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There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound [1] or negative variational free energy) is a useful lower bound on the log-likelihood of some observed data.
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
Fano's inequality can be interpreted as a way of dividing the uncertainty of a conditional distribution into two questions given an arbitrary predictor. The first question, corresponding to the term (), relates to the uncertainty of the predictor.
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
A natural proof is a proof that establishes that a certain language lies outside of C and refers to a natural property that is useful against C. Razborov and Rudich give a number of examples of lower-bound proofs against classes C smaller than P/poly that can be "naturalized", i.e. converted
In two steps, it has been shown that is an upper bound of and that a lower upper bound cannot exist. Therefore s {\displaystyle s} is the supremum of A {\displaystyle A} by definition. Remark
The proof can also be based on Fatou's lemma instead of a direct proof as above, because Fatou's lemma can be proved independent of the monotone convergence theorem. However the monotone convergence theorem is in some ways more primitive than Fatou's lemma.