"This is why I don't blog." —Anonymous

In an earlier post we met the $\lambda$-continuum, a generalization of Laplace’s Rule of Succession. Here is Laplace’s rule, stated in terms of flips of a coin whose bias is unknown. The Rule of Succession Given $k$ heads out of $n$ flips, the probability the next flip will land heads is $$\frac{k+1}{n+2}.$$ To generalize we introduce an adjustable parameter, $\lambda$. Intuitively $\lambda$ captures how cautious we are in drawing conclusions from the observed frequency.... Read more

The Rule of Succession gives a simple formula for “enumerative induction”: reasoning from observed instances to unobserved ones. If you’ve observed 8 ravens and they’ve all been black, how certain should you be the next raven you see will also be black? According to the Rule of Succession, 90%. In general, the probability is $(k+1)/(n+2)$ that the next observation will be positive, given $k$ positive observations out of $n$ total.... Read more

There are four ways things can turn out with two flips of a coin: $$ HH, \quad HT, \quad TH, \quad TT.$$ If we know nothing about the coin’s tendencies, we might assign equal probability to each of these four possible outcomes: $$ Pr(HH) = Pr(HT) = Pr(TH) = Pr(TT) = 1/ 4. $$ But from another point of view, there are primarily three possibilities. If we ignore order, the possible outcomes are $0$ heads, $1$ head, or $2$ heads.... Read more

The classic “Lockean” thesis about full and partial belief says full belief is rational iff strong partial belief is rational. Hannes Leitgeb’s “Humean” thesis proposes a subtler connection. $ \newcommand\p{Pr} \newcommand{\B}{\mathbf{B}} \newcommand{\given}{\mid} $ The Humean Thesis For a rational agent whose full beliefs are given by the set $\mathbf{B}$, and whose credences by the probability function $\p$: $B \in \mathbf{B}$ iff $\p(B \given A) > t$ for all $A$ consistent with $\mathbf{B}$.... Read more

If you look at the little network diagram below, you’ll probably agree that $P$ is the most “central” node in some intuitive sense. This post is about using a belief’s centrality in the web of belief to give a coherentist account of its justification. The more central a belief is, the more justified it is. But how do we quantify “centrality”? The rough idea: the more ways there are to arrive at a proposition by following inferential pathways in the web of belief, the more central it is.... Read more

Today The Open Handbook of Formal Epistemology is available for download. It’s an open access book, the first published by PhilPapers itself. (The editors are Richard Pettigrew and me.) The book features 11 outstanding entries by 11 wonderful philosophers. “Precise Credences”, by Michael G. Titelbaum “Decision Theory”, by Johanna Thoma “Imprecise Probabilities”, by Anna Mahtani “Primitive Conditional Probabilities”, by Kenny Easwaran “Infinitesimal Probabilities”, by Sylvia Wenmackers “Comparative Probabilities”, by Jason Konek “Belief Revision Theory”, by Hanti Lin “Ranking Theory”, by Franz Huber “Full & Partial Belief”, by Konstantin Genin “Doxastic Logic”, by Michael Caie “Conditionals”, by R.... Read more

Here’s a striking result that caught me off guard the other day. It came up in a facebook thread, and judging by the discussion there it caught a few other people in this neighbourhood off guard too. The short version: chances are “self-expecting” pretty much if and only if they’re “self-certain”. Less cryptically: the chance of a proposition equals its expected chance just in case the chance function assigns probability 1 to itself being the true chance function, modulo an exception to be discussed below.... Read more

Location: University of Toronto Dates: June 12–14, 2018 Keynote Speakers: Lara Buchak and Mike Titelbaum Submission Deadline: February 12, 2018 Authors Notified: March 31, 2018 We are pleased to invite papers in formal epistemology, broadly construed to include related areas of philosophy as well as cognate disciplines like statistics, psychology, economics, computer science, and mathematics. Submissions should be: prepared for anonymous review, no more than 6,000 words, accompanied by an abstract of up to 300 words, and in PDF format.... Read more

This post is coauthored with Johanna Thoma and cross-posted at Choice & Inference. Accompanying Mathematica code is available on GitHub. Lara Buchak’s Risk & Rationality advertises REU theory as able to recover the modal preferences in the Allais paradox. In our commentary we challenged this claim. We pointed out that REU theory is strictly “grand-world”, and in the grand-world setting it actually struggles with the Allais preferences. To demonstrate, we constructed a grand-world model of the Allais problem.... Read more

One of my favourite probability puzzles to teach is a close cousin of the Monty Hall problem. Originally from a 1965 book by Frederick Mosteller,1 here’s my formulation: Three prisoners, A, B, and C, are condemned to die in the morning. But the king decides in the night to pardon one of them. He makes his choice at random and communicates it to the guard, who is sworn to secrecy.... Read more