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In this paper, we deal with the classical Statistical Learning Theory's problem of bounding, with high probability, the true risk R(h) of a hypothesis h chosen from a set H of m hypotheses. The Union Bound (U allows one to state that PLR^(h),δqh≤R(h)≤UR^(h),δph≥1-δ where R^(h) is the empirical errors, if it is possible to prove that PR(h)≥L(R^(h),δ)≥1-δ and PR(h)≤U(R^(h),δ)≥1-δ, when h, qh, and ph are chosen before seeing the data such that qh,ph∈[0,1] and ∑h∈H(qh+ph)=1. If no a priori information is available qh and ph are set to 12m,

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