There are several estimators of conditional probability from observed frequencies of features. In this paper, we propose using the lower limit of confidence interval on posterior distribution determined by the observed frequencies to ascertain conditional probability. In our experiments, this method outperformed other popular estimators.
Conditional probability is more popular in the database field than NLP. Apriori @cite_6 is the most practically used method for finding the relationship between features, when the true strength between features is known to be conditional probability. It uses MLE as the measure for the relationship. To overcome the problem of low frequency, Apriori ignores the rare features using a threshold value, which is called minimum supports. Although Apriori is efficient by ignoring low-frequency features, we sometimes need to compare a high frequency but low MLE value relationship with a low frequency but high MLE value relationship. Apriori simply ignores relation of the low frequency but high MLE value relationship.
@cite_6
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Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of n phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise. In applications like location-based services, sensor monitoring and biological databases, the values of the database items are inherently uncertain in nature. An important query for uncertain objects is the probabilistic nearest-neighbor query (PNN), which computes the probability of each object for being the nearest neighbor of a query point. Evaluating this query is computationally expensive, since it needs to consider the relationship among uncertain objects, and requires the use of numerical integration or Monte-Carlo methods. Sometimes, a query user may not be concerned about the exact probability values. For example, he may only need answers that have sufficiently high confidence. We thus propose the constrained nearest-neighbor query (C-PNN), which returns the IDs of objects whose probabilities are higher than some threshold, with a given error bound in the answers. The C-PNN can be answered efficiently with probabilistic verifiers. These are methods that derive the lower and upper bounds of answer probabilities, so that an object can be quickly decided on whether it should be included in the answer. We have developed three probabilistic verifiers, which can be used on uncertain data with arbitrary probability density functions. Extensive experiments were performed to examine the effectiveness of these approaches. Maximum likelihood (ML) is an increasingly popular optimality criterion for selecting evolutionary trees [Felsenstein 1981]. Finding optimal ML trees appears to be a very hard computational task, but for tractable cases, ML is the method of choice. In particular, algorithms and heuristics for ML take longer to run than algorithms and heuristics for the second major character based criterion, maximum parsimony (MP). However, while MP has been known to be NP-complete for over 20 years [Foulds and Graham, 1982; 1986], such a hardness result for ML has so far eluded researchers in the field.An important work by Tuffley and Steel [1997] proves quantitative relations between the parsimony values of given sequences and the corresponding log likelihood values. However, a direct application of their work would only give an exponential time reduction from MP to ML. Another step in this direction has recently been made by Addario- [2004], who proved that ancestral maximum likelihood (AML) is NP-complete. AML “lies in between” the two problems, having some properties of MP and some properties of ML. Still, the AML proof is not directly applicable to the ML problem.We resolve the question, showing that “regular” ML on phylogenetic trees is indeed intractable. Our reduction follows the vertex cover reductions for MP [ 1986] and AML [Addario- 2004], but its starting point is an approximation version of vertex cover, known as gap vc. The crux of our work is not the reduction, but its correctness proof. The proof goes through a series of tree modifications, while controlling the likelihood losses at each step, using the bounds of Tuffley and Steel [1997]. The proof can be viewed as correlating the value of any ML solution to an arbitrarily close approximation to vertex cover. The problem of estimating the phases (angles) of a complex unit-modulus vector @math from their noisy pairwise relative measurements @math , where @math is a complex-valued Gaussian random matrix, is known as phase synchronization. The maximum likelihood estimator (MLE) is a solution to a unit--modulus-constrained quadratic programming problem, which is nonconvex. Existing works have proposed polynomial-time algorithms such as a semidefinite programming (SDP) relaxation or the generalized power method (GPM). Numerical experiments suggest that both of these methods succeed with high probability for @math up to @math , yet existing analyses only confirm this observation for @math up to @math . In this paper, we bridge the gap by proving that the SDP relaxation is tight for @math , and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tight...