eitodennäköisyysotannalla

Eitodennäköisyysotannalla, often translated as inverse probability sampling or rejection sampling in English, is a computational technique used in statistics and machine learning to generate random samples from a probability distribution. It is particularly useful when direct sampling from the target distribution is difficult or impossible. The core idea is to use a simpler, related distribution from which samples can be drawn easily, and then selectively accept or reject these samples based on a specific criterion.

The process begins by choosing a proposal distribution, often denoted as q(x), which is easier to sample from than the target distribution p(x). A candidate sample, x, is then drawn from q(x). Next, a random number u is generated from a uniform distribution between 0 and 1. This candidate sample x is accepted if u is less than or equal to the ratio of the target probability density to a scaled version of the proposal probability density, typically expressed as p(x) / (M * q(x)), where M is a constant chosen such that M * q(x) is always greater than or equal to p(x) for all x. If the condition is met, the sample x is kept; otherwise, it is rejected. This process is repeated until the desired number of samples from the target distribution p(x) is obtained. The value of M plays a crucial role in the efficiency of the method; a smaller M leads to a higher acceptance rate but requires a tighter bounding of the proposal distribution. Eitodennäköisyysotannalla is a fundamental algorithm for Monte Carlo methods, enabling the estimation of complex probabilities and expected values.

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