Characterizing a prior distribution through its Bayes estimator

Let X be a continuous real-valued random variable with a family of possible distributions indexed by the real parameter (theta)(ELEM)(THETA). For the triplet ((delta),(gamma),(pi)), let (delta) denote a real-valued function over the sample space, (gamma) a real-valued function over the parameter space (THETA), and (pi) a non-negative real-valued function over (THETA). We are interested in determining when for a given (delta) and (gamma) one can find a unique prior (pi), such that, (delta) is the Bayes estimator of (gamma) against (pi), for squared error loss. By imposing different sets of conditions on the form of (gamma) and (delta), positive answers to the question are obtained. The solutions can be considered as generalizations of the works of Diaconis and Ylvisaker (1979), Goldstein (1975), and Berger (1980). The implication of these results in the study of the admissibility of an estimator are noticed and other applications are discussed;A second area that is considered is the relationship between the Bayesness of an estimator and the unbiased property of an estimator. When the loss function is squared a dual relationship between the two is noticed. This suggest a general definition of unbiasedness for an arbitrary loss function which generalizes the notion of unbiasedness in the sense of Lehmann (1951). Some consequences of this general definition are noted;Bibliography;Berger, J. O., 1980. Statistical Decision Theory. Springer-Verlag, New York, N.Y. 425 pp;Diaconis, P., and Ylvisaker, D. 1979. Conjugate priors for exponential family. Ann. Statist. 7:269-281;Goldstein, M., 1975. Uniqueness relations for linear posterior expectations, J. Roy. Statist. Soc., Ser. B, 37:402-405;Lehmann, E. L., 1951. A general concept of unbiasedness. Ann. Math. Statist. 22:587-592

Design, recruitment outcomes, and sample characteristics of the Strategies for Prescribing Analgesics Comparative Effectiveness (SPACE) trial

This manuscript describes the study protocol, recruitment outcomes, and baseline participant characteristics for the Strategies for Prescribing Analgesics Comparative Effectiveness (SPACE) trial. SPACE is a pragmatic randomized comparative effectiveness trial conducted in multiple VA primary care clinics within one VA health care system. The objective was to compare benefits and harms of opioid therapy versus non-opioid medication therapy over 12 months among patients with moderate-to-severe chronic back pain or hip/knee osteoarthritis pain despite analgesic therapy; patients already receiving regular opioid therapy were excluded. Key design features include comparing two clinically-relevant medication interventions, pragmatic eligibility criteria, and flexible treat-to-target interventions. Screening, recruitment and study enrollment were conducted over 31 months. A total of 4491 patients were contacted for eligibility screening; 53.1% were ineligible, 41.0% refused, and 5.9% enrolled. The most common reasons for ineligibility were not meeting pain location and severity criteria. The most common study-specific reasons for refusal were preference for no opioid use and preference for no pain medications. Of 265 enrolled patients, 25 withdrew before randomization. Of 240 randomized patients, 87.9% were male, 84.1% were white, and age range was 21–80 years. Past-year mental health diagnoses were 28.3% depression, 17% anxiety, 9.4% PTSD, 7.9% alcohol use disorder, and 2.6% drug use disorder. In conclusion, although recruitment for this trial was challenging, characteristics of enrolled participants suggest we were successful in recruiting patients similar to those prescribed opioid therapy in usual care

Characterizing a prior distribution through its Bayes estimator

Let X be a continuous real-valued random variable with a family of possible distributions indexed by the real parameter (theta)(ELEM)(THETA). For the triplet ((delta),(gamma),(pi)), let (delta) denote a real-valued function over the sample space, (gamma) a real-valued function over the parameter space (THETA), and (pi) a non-negative real-valued function over (THETA). We are interested in determining when for a given (delta) and (gamma) one can find a unique prior (pi), such that, (delta) is the Bayes estimator of (gamma) against (pi), for squared error loss. By imposing different sets of conditions on the form of (gamma) and (delta), positive answers to the question are obtained. The solutions can be considered as generalizations of the works of Diaconis and Ylvisaker (1979), Goldstein (1975), and Berger (1980). The implication of these results in the study of the admissibility of an estimator are noticed and other applications are discussed;A second area that is considered is the relationship between the Bayesness of an estimator and the unbiased property of an estimator. When the loss function is squared a dual relationship between the two is noticed. This suggest a general definition of unbiasedness for an arbitrary loss function which generalizes the notion of unbiasedness in the sense of Lehmann (1951). Some consequences of this general definition are noted;Bibliography;Berger, J. O., 1980. Statistical Decision Theory. Springer-Verlag, New York, N.Y. 425 pp;Diaconis, P., and Ylvisaker, D. 1979. Conjugate priors for exponential family. Ann. Statist. 7:269-281;Goldstein, M., 1975. Uniqueness relations for linear posterior expectations, J. Roy. Statist. Soc., Ser. B, 37:402-405;Lehmann, E. L., 1951. A general concept of unbiasedness. Ann. Math. Statist. 22:587-592.</p