A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.
A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a certain class. In this paper, we forge a connection between random forests and kernel regression. This places random forest probability estimation on more sound statistical footing. As part of our investigation, we develop a model for the proximity kernel and relate it to the geometry and sparsity of the estimation problem. We also provide intuition and recommendations for tuning a random forest to improve its probability estimates.
Simple diffusion processes on networks have been used to model, analyze and predict diverse phenomena such as spread of diseases, information and memes. More often than not, the underlying network data is noisy and sampled. This prompts the following natural question: how sensitive are the diffusion dynamics and subsequent conclusions to uncertainty in the network structure? In this paper, we consider two popular diffusion models: Independent cascades (IC) model and Linear threshold (LT) model. We study how the expected number of vertices that are influenced/infected, given some initial conditions, are affected by network perturbation. By rigorous analysis under the assumption of a reasonable perturbation model we establish the following main results.