What is Hume's problem" or the "problem of induction"?

 The first problem for the inductivist theory intended to justify unbiversal laws by reasoning from particular
observational evidence is known as "Hume's problem" or simply "the problem of induction."  Ever since
David Hume's analysis of synthetic statements (which he called "judgments of matters of fact"), the hopes
for such an inductivist "logic" have been generally frustrated.

 Hume argued, in effect, that our belief in any "general statement" purporting to refer to all members of one class as belonging to another class, i.e. of the logical form "All A are B." derives from sensory experiences of one set of sensations temporally followed by another set of sensations (Hume called them "impressions").  However, no amount of such past evidence can ever establish the truth of a claim that these two sets of sensations are necessarily connected, for that would apply to future, as yet unobserved, sensations.  Nevertheless, on the basis of such experience, we certainly do come to believe that such regularities will persist into the future.  Hume gave the origin of this belief a psychological explanation which expressed as a "habit" of expecting one set of sensations
when one has other sets of sensations which we form because of repeated experience of the two sets
temporally conjoined in the past.  Thus we come to believe in "the principle of the uniformity of nature" (i.e. that the future will be like the past) for psychological, rather than logical, reasons.

For further analysis of Hume's argument click HERE.

     Most contemporary philosophers would concede there is no way to prove a non-trivial general
empirical statement inductively from any finite set of particular observation statements.  The last
nineteenth century inductivist J.S.Mill, thought he succeeded, but while "Mill's methods" or "canons" of
inductive reasoning are useful techniques for arriving at generalizations from particulars, they fail to
establish even any probability of the truth of such generalizations, much less their necessary truth.

     For this reason the defense of inductivism today has abandoned the traditional "quest for
certainty" and embraced "probability" as the grounds for accepting explanations based on laws inductively
justified by appeal to observational evidence.  Contemporary research in inductive logic no longer
searches for rules by which a general conclusion could be proved to be necessarily true based on
particular observation statements.  Instead it rests content with an examination of the probability of the
truth of a conclusion, i.e., of a "law" or theory, given a certain set of premises, i.e., a body of statements
expressing the observational evidence.
 Note: Such a "logic of inductive probability" of the truth of a conclusion based on a body of evidence is quite distinct from the mathematical theory of probability which allows one to deduce the likelihood of the occurrence of a particular event given a range of possible events.
 Much recent technical research in this direction makes use of the mathematical theory of probability to calculate, given a fixed range of "background knowledge" regarding a finite set of possible events, how the probability of a hypothesis is increased or decreased by occurrence or non-occurrence of a new piece of evidence.  Because this approach makes significant technical use of a theorem of probability known as "Bayes's rule," contemporary inductivist views are known as "Bayesian" views.   In particular when the scientist is employed a calculating statistical generalizations from individual cases, the theory of probability can illuminate the nature of the justificatory relationship between observation and theory.  While this certainly represents an active option in contemporary philosophy of science, for many in the field, post Kuhnian developments have turned attention away from the narrow idea that there is such a "logic of justification" and thus made this issue much less crucial to contemporary images of science than it was to holders of the consensus.
What other problem does Inductivism confront?