Practical Applications of Bayesian Reliability

Appendix DJeffreys Prior

The Jeffreys prior is a non‐informative prior distribution that is invariant under transformation (reparameterization). The Jeffreys prior is proportional to the square root of the determinant of the expected Fisher Information Matrix of the selected model

where I(θ) is the expected Fisher Information Matrix, i.e.

where f(X ∣ θ) is the likelihood function of the data X given the parameter vector θ.

Let p(θ) be the Jeffreys prior and ω = h(θ) be a 1‐1 transformation for a single‐parameter case. This prior is invariant under reparameterization, which means that

The proof of the above equation is as follows.

According to chain rule, if y = f(g(x)), then

and according to the product rule,

ω = h(θ), thus θ = h⁻¹(ω). Based on the above equation (θ = h⁻¹(ω) is equivalent to g(x), and ω is equivalent to x in the above equation),

So

Since the expectation of the score ...