Prior Predictive Elicitation for Subjective Reliabi lity Analysis

Reliability applications are notorious for their la ck of data, resulting in poor parameter estimates a nd inaccurate decisions. Subjective methods enable exp ert knowledge to enhance empirical observations. However, the problem of specifying prior distributi ons for model parameters remains unresolved. We investigate methods of prior predictive elicitation to determine hyperparameters of subjective priors that arise in reliability analysis. The distributions co 1. Methodology Inn the context of reliability analysis,, we consider stochasti c decisionn problems involving one or more random variables X with probability density (mass) functionn () θ x f .. This distributionn depends on one or more unknownn parameters θ ,, existing knowledge of which cann be expressedd by a prior density () θ g .. Any data D that become available cann be representedd by a likelihoodd functionn The Bayesiann approachh to inference thenn evaluates the poster ior density () () () θ θ ∝ θ g D L D g ; (1) for direct inference about θ from the model.. We cann alsoo use the posterior density too determi ne the posterior predictive density (mass) function () () () ∫ ∞ ∞ − θ θ θ = d D g x f D x f (2) too make direct inference about X from the model.. This posterior predictive distribution cann now be combinedd withh a suitable cost functionn too determine the posterior expectedd cost () { } () () ∫ ∞ ∞ − = dx D x f x c D x c E , (3) whichh is encounteredd inn many stochastic decisionn problems,, where th e recommendedd strategy is that whichh minimizes this expectation. The frequentist approachh dominatedd statistics inn the twentiethh century but the hypothesis tests and confidence intervals it generates have limitedd use and are of tenn misinterpreted.. Inn this context,, such analysis typically involves evaluating the approximations