Targeted Learning in Data Science pp 303-315 | Cite as

# Online Super Learning

## Abstract

We consider the case that the data *O*^{ n } = (*O*(1), *…*, *O*(*n*)) ∼ *P*_{0}^{ n } is generated sequentially by a conditional probability distribution *P*_{θ, t} of *O*(*t*), given certain summary measures of the past \(\bar{O}(t - 1) = (O(1),\ldots,O(t - 1))\), and where this *t*-specific conditional probability distribution *P*_{θ, t} is identified by a common parameter *θ* ∈ *Θ*. For example, the experiment at time *t* generates a new observation *O*(*t*) from a probability distribution \(\bar{P}_{0}(\cdot \mid z)\) determined by a fixed dimensional summary measure *Z* of the past *O*(1), *…*, *O*(*t* − 1), and one would assume that this conditional distribution \(\bar{P}_{0}\) is an element of some semiparametric model. An important special case is that the sample can be viewed as independent and identically distributed observations from a fixed data generating distribution that is known to belong to some semiparametric statistical model, such as the nonparametric model: in this case \(\bar{P}_{0}(\cdot \mid z) =\bar{ P}_{0}(\cdot )\) does not depend on the past. Another important case is that the data is generated by a group sequential adaptive design in which the randomization and or censoring mechanism is a function of summary measures of the observed data on previously sampled groups (van der Laan 2008b; Chambaz and van der Laan 2011a,b,c). More generally, this covers a whole range of time series models.

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