4 91 TRANSFERRING TARGETED MAXIMUM LIKELIHOOD ESTIMATION INTO SPORT SCIENCE ( , ) = ⋅ (7) The weights in this case are calculated in such a way that they minimize the risk of the SL c. Targeted Maximum Likelihood Estimation After the initial estimation step is completed, the next step is to perform the Targeted Maximum Likelihood Estimation (TMLE) step [2], [13]. The goal of TMLE is to reduce the bias of the estimation of the target parameter [33]. Figure 4 presents an abstract representation of TMLE and its goal. In this graph the circle depicts ℳ, the set of all possible probability distributions. As can be seen, P0 ∈ ℳ, which maps to the target parameter Ψ( P0). Our aim is to use Pn ∈ ℳ with the corresponding Ψ( Pn *) to create a targeted estimate closer to the true target parameter. Figure 4. Graphical depiction of TMLE [2]. The definition of the ATE TMLE estimator ∗ is given by ∗ = ∑ Q [ ` ∗( , )− ` ∗( , )] (8) Which is the targeted version of ψ (Equation (1)). We use the notation \.;( , ) to denote the initial estimate of [ ∣ , ] and, \.∗( , ) to denote its targeted counterpart. Targeting \.;( , ) involves the two new nuisance parameters: the treatment
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