Matching the Paper's Model¶

First, I will focus on what exists in the paper to attempt to structure this code arround.

Equations¶

Below are a list of the equations in the model. Each of these equations are from the origional paper, and the variable explainations were automatically generated by an AI refereancing the paper. I have corrected any equation explainations that I noticed an error in.

Equation 1¶

\[p(h|D) \propto p(D|h)p(h)\]

Bayes' rule for updating beliefs over hypotheses after observing data.

Variable	Description
\(h\)	A hypothesis — a candidate decision rule (defined by a boundary orientation and position) about how to classify observations
\(D\)	The observed data collected so far
\(p(h)\)	The prior probability of hypothesis \(h\) before seeing any data
\(p(D \mid h)\)	The likelihood of observing data \(D\) given that hypothesis \(h\) is true
\(p(h \mid D)\)	The posterior belief in hypothesis \(h\) after observing \(D\); its magnitude reflects the relative strength of belief that this hypothesis is correct

Equation 2¶

\[P(x^t = A|\theta, b, \sigma) = \frac{1}{1 + \exp\!\left(-\sigma\left(x_1^t \cdot \cos(\theta) + x_2^t \cdot \sin(\theta) - b\right)\right)}\]

The probability that stimulus \(x^t\) belongs to Category A, given a linear decision boundary defined by \(\theta\), \(b\), and \(\sigma\). Because classification is binary, \(P(x^t = B \mid \theta, b, \sigma) = 1 - P(x^t = A \mid \theta, b, \sigma)\).

Variable	Description
\(x^t\)	The stimulus (observation) on trial \(t\), a point in the 2D stimulus space
\(x_1^t\)	The value of stimulus \(x^t\) on dimension 1 (e.g., antenna length) on trial \(t\)
\(x_2^t\)	The value of stimulus \(x^t\) on dimension 2 (e.g., antenna angle) on trial \(t\)
\(A\)	Category A — one of the two binary classification outcomes
\(\theta\)	The angle of the decision boundary vector in the stimulus space; controls its orientation (\(\theta = 0\) gives a vertical boundary aligned with one axis)
\(b\)	The bias term; the offset of the decision boundary from the center of the stimulus space
\(\sigma\)	The determinism of the boundary; controls the sharpness of the sigmoidal response (higher \(\sigma\) = sharper boundary)

Equation 3¶

\[r = \frac{2}{\pi}\left\lfloor \theta \bmod \frac{\pi}{2} \right\rfloor\]

Converts a boundary angle \(\theta\) into a normalized distance from the nearest axis-aligned rule, used to define the prior over boundary orientations.

Variable	Description
\(r\)	The relative distance of the decision boundary from the nearest orthogonal (axis-aligned) direction; bounded between 0 and 1, where 0 means the boundary is perfectly axis-aligned
\(\theta\)	The angle of the decision boundary in the stimulus space
\(\bmod\)	The arithmetic modulo operator

Equation 4¶

\[r \sim \text{Beta}(\alpha, \alpha)\]

The prior distribution over \(r\), which encodes the learner's bias toward axis-aligned (unidimensional) decision boundaries.

Variable	Description
\(r\)	The relative distance from the nearest axis-aligned boundary (same as Equation 3)
\(\alpha\)	The shape parameter of the Beta distribution; acts as an abstract weight controlling preference for unidimensional rules. When \(\alpha \ll 1\), the prior strongly favors axis-aligned rules; when \(\alpha = 1\), all orientations are equally likely

Equation 5¶

\[\theta = \frac{\pi}{2}(k + r)\]

Generates a new hypothesis angle \(\theta\) from a randomly chosen quadrant and a sampled relative distance \(r\).

Variable	Description
\(\theta\)	The newly generated decision boundary angle in the stimulus space
\(k\)	An integer between 0 and 3 specifying which quadrant (i.e., which axis-aligned reference direction) the new hypothesis is drawn relative to
\(r\)	The relative distance from the nearest orthogonal axis, sampled from \(\text{Beta}(\alpha, \alpha)\) per Equation 4

Equation 6¶

\[ \begin{align} d_1 &\sim N(b^t, S) \\ d_2 &\sim U(0, 1) \\ \mathbf{x} &= d_1 \mathbf{u}_1 + d_2 \mathbf{u}_2 \end{align} \]

The hypothesis-dependent sampling procedure — how a selection learner generates a new stimulus to query, biased toward the region near their current decision boundary.

Variable	Description
\(\mathbf{x}\)	The new stimulus selected by the learner for the next trial
\(d_1\)	The coordinate in the direction orthogonal to the current decision boundary; sampled from a Normal distribution centered on the boundary
\(d_2\)	The coordinate in the direction parallel to the current decision boundary; sampled uniformly so that selections are spread along the boundary
\(b^t\)	The bias term of the current hypothesis on trial \(t\); serves as the mean of the normal distribution, anchoring selections to the current boundary location
\(S\)	The sampling spread parameter; controls the average orthogonal distance of new selections from the current boundary. Larger \(S\) = weaker bias (selections further from the boundary); smaller \(S\) = stronger bias (selections tightly clustered on the boundary)
\(\mathbf{u}_1\)	Unit vector orthogonal to the current decision boundary
\(\mathbf{u}_2\)	Unit vector parallel to the current decision boundary