# Probability Interval for F(x) with Parameter Estimates from Bayesian Analyses

Problem:

I estimated the shape $\alpha$ and scale $\lambda$ parameter of the Weibull distribution using Bayesian methods. That gave me a marginal posterior distributions for both parameters. Calculating the highest posterior density (HPD) I got two credibility intervals and a point estimate for both paramters.

Using the point estimates I can calculate the probability that a Weibull distributed variable $X$ will yield a value smaller than $x$.

$F(x) = 1 - exp(- (x/\lambda)^\alpha)$

Now I also want to create a credibility interval for that probability.

Question:

How can I calculate the credibility interval for $F(x)$?

Example

Lets say that using the HPD I got credibility intervals for $\alpha$ and $\lambda$ that are [0.52 ; 0.74] and [156.3 ; 187.8] respectively. The point estomates are 0.61 and 175.5.

Using the point estimates and $x = 25$ one gets $F(25) = 0.262584$

What I did so far:

I used the credibility intervals of the parameters and calculated $F(25)$ for all possible combinations of the interval values and point estimates.

• 0.52 & 156.3 $\rightarrow$ $F(25) = 0.3199171$
• 0.52 & 175.5 $\rightarrow$ $F(25) = 0.3044124$
• 0.52 & 178.8 $\rightarrow$ $F(25) = 0.3019741$
• ...
• 0.74 & 178.8 $\rightarrow$ $F(25) = 0.2080028$

I then used the minimum of these values as the lower probability interval and the maximum as the upper probability interval. But I don't know if that is a valid thing to do.

• What you did is definitely not a valid approach. I answered the question here: stats.stackexchange.com/questions/88515/… in a different context. – jaradniemi Feb 20 '15 at 17:13
• You cannot use point estimates as pluggins in a Bayesian approach but need to infer the posterior on $F(25)$ from the posterior on $\alpha$ and $\lambda$. – Xi'an Feb 20 '15 at 18:44
• Thanks for your input. Because I have to simulate the posterior using the R package LaplacesDemon I will add the estimation of F(x) for each iteration of $\alpha$ and $\lambda$. That should give me a distribution for the F(x) value from which I can get the credibility interval. – elevendollar Feb 22 '15 at 12:42
• I just realized that the approach is not feasible as I have to estimate $F(X)$ for 3650 values of $X$ and also calculate the credibility interval for each value of $X$. That will take way to long considering the time it already takes to simulate the posterior of $\alpha$ and $\lambda$. @Xi'an: Can you give me a hint on how to infer the posterior of $F(25)$ from the posterior of $\alpha$ and $\lambda$? – elevendollar Feb 23 '15 at 8:19