Manuscript Psy03-D@1 · Psy03-D-v1.tex · section · 3e18951d

Points, rules, and observations

An explicitly specified hidden-rule task › Points, rules, and observations · lines 256–312

Psy03-D@1#eq:ruleprocess, Psy03-D@1#eq:rules

Let

μ_{0} = (m, m), μ_{1} = (- m, m), μ_{2} = (- m, - m), μ_{3} = (m, - m), m > 0, s > 0.

At each step,

X_{t} = μ_{K_{t}} + s Z_{t},

where $K_{t}$ is uniform on ${0, 1, 2, 3}$ , $Z_{t} \sim N (0, I_{2})$ , and the point draws are independent. The public task uses $m = 1.5$ , $s = 0.8$ CheungPublicCode. The calculations below apply to every $m > 0, s > 0$ .

Let $B \subseteq {1, \dots, T - 1}$ be a switch schedule. It can be deterministic or random, but is independent of the point draws and of the initial rule. Let

R_{0} \sim Unif {0, 1, 2, 3} .

At a switch, add an independent uniform increment from ${1, 2, 3}$ modulo four:

R_{t} = {R_{t - 1}, R_{t - 1} + J_{t} (mod 4), t \in / B, t \in B .

For $x = (x_{1}, x_{2})$ , define

f_{0} (x) f_{1} (x) f_{2} (x) f_{3} (x) = 21 {x_{1} \geq 0} + 1 {x_{2} \geq 0}, = 21 {x_{2} \geq 0} + 1 {x_{1} \geq 0}, = 1 {x_{1} \geq 0} + 21 {x_{2} \geq 0}, = 21 {x_{2} \geq x_{1}} + 1 {x_{2} \geq - x_{1}} .

These are the task mappings aligned with the public apply_rule record CheungPublicCode. The target is

Y_{t} = f_{R_{t}} (X_{t}) .

Optional ritual overwriting is disabled.

The evaluator may retain $R_{t}, Y_{t}, B$ for scoring. The predictor receives the point sequence, not these annotations.

Displayed formulas in this passage

Psy03-D@1#eq:rules · line 295
$f_{0} (x) f_{1} (x) f_{2} (x) f_{3} (x) = 21 {x_{1} \geq 0} + 1 {x_{2} \geq 0}, = 21 {x_{2} \geq 0} + 1 {x_{1} \geq 0}, = 1 {x_{1} \geq 0} + 21 {x_{2} \geq 0}, = 21 {x_{2} \geq x_{1}} + 1 {x_{2} \geq - x_{1}} .$
unlabeled · line 259
$μ_{0} = (m, m), μ_{1} = (- m, m), μ_{2} = (- m, - m), μ_{3} = (m, - m), m > 0, s > 0.$
unlabeled · line 267
$X_{t} = μ_{K_{t}} + s Z_{t},$
unlabeled · line 280
$R_{0} \sim Unif {0, 1, 2, 3} .$
Psy03-D@1#eq:ruleprocess · line 285
$R_{t} = {R_{t - 1}, R_{t - 1} + J_{t} (mod 4), t \in / B, t \in B .$
unlabeled · line 305
$Y_{t} = f_{R_{t}} (X_{t}) .$

Points, rules, and observations

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