Disagreement is a ubiquitous feature of human life, and philosophers have dutifully attended to it. One important question related to disagreement is epistemological: How does a rational person change her beliefs (if at all) in light of disagreement from others? The typical methodology for answering this question is to endorse a steadfast or conciliatory disagreement norm (and not both) on a priori grounds and selected intuitive cases. In this paper, I argue that this methodology is misguided. Instead, a thoroughgoingly Bayesian strategy is what's needed. Such a strategy provides conciliatory norms in appropriate cases and steadfast norms in appropriate cases. I argue, further, that the few extant efforts to address disagreement in the Bayesian spirit are laudable but uncompelling. A modelling, rather than a functional, approach gets us the right norms and is highly general, allowing the epistemologist to deal with (1) multiple epistemic interlocutors, (2) epistemic superiors and inferiors (i.e. not just epistemic peers), and (3) dependence between interlocutors.

Beta retinal ganglion cells (RGCs) of the cat are classified as either on-center or off-center, according to their response to light. The cell bodies of these on- and off-center RGCs are spatially distributed into regular patterns, known as retinal mosaics. In this paper, we investigate the nature of spatial dependencies between the positioning of on- and off-center RGCs by analysing maps of RGCs and simulating these patterns. We introduce principled approaches to parameter estimation, along with likelihood-based techniques to evaluate different hypotheses. Spatial constraints between cells within-type and between-type are assumed to be controlled by two univariate interaction functions and one bivariate interaction function. By making different assumptions on the shape of the bivariate interaction function, we can compare the hypothesis of statistical independence against the alternative hypothesis of functional independence, where interactions between type are limited to preventing somal overlap. Our findings suggest that the mosaics of on- and off-center beta RGCs are likely to be generated assuming functional independence between the two types. By contrast, allowing a more general form of bivariate interaction function did not improve the likelihood of generating the observed maps. On- and off-center beta RGCs are therefore likely to be positioned subject only to homotypic constraints and the physical constraint that no two somas of opposite type can occupy the same position.

We examine how the binomial distribution B(n,p) arises as the distribution Sn = ∑i=1nXi of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which Sn follows B(n,p), given that Sn-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.