We investigate the choice of default prior for use with likelihood to facilitate Bayesian and frequentist inference. Such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters not of interest and accordingly providing a density type assessment for a parameter of interest. For regular models with independent coordinates we develop a secondorder prior for the full parameter based on an approximate location relation from near a parameter value to near the observed data point; this derives directly from the coordinate distribution functions and is closely linked to the original Bayes approach. We then develop a modified prior that is targetted on a component parameter of interest and avoids the marginalization paradoxes of Dawid, Stone and Zidek (1973); this uses some extensions of Welch-Peers theory that modify the Jeffreys prior and builds more generally on the approximate location property. A third type of prior is then developed that targets a vector interest parameter in the presence of a vector nuisance parameter and is based more directly on the original Jeffreys approach. Examples are given to clarify the computation of the priors and the flexibility of the approach.

Bayes (1763) introduced the observed likelihood function to statistical inference and provided a weight function to calibrate the parameter; he also introduced a confidence distribution on the parameter space but restricted attention to models now called location models; of course the names likelihood and confidence did not appear until much later: Fisher (1922) for likelihood and Neyman (1937) for confidence. Lindley (1958) showed that the Bayes and the confidence results were different when the model was not location. This paper examines the occurrence of true statements from the Bayes approach and from the confidence approach, and shows that the proportion of true statements in the Bayes case depends critically on the presence of linearity in the model; and with departure from this linearity the Bayes approach can be seriously misleading. Bayesian integration of weighted likelihood provides a first order linear approximation to confidence, but without linearity can give substantially incorrect results. 1

We propose a new Metropolis-Hastings algorithm for sampling from smooth, unimodal distributions; a restriction to the method is that the target be optimizable. The method can be viewed as a mixture of two types of MCMC algorithm; specifically, we seek to combine the versatility of the random walk Metropolis and the efficiency of the independence sampler as found with various types of target distribution. This is achieved through a directional argument that allows us to adjust the thickness of the tails of the proposal density from one iteration to another. We discuss the relationship between the acceptance rate of the algorithm and its efficiency. We finally apply the method to a regression example concerning the cost of construction of nuclear power plants, and compare its performance to the random walk Metropolis algorithm with Gaussian proposal.

La revue canadienne de statistique

summary In the Bayes approach the model-data summary is arguably the observed likelihood function; in the frequentist approach it is arguably a p-value function assessing a least squares or maximum-likelihood departure; and in the higherorder likelihood approach it is the observed likelihood together with a canonical reparameterization. For likelihood the obvious method of combining is to add log-likelihoods from independent sources: this is in the nature of likelihood itself and is also an implicit Bayes imperative as only likelihood is used in the Bayesian argument. For the familiar frequentist approach the combining of p-values is often ad hoc: we discuss first a Fisher proposal and then offer a likelihood based alternative. For the higher order likelihood approach the combining begins with the standard summary, which is likelihood plus a canonical reparameterization: we develop the appropriate higher order combining procedure. For the p-value summary, Fisher (1973) proposed a quick and easy method for combining p-values from independent investigations: multiply them together and use chi-square tables.

Fisher (1948) proposed a quick and easy method for combining p-values from independent investigations: multiply them together and use chi-square tables. The proposal received criticism that it did not address power and other conventional criteria, but he had clearly assumed that the related background concerning the investigations was unavailable. First order likelihood gives a simple modification for use when observed information is available: the p-values are converted to likelihood values and the likelihood values to observed likelihood functions; in turn these are combined to give the new composite p-value. Higher order likelihood offers further refinement in the presence of a regular model and observed data where full third order inference is available using the observed log-likelihood ℓ(θ) and an observed likelihood gradient ϕ(θ) which is treated as a reparameterization. The combining of results from independent investigations then amounts to the combining of the observed log-likelihoods ℓi(θ) and the the combining of the observed reparameterizations ϕi(θ). We develop this information combining procedure which is to add log-likelihoods and to weight and add canonical parameters. Examples are given. 1

Bayesian and frequentist methodologies when applied to the same model–data information can lead to different statistical inference results. A prominent example involves a rotationally symmetric normal error distribution located at an arbitrary point (θ1,θ2) on the plane. The radial distance ρ = (θ 2 1 + θ2 2)1/2 from the origin has a Bayes posterior survival value s(ρ) that is uniformly greater than the frequentist p-value p(ρ), can be expressed in terms of the noncentral chi-square distribution function with 2 degrees of freedom, and can attain 8 percentage points when ˆρ = 5. We use this Bayes–frequentist difference as a reference to explore the Bayesian bias attributable to parameter curvature. For this, we consider a two parameter regular statistical model and define a curvature measure for an interest parameter; the curvature measure is a mofication of the Efron measure and targets Bayesian adjustment rather than departure from the information lower bound as considered by Efron. Examples are given and simulations are provided.

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