Suppose one observes a sample of independent and identically distributed observations from a particular data generating distribution. Suppose that one is concerned with estimation of a particular pathwise differentiable Euclidean parameter. A substitution estimator evaluating the parameter of a given likelihood based density estimator is typically too biased and might not even converge at the parametric rate: that is, the density estimator was targeted to be a good estimator of the density and might therefore result in a poor estimator of a particular smooth functional of the density. In this article we propose a one step (and, by iteration, k-th step) targeted maximum likelihood density estimator which involves 1) creating a hardest parametric submodel with parameter epsilon through the given density estimator with score equal to the efficient influence curve of the pathwise differentiable parameter at the density estimator, 2) estimating epsilon with the maximum likelihood estimator, and 3) defining a new density estimator as the corresponding update of the original density estimator. We show that iteration of this algorithm results in a targeted maximum likelihood density estimator which solves the efficient influence curve estimating equation and thereby yields a locally efficient estimator of the parameter of interest, under regularity conditions. In particular, we show that, if the parameter is linear and the model is convex, then the targeted maximum likelihood estimator is often achieved in the first step, and it results in a locally efficient estimator at an arbitrary (e.g., heavily misspecified) starting density.We also show that the targeted maximum likelihood estimators are now in full agreement with the locally efficient estimating function methodology as presented in Robins and Rotnitzky (1992) and van der Laan and Robins (2003), creating, in particular, algebraic equivalence between the double robust locally efficient estimators using the targeted maximum likelihood estimators as an estimate of its nuisance parameters, and targeted maximum likelihood estimators. In addition, it is argued that the targeted MLE has various advantages relative to the current estimating function based approach. We proceed by providing data driven methodologies to select the initial density estimator for the targeted MLE, thereby providing data adaptive targeted maximum likelihood estimation methodology. We illustrate the method with various worked out examples.

Summary Neurons in sensory cortex integrate multiple influences to parse objects and support perception. Across multiple cortical areas, integration is characterized by two neuronal response properties: (1) surround suppression: modulatory contextual stimuli suppress responses to driving stimuli; (2) “normalization”: responses to multiple driving stimuli add sublinearly. These properties depend on input strength: for weak driving stimuli, contextual influences more weakly suppress or facilitate and summation becomes linear or supralinear. Understanding the circuit operations underlying integration is critical to understanding cortical function and disease. We present a simple, general theory. A wealth of integrative properties including the above emerge robustly from four properties of cortical circuitry: (1) supralinear neuronal input/output functions; (2) sufficiently strong recurrent excitation; (3) feedback inhibition; (4) simple spatial properties of intracortical connections. Integrative properties emerge dynamically as circuit properties, with excitatory and inhibitory neurons showing similar behaviors. In new recordings in visual cortex, we confirm key model predictions.

Cystic fibrosis is caused by mutations in the gene encoding the cystic fibrosis transmembrane conductance regulator (CFTR). The most common mutation, DeltaF508, results in the production of a misfolded CFTR protein that is retained in the endoplasmic reticulum and targeted for degradation. Curcumin is a nontoxic Ca-adenosine triphosphatase pump inhibitor that can be administered to humans safely. Oral administration of curcumin to homozygous DeltaF508 CFTR mice in doses comparable, on a weight-per-weight basis, to those well tolerated by humans corrected these animals' characteristic nasal potential difference defect. These effects were not observed in mice homozygous for a complete knockout of the CFTR gene. Curcumin also induced the functional appearance of DeltaF508 CFTR protein in the plasma membranes of transfected baby hamster kidney cells. Thus, curcumin treatment may be able to correct defects associated with the homozygous expression of DeltaF508 CFTR.