What is a “robust” explanation in philosophy?

The phrase "robust explanation" does not have a particular definition in philosophy. One would have to ask any given speaker to clarify what they mean in the circumstance.

The ordinary connotation of "robust" though would suggest that a robust explanation is one that would withstand minor deviations of the relevant facts, but this would have to be based on some shared understanding of "minor" and "relevant" in the circumstances.

A coin landing on heads 100 times in a row might only "call out" for an explanation if one has some existing beliefs about the coin, creating the (failed) expectation that it would land on heads roughly half the time. Without such beliefs or expectations, neither explanation "calls out" for an explanation any more than the other.

What's actually going on is that we have existing beliefs (or hypotheses), based on experience, that:

• The coin is fair (based on coins usually, but not always, being fair).
• Tossing a coin is chaotic enough to result in a random distribution.

If the above were true, it would be an unlikely outcome to get heads many times in a row (we can model this via confidence intervals). So we might suppose that either or both claims are false, and we can consider whether and how competing claims may better explain the evidence (be that the coin being weighted or a coin-toss not actually being as chaotic as believed).

Though note that a coin being fair and a coin being weighted are both designed, because essentially all coins are designed. Calling the latter "rigged" suggests intent (i.e. "more" design). In the case of a coin, being weighted may be intentional (whereas being fair may merely be a side effect of design*). But this isn't, as such, a principle we can generalise to determine whether something was designed (though it may relate to other such principles). Naturally-forming things could lean towards some outcomes above others due to simple asymmetries in how those things formed. Coins are usually designed to be roughly symmetrical (fair), but any given natural forming process may or may not produce symmetry. And some natural things (e.g. physical laws or physical constants) may not have formed at all - they may just always have been that way.

* If we compare this to e.g. dice, those are usually designed specifically to be fair, since producing each side with equal probability is the main use of a dice (whereas the main use of a coin is as currency).

So, what is a "robust" explanation, then?

The concept of a "robust" explanation is not particularly robustly defined.

Though we could define it in terms of having high explanatory power, such as having greater predictive power, fewer surprising facts and fewer additional claims to retrofit the explanation to the evidence.

In the case of flipping a coin, the fair-chaotic-coin explanation would predict that we're get a roughly-even distribution between heads and tales across many flips, and it would be surprising if we get either heads or tales exclusively or many times in a row. With a weighted coin, this is less surprising and we may correctly predict this outcome going forward, if indeed it's weighted.

There may be additional claims required to explain how you ended up with a weighted coin, or a fair coin for that matter, in the first place. But both of these things are fairly common occurrences*, so neither would be too extraordinary of a claim to add to our explanation. If the existence of a weighted coin were near impossible, it may still be more likely that what actually happened was the exceedingly unlikely possibility that a fair coin landed on heads many times in a row.

_{* A coin being fair is more common than it being weighted. The former would often be the default explanation, the null hypothesis that we'd need to reject, based on the evidence, to accept that the coin is weighted. Though in certain settings, weighted coins may be more likely and there may be reasons to mitigate that possibility, e.g. a casino wouldn't let you bring your own coins (or dice, rather), because they'd lose a lot of money if it ends up being weighted. And you'd win a lot of money, which would make it more likely for someone to bring weighted dice.}

Never mind that we also need to consider the ways we could end up with a weighted coin. For a coin in particular, we generally broadly have 2 main possibilities:

• The coin was intentionally made to be weighted.
• There was an inconsistency in the production process (or being weighted was a side effect of patterns on the coin).

The former may be more likely, but neither of these are particularly extraordinary. Note that the latter doesn't entail intent (at least not in the design of the coin, specific to it being weighted).

We wouldn't say e.g. the coin being fair is unlikely, therefore it was zapped into existence by a unicorn (unless you already have strong evidence that said unicorn exists and has a habit of zapping coin-like things into existence). We wouldn't say that even if we didn't know how weighted coins could come about.

For other things for which people say "this would be unlikely, therefore this other explanation [of it being designed this way] is correct" (in particular in relation to a god's existence, i.e. fine tuning, which has other issues):

• We need to consider how extraordinary (or otherwise justified) this explanation is.
• We also need to consider competing ways that this "weighting" could've happened.

Just because someone comes up with an explanation to avoid some improbability, doesn't mean they're correct.

As a big fan of Quine’s web of belief theory, I see it this way.

Each of us has an interconnected system of beliefs, made up of links and self-reinforcing relationships. At the core of the web are self-evident, almost indubitable factual beliefs (I exist)... Moving outward toward the periphery are very well-corroborated theories (Mars and the Earth are planets), in the sense that they are not only strongly demonstrated within their own domain, but are also highly compatible with many other parts of the web of beliefs. These are very difficult to change, even though it is possible through a paradigm shift. But the effect can be potentially cascading; if I discovered that the Sun and the Earth are not planets, I would have to call into question my entire education, the reliability of everything I’ve read in school books, my map of the cosmos, and so on. At the periphery there is an mist of extremely fragile theories and beliefs, connected to the web by only a few thin links. These are very easy to change; it only takes a small shift in a variable. For example, I might think that a certain politician is competent, but I can quickly change my mind. I might think that Oppenheimer is a great movie, but a friend might point out some sharp observations that convince me it’s overrated.

