Rational Rules is a model of concept learning. Concepts are represented as compositional rule formulas. The version below is a simplified version of the model used in Ref:Goodman2008ub.

``````;;first set up the training (cat A/B) and test objects:
(define num-features 4)

(define A-objects (list '(0 0 0 1) '(0 1 0 1) '(0 1 0 0) '(0 0 1 0) '(1 0 0 0)))

(define B-objects (list '(0 0 1 1) '(1 0 0 1) '(1 1 1 0) '(1 1 1 1)))

(define T-objects (list '(0 1 1 0) '(0 1 1 1) '(0 0 0 0) '(1 1 0 1)
'(1 0 1 0) '(1 1 0 0) '(1 0 1 1)))

;;here are the human results from Nosofsky et al, for comparison:
(define human-A '(0.77 0.78 0.83 0.64 0.61))
(define human-B '(0.39 0.41 0.21 0.15))
(define human-T '(0.56 0.41 0.82 0.40 0.32 0.53 0.20))

;;two parameters: stopping probability of the grammar, and noise probability:
(define tau 0.3)
(define noise-param (exp -1.5))

;;a generative process for disjunctive normal form propositional equations:
(define (get-formula)
(if (flip tau)
(let ((c (Conj))
(f (get-formula)))
(lambda (x) (or (c x) (f x))))
(Conj)))

(define (Conj)
(if (flip tau)
(let ((c (Conj))
(p (Pred)))
(lambda (x) (and (c x) (p x))))
(Pred)))

(define (Pred)
(let ((index (sample-integer num-features))
(value (sample-integer 2)))
(lambda (x) (= (list-ref x index) value))))

(define (noisy-equal? a b) (flip (if (equal? a b) 0.999999999 noise-param)))

(define samples
(mh-query
1000 10

;;infer a classification formula
(define my-formula (get-formula))

;;look at posterior predictive classification
(map my-formula (append T-objects A-objects B-objects))

;;conditioning (noisily) on all the training eamples:
(and (all (map (lambda (x) (noisy-equal? true (my-formula x))) A-objects))
(all (map (lambda (x) (noisy-equal? false (my-formula x))) B-objects)))))

;;now plot the predictions vs human data:
(define (means samples)
(if (null? (first samples))
'()
(pair (mean (map (lambda (x) (if x 1.0 0.0)) (map first samples)))
(means (map rest samples)))))

(scatter (map pair (means samples) (append human-T human-A human-B))
"model vs human")
``````

References:

• Cite:Goodman2008ub
• Cite:ProbMods