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In analysis of algorithms, probabilistic analysis of algorithms is an approach to estimate the computational complexity of an algorithm or a computational problem. It starts from an assumption about a probabilistic distribution of the set of all possible inputs.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are ...
To apply the method to a probabilistic proof, the randomly chosen object in the proof must be choosable by a random experiment that consists of a sequence of "small" random choices. Here is a trivial example to illustrate the principle. Lemma: It is possible to flip three coins so that the number of tails is at least 2. Probabilistic proof.
One can compute this directly, without using a probability distribution (distribution-free classifier); one can estimate the probability of a label given an observation, (| =) (discriminative model), and base classification on that; or one can estimate the joint distribution (,) (generative model), from that compute the conditional probability ...
An example calibration plot. Calibration can be assessed using a calibration plot (also called a reliability diagram). [3] [5] A calibration plot shows the proportion of items in each class for bands of predicted probability or score (such as a distorted probability distribution or the "signed distance to the hyperplane" in a support vector ...
For the following definitions, two examples will be used. The first is the problem of character recognition given an array of bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative.
Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in the sample space is less than the expected value, a contradiction.
Bayesian optimization algorithms operate by maintaining a probabilistic belief about throughout the optimization procedure; this often takes the form of a Gaussian process prior conditioned on observations. This belief then guides the algorithm in obtaining observations that are likely to advance the optimization process.