Contemporary high-throughput technologies provide measurements of very large numbers of variables but often with very small sample sizes. see Figure 1 conveying the prior network knowledge of the label-conditional distribution = 0 1 which includes a term contributed by the uncertainty class: λ∈ [0 1 reflects the uncertainty of the labeled training data compared to the total amount of uncertainty in our prior knowledge and is the standard unit (? 1)–simplex and is any uncertainty class containing |Πbetween a given and the uncertainty class. If the objective function in (4) is a convex function then the optimization problem can be solved efficiently. Since the log-likelihood of the multinomial distribution is concave (i.e. the negative log-likelihood function for = 1 … (i.e. the regularizer term) in (4) to make it a convex programming problem. We use is the (KL-distance). Lemma 1 (RML Classifier). = 1 2 … |ΠΠ∈ {0 1 and → 0 and → 1 and ∈ [0 1 is the degree of contamination and is one of a finite number of randomly chosen densities from corresponds to increasing the variance of prior knowledge about the true distribution. We assume a uniform distribution for the contamination part whose domain is the relative interior of Jatropholone B the volume under the (? 1)-simplex. Since our application of interest is related to steady-state classifiers we assume that in the simplex the corners and axes have measure zero. 2 p-point uncertainty class The is the actual steady-state distribution form a partition of the state space denoted by ∈ is any density function. We will use the following notation throughout the paper for the probability mass cumulated in each partition: = 1 means that we only know that the label-conditional probabilities for the bins sum up to 1 which corresponds to a minimal amount of prior knowledge. On the other hand = ∈ {1 … ∈ {0 1 means that we are certain about the label-conditional distributions because we are given all bin probabilities – hence minimal variance in the uncertainty class (for more details refer to Section 1 of the supplementary materials on the companion website). 3 Moments for the true error For a classifier trained on the sample data |[Pr(|and the uncertainty-class space. They take the form ∈ (0 1 The cases λ∈ {0 1 can be handled with a slight modification to the proof. Theorem 1 (First-Order Moment of the True Error: Π1 = 1 … b ?= 0 .. ∈ {0 1 denotes1 ? Π1 = 1 … 0 1 = 1 … b = 0 .. and (similarly for and and (similarly for and in (4) should be adjusted based on the relative uncertainty between the training data and the prior knowledge. We propose three approaches for tuning the regularization parameter. 4.1 Minimizing the expected true error The optimal value of the regularization parameter based on expected true error can be found by solving the following optimization problem: ∈ {0 1 approximated in Theorems 1 and 2. (22) is a constrained nonlinear programming problem whose global minimum is not guaranteed to be found by classic gradient-based methods. 4.2 SURE-tuning of regularization parameter One way to evaluate the performance of the estimator in Lemma 1 is to use the mean-squared error (MSE) of the estimator. In the problem of multinomial distribution estimation the MSE can be expanded as follows to show that the estimate depends on λin (23) [43 44 however MSEdepends on the Jatropholone B parameter for estimating λ ≥ 2. Corollary 1 (SURE-Optimal Regularization Parameter). → ∞ we obtain = 1 (a detailed description of the Zipf Jatropholone B model will be provided in Section 5). We observe the behavior of using Monte-Carlo expectation over 4000 training data sets (for Jatropholone B each fixed sample size) and 500 uncertainty classes. We consider different values for ∈ [0 1 and sample size as the x-axis and as the y-axis. As → 1 (uncertainty Rabbit polyclonal to AnnexinA10. is increased) for a fixed sample size decreases as in equation (26). Figure 2 Illustrating the expected value of Jatropholone B for different amount of sample and uncertainty sizes. The result is for as = 1 … |Πgiven in (3). The exact expression for E(pairs of uncertainty classes denoted by = 1 … = 0 1 we generate sample sets with size denoted by = 1 … (which was used to generate the sample. We denote this error by and in (27) as = 0 1 follow a Zipf model is a normalizing constant. The Zipf distribution introduced by G.K. Zipf to model the frequency of words in common text [46] is a well-known power-law discrete distribution encountered in many.