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. Author manuscript; available in PMC: 2013 Jan 16.

Published in final edited form as: Electron J Stat. 2012;6:1059–1099. doi: 10.1214/12-EJS703

Fig 1 — Illustration of the TMLE procedure (with its general one-step updating procedure). We intentionally represent the initial estimator $P_{n}^{0}$ closer to P₀ than its kth and (k +1)th updates $P_{n}^{k}$ and $P_{n}^{k + 1}$ , heuristically because $P_{n}^{0}$ is as close to P₀ as one can possibly get (given P_n and the specifics of the super-learning procedure) when targeting P₀ itself. However, this obviously does not necessarily imply that $Ψ (P_{n}^{0})$ performs well when targeting Ψ (P₀) (instead of P₀), which is why we also intentionally represent $Ψ (P_{n}^{k + 1})$ closer to Ψ (P₀) than $Ψ (P_{n}^{0})$ . Indeed, $P_{n}^{k + 1}$ is obtained by fluctuating its predecessor $P_{n}^{k}$ in the direction of Ψ ”, i.e., taking into account the fact that we are ultimately interested in estimating Ψ (P₀). More specifically, the fluctuation { $P_{n}^{k} (ε) : ∣ ε ∣ < η_{n}^{k}$ } of $P_{n}^{k}$ is a one-dimensional parametric model (hence its curvy shape in the large model ) such that (i) $P_{n}^{k} (0) = P_{n}^{k}$ , and (b) its score at ε = 0 equals the efficient influence curve $D^{★} (P_{n}^{k})$ at $P_{n}^{k}$ (hence the dotted arrow). An optimal stretch $ε_{n}^{k}$ is determined (e.g. by maximizing the likelihood on the fluctuation), yielding the update $P_{n}^{k + 1} = P_{n}^{k} (ε_{n}^{k})$ .

Inline graphic — Illustration of the TMLE procedure (with its general one-step updating procedure). We intentionally represent the initial estimator $P_{n}^{0}$ closer to P₀ than its kth and (k +1)th updates $P_{n}^{k}$ and $P_{n}^{k + 1}$ , heuristically because $P_{n}^{0}$ is as close to P₀ as one can possibly get (given P_n and the specifics of the super-learning procedure) when targeting P₀ itself. However, this obviously does not necessarily imply that $Ψ (P_{n}^{0})$ performs well when targeting Ψ (P₀) (instead of P₀), which is why we also intentionally represent $Ψ (P_{n}^{k + 1})$ closer to Ψ (P₀) than $Ψ (P_{n}^{0})$ . Indeed, $P_{n}^{k + 1}$ is obtained by fluctuating its predecessor $P_{n}^{k}$ in the direction of Ψ ”, i.e., taking into account the fact that we are ultimately interested in estimating Ψ (P₀). More specifically, the fluctuation { $P_{n}^{k} (ε) : ∣ ε ∣ < η_{n}^{k}$ } of $P_{n}^{k}$ is a one-dimensional parametric model (hence its curvy shape in the large model ) such that (i) $P_{n}^{k} (0) = P_{n}^{k}$ , and (b) its score at ε = 0 equals the efficient influence curve $D^{★} (P_{n}^{k})$ at $P_{n}^{k}$ (hence the dotted arrow). An optimal stretch $ε_{n}^{k}$ is determined (e.g. by maximizing the likelihood on the fluctuation), yielding the update $P_{n}^{k + 1} = P_{n}^{k} (ε_{n}^{k})$ .