Hypothesis in epistemological philosophy: Radical Probabilism
Radical probabilism stands as a rather inconvenient hypothesis within the venerable halls of philosophy, specifically carving out a contentious niche in epistemology and the often-misunderstood domain of probability theory. Its central, and frankly rather unsettling, tenet is the assertion that absolutely no facts can ever be known with absolute certainty. This isn't a mere academic quibble; such a view carries profound, almost unsettling, implications for the entire edifice of statistical inference, threatening to dissolve the very ground upon which we imagine our certainties are built. The philosophy is most closely and famously associated with the late Richard Jeffrey, who, with a characteristic blend of intellectual rigor and dry wit, encapsulated this discomfiting notion with the memorable dictum: "It's probabilities all the way down." A sentiment that, if you truly grasp it, tends to make one feel a little less stable than before.
Background
Main article: Subjective probability
Before one can truly appreciate the radical implications of probabilism, it's perhaps necessary to briefly revisit the bedrock upon which much of this discussion rests: Bayes' theorem. This elegant mathematical statement provides a formal rule for updating a probability in light of new, relevant information. It quantifies how the degree of belief in a hypothesis changes as evidence accumulates. However, in 1967, the philosopher and historian of science Ian Hacking meticulously argued that, in its most commonly understood static form, Bayes' theorem merely establishes a relationship between probabilities held simultaneously at a single point in time. It describes how one could relate a prior probability to a posterior probability given certain conditional information. What it does not inherently dictate, Hacking pointed out, is how a learner should dynamically update their probabilities when genuinely new evidence becomes available and observed over a temporal sequence, contrary to what many contemporary Bayesians at the time were suggesting or implicitly assuming. This distinction is crucial; one is a logical relation, the other an instruction for action in the face of evolving knowledge.
According to Hacking's incisive critique, there exists a compelling, almost irresistible, temptation for a learner to adopt Bayes' theorem by analogy when confronted with new information. Imagine, for a moment, a learner who has initially formed a set of probabilities, such as P old ( A & B ) = p and P old ( B ) = q. If this learner subsequently acquires the knowledge, with supposed certainty, that B is, in fact, true, then nothing within the fundamental axioms of probability or any of the logical results derived directly from them explicitly dictates how this individual should adjust their beliefs. The impulse, the "temptation" as Hacking called it, is to directly apply the conditional probability formula and set their P new ( A ) = P old ( A | B ) = p / q. This step, often referred to as Bayes' rule of updating or simple conditioning, seems intuitively correct, a natural progression of rational belief.
In actuality, this specific step—the adoption of Bayes' rule for updating beliefs in a dynamic context—can indeed be justified. This justification doesn't spring solely from the foundational axioms of probability but rather from an additional argument, specifically a dynamic Dutch book argument. A Dutch book is a scenario where a set of bets, regardless of the outcome of the events, guarantees a net loss for the bettor. A dynamic Dutch book argument extends this concept over time, demonstrating that an agent who fails to update their probabilities according to Bayes' rule, given certain knowledge, could be subject to a series of bets that would guarantee their financial ruin. This argument posits that adherence to Bayesian updating is not merely rational but a necessary condition for avoiding such an exploitative situation. This powerful justification was first articulated by David Lewis in the 1970s, though, with typical philosophical irony, he never formally published the full argument himself.
However, even this seemingly robust defense has not escaped the relentless scrutiny of the philosophical community. The dynamic Dutch book argument for Bayesian updating has faced significant criticisms from a cohort of distinguished thinkers. Ian Hacking himself, who initiated much of this discourse, questioned its universal applicability [1]. Henry E. Kyburg Jr. raised concerns about the idealizations inherent in the argument [3]. David Christensen probed the assumptions concerning "clever bookies" and the nature of coherent beliefs [4]. And Patrick Maher offered his own detailed objections, particularly regarding the concept of diachronic rationality [5] [6]. Despite these criticisms, the argument has found staunch defenders, notably Brian Skyrms, who has vigorously defended its coherence and implications [2]. The debate underscores a fundamental tension: Is rationality merely about internal consistency at a point in time, or does it also demand a specific, prescribed method for evolving beliefs over time?
Certain and uncertain knowledge
The preceding discussion largely operates under the assumption that the "new data" or "evidence" that triggers an update in belief is itself known with absolute certainty. This assumption, however, brings us to a critical juncture. The eminent philosopher C. I. Lewis famously posited that "If anything is to be probable then something must be certain" [7]. On Lewis's account, and indeed on many foundationalist epistemological views, there must exist some bedrock of certain facts, some incorrigible truths, upon which all other probabilities and beliefs are ultimately conditioned. Without such a foundation, the entire structure of knowledge, particularly probable knowledge, would, he argued, simply float unmoored. It's an understandable human desire, this search for an absolute anchor in a sea of uncertainty.
Yet, this very notion of absolute certainty in empirical matters is directly challenged by a principle known as Cromwell's rule. This rule, often attributed to Oliver Cromwell's admonition to "think it possible you may be mistaken," declares that, with the sole exception of purely logical laws (and even that is debatable for some), one should never assign a probability of 0 or 1 to any empirical proposition. To do so would be to effectively close one's mind to any future evidence, no matter how compelling, that might contradict that proposition. It would be to declare absolute certainty in a world where experience constantly teaches us the limits of our knowledge.
