I just had another idea (loosely inspired on the Glicko rating system): suppose that a person on Day 0 has an unknown “true weight” W_0, but because of measurement errors and unknown amount of body water etc. the scale reads w_0, which is normally distributed with mean W_0 and variance σ^2; suppose also that if we knew W_0 we would assign W_1 (the “true weight” on Day 1) a probability distribution with mean W_0 and variance c^2(t_1 − t_0). Now, if our probability distribution for W_n is a Gaussian with mean u_n and variance σ_n^2, on knowing the measured weight w_n we would update it to mean (u_n/σ_n^2 + w_n/σ^2)/(1/σ_n^2 + 1/σ^2) and variance 1//(1/σ_n^2 + 1/σ^2). Hence:

sigma_sq = 1
c_sq = 0.01111111111111111111
smoothweight = 0
sigman_sq = infinity
every day:
sigman_sq = sigman_sq + c_sq
if user reports weight:
smoothweight = (smoothweight/sigman_sq + today.weight/sigma_sq)/(1./sigman_sq + 1./sigma_sq)
sigman_sq = 1./(1./sigman_sq + 1./sigma_sq)

(Multiplying sigma_sq and c_sq by the same constant doesn't affect the values of smoothweight, as far as I can tell.) In the limit of data on a large number of consecutive days, sigman_sq approaches 0.1 and the algorithm becomes equivalent to the other ones. I've tried this with my own gapped data and the trend line changes faster than with the Hacker's Diet algorithm but not as fast as with my old algorithm. But I now prefer this because it has a rationale resembling more a derivation from first principles than someone pulling stuff out of their ass.

(Is there a way of getting real subscripts and superscripts?)