_c)^2\), and the Gaussian log-likelihood with the variance profiled out is used for comparison,

\[ \log L = -\frac{n}{2}\Bigl[\log\Bigl(2\pi\,\frac{\mathrm{SSE}}{n}\Bigr) + 1\Bigr]. \]

Fitting and comparison

\(\hat\theta = \arg\max_\theta \log L(\theta)\) by Nelder-Mead from several random starts (and over all discrete structures, such as projector ranks). Models are ranked by

\[ \mathrm{AIC} = 2k - 2\log L(\hat\theta), \qquad \mathrm{BIC} = k\log n - 2\log L(\hat\theta), \]

with \(k\) fitted parameters and \(n\) observations (lower is better). The BIC weight of model \(m\) is

\[ w_m = \frac{\exp(-\tfrac12\Delta_m)}{\sum_{m'}\exp(-\tfrac12\Delta_{m'})}, \qquad \Delta_m = \mathrm{BIC}_m - \min_{m'}\mathrm{BIC}_{m'} . \]

Model recovery. For each generating model \(g\), simulate \(R\) data sets, fit every candidate, and record \(\Pr(\text{selected} = m \mid g)\). A design can tell the models apart when this matrix is close to the identity.

Individual fits. For people \(i = 1, \dots, I\), the group criterion of a model is \(\sum_i \mathrm{BIC}_i\) (each person has their own parameters), compared with the BIC of one pooled fit.

Distances between distributions. \(\mathrm{KL}(p\Vert q) = \sum_i p_i\log(p_i/q_i)\) and \(\mathrm{TVD}(p, q) = \tfrac12\sum_i\lvert p_i - q_i\rvert\).

Reference: Burnham & Anderson (2002); Busemeyer & Bruza (2024), chapters 2-3.

Extensions for every model

These three tools work with every multinomial model in the library (the bootstrap also with least-squares models). They are what Quantum Mind adds to the published models: the papers report point estimates fitted once to a group, while a robot needs uncertainty, per-person adaptation and a way to choose its next question.

Parametric bootstrap (bootstrap()). From the fit \(\hat\theta\), simulate \(b = 1, \dots, B\) data sets with the observed totals \(n_c\),

\[ \tilde y^{(b)}_c \sim \mathrm{Multinomial}\bigl(n_c,\; p_c(\hat\theta)\bigr) \quad\text{or}\quad \tilde y^{(b)}_c = \hat y_c + \varepsilon,\ \varepsilon \sim \mathcal N\bigl(0, \mathrm{SSE}/n\bigr), \]

refit each to get \(\hat\theta^{(b)}\), and report the percentile interval \(\bigl[q_{\alpha/2}, q_{1-\alpha/2}\bigr]\) of \(\{\hat\theta^{(b)}\}\) for each parameter and of \(\{p_c(\hat\theta^{(b)})\}\) for each predicted probability.

Online per-person model (OnlinePersonModel). Particles \(x_i \sim \mathcal N(x_0, \Sigma_0)\) on the unconstrained scale (\(\Sigma_0 = s^2 I\) or a population covariance), \(i = 1, \dots, N\), with \(x_0\) the population estimate. After outcome \(k\) in condition \(c\),

\[ \log w_i \leftarrow \log w_i + \log p_{c,k}(x_i), \qquad \mathrm{ESS} = \Bigl(\sum_i \bar w_i^2\Bigr)^{-1}, \]

with \(\bar w\) the normalised weights. When \(\mathrm{ESS} < \tfrac12 N\), the particles are resampled systematically and each is moved by Metropolis-Hastings steps with a Gaussian random-walk proposal \(x' = x_i + \mathcal N\bigl(0, \tfrac{2.38^2}{k} V\bigr)\) (\(V\) the particle covariance, \(k\) the number of parameters), accepted with probability \(\min\{1, \pi(x')/\pi(x_i)\}\) where \(\log\pi(x) = -\tfrac12 (x - x_0)^T \Sigma_0^{-1} (x - x_0) + \sum_c\sum_k y_{c,k}\log p_{c,k}(x)\) is the log posterior given every answer so far. These moves leave the posterior unchanged and restore the diversity that resampling removes (resample-move). Predictions are posterior predictive, \(\hat p_c = \sum_i \bar w_i\, p_c(x_i)\).

Choosing the most informative condition (information_gain()). For models \(m\) with probabilities \(w_m\), one observation in condition \(c\) carries

\[ I(c) = H\Bigl(\sum_m w_m\, p_m(c)\Bigr) - \sum_m w_m\, H\bigl(p_m(c)\bigr) \quad\text{bits}, \qquad 0 \le I(c) \le \log_2 M, \]

the mutual information between the model identity and the outcome (the weighted Jensen-Shannon divergence; with BIC weights it equals the ensemble disagreement of HumanModelEnsemble). After observing counts \(y\), the model probabilities become \(w_m \propto w_m \prod_c \prod_k p_{m,c,k}^{\,y_{c,k}}\) (model_posterior()).

References: Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall. Chopin, N. (2002). A sequential particle filter method for static models. Biometrika, 89(3), 539-551. Myung, J. I., & Pitt, M. A. (2009). Optimal experimental design for model discrimination. Psychological Review, 116(3), 499-518.