It was precisely this Lewisian dictum, this insistence on a foundation of certainty, that Richard Jeffrey famously and unequivocally rejected [8]. Jeffrey saw no need for such an unassailable bedrock. He recognized the inherent fragility of all empirical knowledge and embraced it, proposing that our beliefs, even those we hold most dear, are always, and only, probable. He later quipped, with that characteristic blend of weariness and insight, "It's probabilities all the way down," a direct and knowing reference to the "turtles all the way down" metaphor. This ancient anecdote, used to illustrate the problem of infinite regress when attempting to find an ultimate foundation for the world, perfectly captures Jeffrey's vision. There is no ultimate, certain foundation for our probabilities; there are only probabilities resting on other probabilities, indefinitely. He termed this audacious position radical probabilism [9]. It is a view that demands a certain intellectual humility, a constant readiness to acknowledge the provisional nature of all belief.
Conditioning on an uncertainty – probability kinematics
If, as radical probabilism asserts, nothing is truly certain, then the standard application of Bayes' rule, which hinges on the assumption of certain new evidence, begins to falter. In such a scenario, where the "new data" itself arrives not as an absolute truth but as a mere subjective shift in the probability of some critical fact, Bayes' rule as typically applied isn't quite able to capture the nuance. The new evidence might not have been fully anticipated, or it might even prove difficult to articulate precisely or formalize completely after the event has occurred. It's not about learning 'B is true', but about learning 'my probability for B has changed from q to q_new'. This is a more subtle, yet far more common, epistemic predicament.
Given this pervasive uncertainty, it seems eminently reasonable, as a foundational starting position for updating beliefs, to extend the logical framework of the law of total probability and apply it to the dynamic process of updating, much in the same conceptual way that Bayes' theorem itself was extended from its static form. The law of total probability allows us to calculate the total probability of an event by considering all the mutually exclusive ways it can occur. When adapting this for belief updating, we account for how a change in the probability of one event (B) affects the probability of another event (A), without requiring B to become certain.
The rule proposed for this more nuanced updating is expressed as:
P new ( A ) = P old ( A | B ) P new ( B ) + P old ( A | not- B ) P new (not- B )
This formulation essentially states that the new probability of A is a weighted average of the old conditional probabilities of A given B and A given not-B, with the weights being the new probabilities of B and not-B respectively. It acknowledges that while our belief in B has shifted, our conditional beliefs—what we would believe about A if B were true, or if B were false—remain stable, at least in this specific type of update.
Adopting such a rule for updating is sufficient to avoid the dreaded Dutch book scenario, thereby preserving a form of coherence in one's beliefs over time. However, and this is a crucial distinction, it is not strictly necessary to avoid a Dutch book [2]. This implies that there might be other coherent updating rules, or that the conditions under which this rule is applied are specific. Nevertheless, it was this rule that Richard Jeffrey passionately advocated as the appropriate method for updating beliefs under the philosophical stance of radical probabilism, and he famously christened this approach probability kinematics [10]. In recognition of his pioneering work, this method is also frequently referred to as Jeffrey conditioning. It provides a mechanism for rational belief change even when the evidence itself is probabilistic rather than absolute.
Alternatives to probability kinematics
While probability kinematics offers a compelling and coherent framework for updating beliefs in a world devoid of certainties, it is by no means the sole sufficient updating rule compatible with the tenets of radical probabilism. The philosophical landscape, ever fertile for alternative approaches, has yielded several other notable methodologies, each with its own merits and theoretical underpinnings.
One prominent alternative was championed by E. T. Jaynes in the form of his maximum entropy principle (MAXENT). Jaynes argued that when updating a probability distribution in light of new information (or constraints), one should choose the distribution that maximizes the Shannon entropy subject to those constraints. In simpler terms, MAXENT dictates that one should select the probability distribution that is "least informative" or "maximally ignorant" given the available data, thus avoiding the introduction of any additional assumptions or biases beyond what the evidence strictly compels. It's a principle of epistemic parsimony, ensuring that one only believes what one must, and nothing more.
Another significant contribution comes from Brian Skyrms with his principle of reflection. This principle essentially dictates that one's current belief about a future probability should be equal to the expected value of that future probability. More precisely, it suggests that one should regard one's future beliefs as coherent with one's present beliefs, meaning that one should not expect to rationally change one's mind without good reason. It’s a statement about the consistency of beliefs over time, connecting an agent's current credence with their anticipated future credences.
Intriguingly, it turns out that probability kinematics, for all its distinctiveness, can be demonstrated to be a special case of maximum entropy inference [11]. This means that under certain specific conditions and constraints, applying Jeffrey conditioning will yield the same result as applying the maximum entropy principle. This connection highlights a deeper mathematical and conceptual unity among these updating rules. However, it is crucial to note that maximum entropy is not a grand generalization of all such sufficient updating rules. While it encompasses probability kinematics, there are other coherent updating mechanisms that do not necessarily fall under the MAXENT umbrella, suggesting that the landscape of rational belief revision is richer and more complex than any single principle might fully capture. The ongoing exploration of these alternatives continues to refine our understanding of how rational agents should navigate a world where certainty is a luxury we simply cannot afford.