A Convergence Result for the Baby Feeding Problem
Kris Hauser

Abstract— We present new asymptotic convergence results
for a variant of the canonical baby feeding problem. Consider
a hyperactive squirming baby at feeding time, with a dab ofpuréed carrots on her chin. The baby proceeds to squirm. At
each squirm at t = 1, . . . , ∞, she contacts one area of her bodywith another. We then pose the question, how many squirms
are needed such that carrots cover a 1- fraction of her body?
In this paper we model the baby as a bounded compact set
in a Hauserdorff space, and take a Markovian, nonadversarial
squirming model. Our analysis proves that convergence is upperbounded by an exponentially decreasing function. Experimental
results with a 6 month old infant support the conclusions ofour model. We also present extensions of this result to the rateof coverage of the bib, high chair, parents, floors, walls, andceiling. Future work will consider adversarial squirming, andgame-theoretic analysis of the scenario as the parent attempts
to clean the baby.

I. P ROBLEM S TATEMENT
Let B represent the surface of the baby. Assume a sphericalbaby. We assume B ⊆ X is a bounded compact submanifold
in a measure space (X , µ). Without loss of generality, we
scale µ appropriately such that µ(B) = 1. At t = 0, asubset C of B is covered in carrot purée, or mashed potatoes,or smushed peas, or some other sticky, fluid, nutritious
substance which can be spread infinitely thin upon any
surface.
Suppose at each time t, the baby squirms. A squirm(ft , At , Bt ) is a tuple containing two subsets At , Bt ∈ B,
and a bijection ft : At 7→ Bt . The action of the squirm issuch that any food at a point x ∈ At is smeared on ft (x),and any food at a point y ∈ Bt is smeared on ft−1 (y). Thus,
at time t, we can define the sequence C0 , . . . , Ct , such that
C0 = C, and Ct+1 is defined by the squirm at t as follows:
Ct+1 = {x|x ∈ Ct ∨ x ∈ ft (At ) ∨ x ∈ ft−1 (Bt ).

(1)

To squirm, the baby chooses from a set of candidate
squirms S. The probability that the baby chooses S ∈ S
at time t is given by the policyπ(t, X, S) : R × 2B × S 7→ R(2)Let αt = µ(Ct ) denote the random variables indicating thefraction of B covered with food at time t. The baby-feedingproblem seeks to ask the following questions:
1) For x ∈ B, at what t does the probability that x ∈ Ct
exceed p > 0?
2) Does the distribution of αt converge to 1 as t → ∞?
3) If so, at what time t does µ(Ct ) exceed 1 −  withprobability γ, for constants , γ > 0?
Kris Hauser is with the Department of Electrical Engineering andComputer Science, University of California at Berkeley, Berkeley, CA, USA

hauser@berkeley.edu(a) Before experiment.

(b) After the experiment.
Fig. 1. Subject of experiment, a six month old female infant. Squash puréecovered 25.2% of her body and 17.7% of her highchair after three minutesof squirming.

II. A T IME -I NVARIANT M ARKOVIAN BABYOur analysis addresses the case where µ(At ) = µ(Bt ) =L, and the choices of squirm S ∈ S are identically and independently distributed. We assume that S is chosen uniformly
at random. This defines an infinite-state Markov chain on 2B ,with initial state C and the transition function is defined suchthatP r(x ∈ Ct+1 |x ∈/ Ct ) = αk L(3)