OH BOY I GET TO WRITE A RANT ON EPISTEMOLOGY OF SCIENCE AND INTERPRETATIONS OF STATISTICS! Eh-hem...

Anyway, Popper believed that you could never really give affirmative support to a hypothesis, only fail to falsify it. A good example of why he believed this is that, after all, Newtonian gravitational mechanics passed hundreds of experimental tests for centuries at a time, but was still not actually as true as Einstein's later Theory of General Relativity. Popper also held that the actual process of coming up with theories required a Real Human Thinker somewhere in the mix, and that the experimental procedure of forming hypotheses, testing them, and converging to truth required the magic secret sauce of Real Human Thought.

This combined very well with Fisher and Neyman-Pearson statistical testing, under which one infers, "the conditional probability (likelihood) of my evidence given absolute belief in my null hypothesis is very small", and thus treats the null hypothesis as falsified in Popperian fashion. Of course, in actual statistical testing, the critical values of the test statistic will take a particular alternate hypothesis (parameter value or distribution) into account, and thus a very low likelihood of the evidence, given the null hypothesis, is taken (note the italics: this isn't really a valid probabilistic inference) as support for the built-in alternative hypothesis.

This is, of course, complete bunk: we've done no quantitative reasoning whatsoever about the hypotheses themselves, null or alternative. Which leaves us with exactly the same problem in frequentist statistics as in Popperian philosophy of science: there's a big box marked "And then the magic of scientific thought happens, in the magical mind of an actual scientist!" Well, we might ask, if the philosophy of science is supposed to guide scientists as a genuine epistemological tool, what does Popper advise that the scientist should think and believe? And the answer is: conjecture things and test them, hoping that intuition and logic will guide the scientist to conjecture things which later turn out easy to test but difficult to falsify, this being taken as approximating eventual truth.

This gets rid of a problem that only exists in the academic study of epistemology: the Problem of Induction. But look what a price we've paid to sidestep it!

Scientific epistemology as Bayesian inference, on the other hand, has considerably fewer such problems. The two jobs remaining for the scientist are to invent coherent, testable hypotheses and to assign priors to them. Testability is given a rigorous meaning: a hypothesis is testable when it generates a non-uniform likelihood distribution over evidence. Priors are grounded in subjectivity, which sounds dirty but only really matters at small sample sizes (where frequentist inference would be weak anyway). Inductive inference is then given a rigorous meaning: model the set of hypotheses as a set of mutually exclusive propositions yielding nonuniform likelihoods over the evidence, collect data, and then use Bayes' Rule to move information from your (assumed) belief in the data to your (malleable) belief in the various hypotheses. This is rigorous probabilistic inference, since Bayes' Theorem is derivable directly from the definitions of conditional and joint probability.

(Philosophically speaking, this is also how you can escape the trap of trusting only in a priori reasoning: Bayes' Theorem and Cox's Theorem give a solid a priori argument that you will always "lose at life" if you don't reason using evidence and probability theory, compared to someone who does, therefore you really, actually should and we're not just making this up.)

And then we can do what /u/murbard2 is alluding to, and go for "hardcore mode" on Bayesianism: Objective Informative Bayesianism (better referred to as Algorithmic Bayesianism). This is the circumstance in which we explicitly treat statistical inference as a way of moving information from belief in data to belief in hypotheses (Bayesian) or from belief in hypotheses to belief in data (frequentist), treat models/hypotheses as computational objects, and treat probabilities as measuring belief in terms of information (usually measured in "decibels" if you're being casual or "bits" if you're a Real Information Theorist). Or, in fact, this approach lets us dissolve the very notion of belief: information/evidence becomes something that weighs upon a hypothesis space to locate the truth, like how mass weighs upon space-time to create gravity. You can thus view the real physical system under examination as emitting information when an experiment takes place, with some anti-information (randomness) mucking it up slightly. Your inductive process starts with a hypothesis space "weighed up" or "evened out" with anti-information (ignorance) which then comes to reflect the real world as more and more space is "weighed down" by evidence-information the world emitted. The primary problem, then, is how to assign priors: how to decide how much information must "weigh down" a particular hypothesis before it becomes "heavy" (believable with confidence).

Notably, when we phrase it this way, probabilistic phrasings of Occam's Razor become intuitive to the point of obviousness: a simpler theory will have a higher prior, meaning we need less information to make us confident in it. Thus, given a set of theories which all explain exactly the same data exactly as well (generate the same likelihoods for that data), and an assignment of priors such that simpler theories have greater priors, the theory about which we are most informed by the evidence, and in which we can be most confident, must therefore be the simplest.

(Of course, you could assign perverse priors that order your theories from the most complex to the simplest (in descending order of belief), but this just means you will require more evidence to arrive to the same answers.)

Algorithmic information theory then gives ways to formalize Occam's Razor by assigning prior probabilities to all possible Turing Machines based on their algorithmic complexity. This isn't very useful in real life, but does actually solve the Problem of Induction and give a provably optimal way to generate predictive probabilities for anything computable at all (ie: just about anything).