From Wikipedia:
Research programmes
Lakatos's second major contribution to the philosophy of science was his model of the "research programme", which he formulated in an attempt to resolve the perceived conflict between Popper's falsificationism and the revolutionary structure of science described by Kuhn. Popper's standard of falsificationism was widely taken to imply that a theory should be abandoned as soon as any evidence appears to challenge it, while Kuhn's descriptions of scientific activity were taken to imply that science is most fruitful during periods in which popular, or "normal", theories are supported despite known anomalies. Lakatos's model of the research programme aims to combine Popper's adherence to empirical validity with Kuhn's appreciation for conventional consistency.
A Lakatosian research programme is based on a hard core of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. More modest and specific theories that are formulated in order to explain evidence that threatens the "hard core" are termed auxiliary hypotheses. Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to "protect" the "hard core". Whereas Popper was generally read as hostile toward such ad hoc theoretical amendments, Lakatos argued that they can be progressive, i.e. productive, when they enhance the programme's explanatory and/or predictive power, and that they are at least permissible until some better system of theories is devised and the research programme is replaced entirely. The difference between a progressive and a degenerative research programme lies, for Lakatos, in whether the recent changes to its auxiliary hypotheses have achieved this greater explanatory/predictive power or whether they have been made simply out of the necessity of offering some response in the face of new and troublesome evidence. A degenerative research programme indicates that a new and more progressive system of theories should be sought to replace the currently prevailing one, but until such a system of theories can be conceived of and agreed upon, abandonment of the current one would only further weaken our explanatory power and was therefore unacceptable for Lakatos. Lakatos's primary example of a research programme that had been successful in its time and then progressively replaced is that founded by Isaac Newton, with his three laws of motion forming the "hard core".
The Lakatosian research programme deliberately provides a framework within which research can be conducted on the basis of "first principles" (the "hard core"), which are shared by those involved in the research programme and accepted for the purpose of that research without further proof or debate. In this regard, it is similar to Kuhn's notion of a paradigm. Lakatos sought to replace Kuhn's paradigm, guided by an irrational "psychology of discovery", with a research programme no less coherent or consistent, yet guided by Popper's objectively valid logic of discovery.
Lakatos was following Pierre Duhem's idea that one can always protect a cherished theory (or part of one) from hostile evidence by redirecting the criticism toward other theories or parts thereof. (See Confirmation holism and Duhem–Quine thesis). This aspect of falsification had been acknowledged by Popper.
Popper's theory, falsificationism, proposed that scientists put forward theories and that nature "shouts NO" in the form of an inconsistent observation. According to Popper, it is irrational for scientists to maintain their theories in the face of nature's rejection, as Kuhn had described them doing. For Lakatos, however, "It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and nature may shout INCONSISTENT". The continued adherence to a programme's "hard core", augmented with adaptable auxiliary hypotheses, reflects Lakatos's less strict standard of falsificationism.
Could religions be viewed as Lakatosian research programmes? Rather than viewing religions as theories (as discussed in an earlier question), could we conceptualize them as having a hard core of fundamental assumptions, surrounded by a network of auxiliary hypotheses (such as the various denominations or interpretations within a religion)? If so, how might one modify a religious research programme to enhance its explanatory and/or predictive power, allowing it to be considered progressive rather than degenerative or pseudoscientific (see Imre Lakatos: Pseudoscience)?
NOTE: The inspiration to ask this question came from this answer. A quote from said answer:
Lakatos was talking about science theories, but his Research Programme thinking is very applicable to philosophic worldviews as well. Evaluating the consilience of evidences for or against a worldview, and the overall "life health" of living by a worldview, are how one would apply Lakatos' thinking to worldview competition. Note the "auxiliary assumptions" from Lakatos are the defensive memes of Memeplex theory.
The right way to do an evaluation of competing worldviews is to carefully accumulate the for and against evidences, examine the validity of protective auxiliary assumptions (many protective memes are actually fallacies), and look at the overall utility of the worldviews to live life well. Note that the two commentators agreed that Alex's worldview seemed to be effective and useful in living life, so it scored well on that metric.