Bayesian inference can be used to simultaneously find the order of a polynomial and its coefficients, trading off fit against prior preference for simpler models.

```
;;;fold: my-pi, glp, range
(define my-pi 3.14159265358979323)
(define (glp mean ssq smp)
(let ([diff (- smp mean)])
(- (- (/ (* diff diff) (* 2.0 ssq)))
(* 0.5 (+ (log 2) (log my-pi) (log ssq))))))
(define (xrange i j)
(if (= i j)
'()
(cons i (range (+ i 1) j))))
(define (range i j)
(reverse (xrange i j)))
;;;
(define (make-poly c)
(lambda (x) (apply + (map (lambda (a b) (* a (expt x b)))
c
(iota (length c))))))
(define x-vals (map (lambda (x) (/ x 1)) (range -5 5)))
(define true-coeffs '(-.5 1 .5))
(define true-y-vals (map (make-poly true-coeffs) x-vals))
(define obs-noise 0.2)
(define obs-y-vals
(map (lambda (x) (gaussian x obs-noise)) true-y-vals))
(define (noisy-equals? x y)
(factor (glp x obs-noise y))
#t)
(define samples
(mh-query
10000 5
(define poly-order (sample-integer 4))
(define coefficients
(repeat (+ poly-order 1)
(lambda () (gaussian 0 2))))
(define y-vals
(map (make-poly coefficients) x-vals))
(list poly-order coefficients)
(all (map noisy-equals? y-vals obs-y-vals))))
(hist (map first samples))
```

References:

- Cite:ProbMods