Well, I’d say your example fits quite well into this framework. First of all, a theory is robust and well-reinforced when it is solidly embedded in the web of beliefs. It is justified, and in turn justifies, other nodes and networks. In your example, one of your deep beliefs is that statistically rare events may exist, but that when they occur, especially when humans tools and agency are involved, they are often not the product of pure randomness but of hidden variables/manipulation. However, you might also hold another belief that Bob is honest (and this belief too may be well justified, because maybe you know personally Bob, and have had the chance to verify many times that he is honest). So if Bob flips heads ten times in a row, in a sense you would find yourself in difficulty interpreting this new event: did he have incredible luck, or did honest Bob cheat? “maybe the coin had a tiny defect”?... maybe Bob lost his job and is desperate and forced to cheat... all of these are possible explanations, but they are not robust beliefs. Because ultimately they are conflicting beliefs. None of them would be particularly convincing, and you would be inclined to modify them. On the other hand, if Bob were a well-known dishonest person, whom you have seen lie and deceive multiple times, and he flips heads ten times in a row, suddenly the two things would strongly reinforce each other toward a new belief: there it is, Bob cheated. And it would take further and more convincing evidence to persuade you that it was actually pure chance.

The issue a about Scientific Explanation are many.

An explanation must be nomological (involving laws) and causal.

But a deduction from scientific laws is not enough: it must be produced in a relevant context.

More specifically:

(page 685) many philosophers share the basic thought that explanations should be in a certain way ‘robust’ — they should apply to a wide range of possible situations and not only the very specific situation that actually occurred.

This is knoiwn as rejection of ad hoc hypothesis: "an ad hoc hypothesis is a hypothesis added to a theory in order to save it from being falsified."

The issue is already in the title: coincidences (like miracles) are not "robust".

This is correct: they are not nomological nor causal and probably neither relevant.

See also G. Bamford, Popper's Explications of Ad Hocness: Circularity, Empirical Content, and Scientific Practice (1993).

I would say our most robust reasoning is about applying the continuous symmetries of conservation laws. This is even more robust than causal closure because it applies to quantum as well as classical phenomena.

Heads & tails of a coin have different printed volumes, so weights, similarly for faces of a die marked with different numbers of hollows; but we assume they are smaller than the σ, standard deviation, of initial conditions we would measure to predict from. The case of using lasers to predict where the ball in a roulette wheel will fall (Guardian newspaper article) from during it's spin, further illustrates the difference between idealised & practical randomness. See the 3.8 σ anomaly in Superbowl coin tosses for another practical example of the divergence between our idealised expectation that coin tosses are random, and the practical reality that highly motivated humans can systematically influence outcomes of coins they toss: Explore the intriguing trend of the Super Bowl coin flip and the NFC's unprecedented winning streak that challenges probability (link to Discover magazine article).

We use idealisations and the physics of solid bodies and statistical distributions inferred from truly random inputs acting on them, to model our randomness generators, like discs and cubes and balls spun in wheels. But if you think something is truly random, I'd say, how much would you bet..?

This is Lavarand, a wall of lava lamps used by Internet security company Cloudflare to seed random number generators for internet security cryptography. According to W3Techs, Cloudflare is used by approximately 21.3% of all websites on the Internet as of January 2026. The difference between nearly random and even a tiny bit predictable has significant implications for digital security, and they use this and also a wall of double-pendulums and a Geiger counter, to reduce issues of predictability of their strings.

See eg. Randomness and Pseudorandomness for the issues with generating relatively random strings from effectively classical systems, and the wider philosophical issues. In information theory and the physics of entropy, it may be the the case that better measurements or models of microstructures yield substantially different inferences and insights.

The author is using 'robust' in a particular way. Initially, he says that:

[for explanations to be robust[ they should apply to a wide range of possible situations and not only the very specific situation that actually occurred (pg 686)

then later he develops that into a principle of robustness, saying:

[robustness means] that explanations are better if in more of (that is, a higher proportion of) the physically possible worlds where the explanandum is true, the explanans explains the explanandum.

In both cases, robustness entails the idea that an explanation is more acceptable if it explains a wider range of phenomena, not just the particular phenomena being observed. So, to take the original example, say we see a coin flip heads fifty times in a row. Now we have a choice:

1. we can assert that this is a fair coin that just happened to flip fifty heads in a row, or…
2. we can assert that some manner of cheating is occurring.

Now in the infinite possible worlds in which we flip a fair coin 50 times, only 1 in 2⁵⁰ will produce fifty sequential heads. Therefore this explanation only works as an explanation in 1:2⁵⁰ cases. But in the infinite possible worlds in which people try to cheat, far more than 1:2⁵⁰ cases will create fifty sequential heads (because that is the intention of cheating). The 'cheating' explanation works as an explanation in more cases, therefore it is more robust than the 'fair coin' explanation.

Of course, we can bring anything into our analysis that we like, simply by modifying our set of 'possible worlds'. For instance (using your last example) when we say that Bob is disinclined to cheat, we mean that (as a matter of Bob's character) there are few possible worlds in which he would cheat. Then we have a metaphysical function: the possible worlds based on the statistics of a fair coin intersected with the (reduced) possible worlds in which Bob would cheat. This would give us a new set of possible worlds, and thus a new calculation of robustness. If in fact we can assert that there are no possible worlds in which Bob would cheat, then the 'fair coin' explanation is suddenly the most robust because we've excluded all the cases in which Bob got his results by cheating; all that's left is 'fair coin' cases.

Robustness (like statistical power) is dependent both on the assumptions we are willing to make about our explanations, and on the correctness of those assumptions.