Scalable Variational Bayesian Kernel Selection for Sparse Gaussian Process Regression

This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framework for SGPR models where its posterior belief is learned together with that of the other variational variables (i.e., inducing variables and kernel hyperparameters). In particular, we transform the discrete kernel belief into a continuous parametric distribution via reparameterization in order to apply VI. Though it is computationally challenging to jointly optimize a large number of hyperparameters due to many kernels being evaluated simultaneously by our VBKS algorithm, we show that the variational lower bound of the log-marginal likelihood can be decomposed into an additive form such that each additive term depends only on a disjoint subset of the variational variables and can thus be optimized independently. Stochastic optimization is then used to maximize the variational lower bound by iteratively improving the variational approximation of the exact posterior belief via stochastic gradient ascent, which incurs constant time per iteration and hence scales to big data. We empirically evaluate the performance of our VBKS algorithm on synthetic and massive real-world datasets.

Games with Symmetric Incomplete Information and Asymmetric Computational Resources

We consider random public signals on the state of two-person zero-sum game with incomplete information on both sides (both players do not know the state of the game). To learn the state, each player chooses a finite automaton which receives the public signal; the player only sees the output of the automaton chosen. Supposing that the size of automata available to Player 1 is essentially bigger than that available to Player 2, we give an example of public signal with random length of output strings where the posterior belief of Player 1 is the state and the posterior belief of Player 2 is close to his original belief. Thus, we demonstrate that asymmetric information about the state of a game may appear not only due to a private signal but as a result of a public signal and asymmetric computational resources of players. Besides, for a class of random signals with fixed length of output strings, we estimate the fraction of signals such that some automaton of given size may help Player 2 to significantly reestimate prior probability of the state. We show that this fraction is negligible if the size of automata of Player 2 is sufficiently smaller than length of output strings.

The Effect of Medical Test to Belief Updating and Willingness to Pay for Health Insurance Premium: Evidences from Laboratory Experiment

Purpose: The objective of this study is to analyze the effect of the results of medical tests on three health indicators, i.e. blood pressure, cholesterol level, and blood glucose level, for belief updating and willingness to pay for health insurance. Specifically, this study examined whether individuals update their belief on their health status after being informed the results of their medical tests. This study also investigated whether there is a significant difference between the willingness to pay for the individuals who were informed about the results of their medical tests and of individuals who were not informed about the results of their medical tests. Approach: This study utilizes laboratory experiments. There are two groups in the experiments: the treatment group and the control group. The individuals in the treatment group receive information on the results of the medical tests which cover blood pressure, glucose level and cholesterol level tests. The individuals in the control group do not receive any information. We compare the willingness to pay between the treatment group and the control group. Results: There are significant differences in the value of willingness to pay for health insurance premium based on prior belief (individuals’ belief prior to the medical tests) and on posterior belief (individuals’ belief after the medical tests) between control group and treatment group. Belief updating occurs when there is a difference between prior belief and posterior belief due the presence of an event. Value: This work contributes to the better understanding about the individual decision making on health insurance purchase. Conclusion: The medical tests on blood pressure, cholesterol level, and glucose level significantly affect the willingness to pay for health insurance premium. There are significant changes in individual’s posterior belief due to the information provided by the medical tests. An individual’s willingness to pay for health insurance premium may change due to a change in his or her health status belief.

What is Bayesian statistics?

Bayesian statistics is included in few elementary statistics courses, and many mathematicians have heard of it, perhaps through collateral readings from popular literature or [1], selected as an Editor's Choice in the New York Times Book Review. ‘Bayesian statistics’ provides for a way to incorporate prior beliefs, experience, or information into the analysis of data. Bayesian thinking is natural, and that is an advantage. For example, on a summer morning, if we see dark rain clouds up in the sky, we leave home for work with an umbrella because prior experience tells us that doing so is beneficial. In general, the idea is simple; schematically, it looks like this:(prior belief) + (data: new information) ⇒ (posterior belief).Thus, we begin with a prior belief that we allow to be modified or informed by new data to produce a posterior belief, which then becomes our new prior, and this process is never-ending. We are always willing to update our beliefs according to new information.

EVALUATION OF THE QUANTITATIVE PREDICTION OF A TREND REVERSAL ON THE JAPANESE STOCK MARKET IN 1999

In January 1999, the authors published a quantitative prediction that the Nikkei index should recover from its 14-year low in January 1999 and reach ≈20 500 a year later. The purpose of the present paper is to evaluate the performance of this specific prediction as well as the underlying model: the forecast, performed at a time when the Nikkei was at its lowest (as we can now judge in hindsight), has correctly captured the change of trend as well as the quantitative evolution of the Nikkei index since its inception. As the change of trend from sluggish to recovery was estimated quite unlikely by many observers at that time, a Bayesian analysis shows that a skeptical (resp. neutral) Bayesian sees prior belief in our model amplified into a posterior belief 19 times larger (resp. reach the 95% level).