This is just an Excerpt from a larger document, click here to view the entire document.Normal Distribution Case
In practice, the Prior distribution f(θ) used to describe the variation pattern (or belief) of the unknown parameter θ, is often the Normal distribution. When applicable, the Normal greatly simplifies all the preceding derivations, as we will show through another example.
Assume that the unknown parameter of interest θ is now the mean of a Normal (?, σ2) population. Assume that the Prior distribution f(θ) of parameter is, itself, also Normal with mean μ and known standard deviation . Assume that we have obtained the sample average from a random sample of size "n" from the population of interest (i.e., of the Normal (θ, σ2)).
Then, the Posterior Distribution of the population mean is also the Normal, and its parameters μ* and σ*, respectively, will be given by the formulas:
Graphical depictions of the Prior and Posterior distributions θ of are shown in Figure 3.
Figure 3. Prior and Posterior Distributions for the parameter Population Mean (Click to Zoom)
Therefore, a 100(1 - α)% Bayesian confidence interval (CI) for the true mean θ (say, an 95% CI for an α = 0.05 and zα/2 = 1.96) can be constructed from the Posterior distribution of using the following equation.
μ* ± zα/2 x σ* = (μ* - zα/2 x σ* , μ* + zα/2 x σ*)
We will now illustrate the use of all this information with a practical reliability example. Assume that a manufacturer of an electronic amplifier wants to estimate the length of the life X of its device. Assume that, for historical reasons, it is believed that the distribution of the life is Normal, with unknown mean and known standard deviation σ = 150 hours.
Assume that, based on the same historical reasons, this electronic amplifier manufacturer believes that the Prior Distribution for the unknown mean life is also Normal, with known mean μ =
1000 hours and standard deviation δ = 100 hours.
Assume now that the manufacturer measures the lives from a random sample of n = 25 electronic amplifiers (if the sample size is n ≥ 30, then the distribution of life X doesn't even need to be Normal and the unknown variance σ2 can now be approximated by the sample variance s2). Assume that this sample yields an average life x = 900 hours. Using all the information, we derive a 100(1 - α)% Bayesian CI for the amplifier mean life θ, as:
μ* - zα/2 x σ* < θ < μ* + za/2 x σ*
According to the preceding, the resulting Normal Posterior Distribution will have the following parameters.
Therefore, for α = 0.05 and zα/2 = 1.96, a 95% Bayesian CI for the unknown mean life θ is:
Had we chosen to ignore (or had no access to) this information about the Prior Distribution, we could have still obtained a 95% Classic CI for θ as:
x ± zα/2 x σ / √n = 900 ± 1.96 x (150 /√25 = 9000 ± 58.8
= (841.2, 958.8)
As we have advised several times in the previous sections, if the assumed Prior Distribution is appropriate and the Prior's parameters are accurate, then the Bayesian CI is more efficient. Such efficiency becomes evident in the fact that the Bayesian CI is narrower (notice its smaller half-width, 56.32) than the Classical CI (half-width is 58.8). This reduction in the CI width occurs because the Bayesian approach uses more information than the Classical (e.g., the shape of the Prior distribution as well as its parameters).
In addition, the Bayesian point estimator (908.26) of the parameter (θ) differs from the Classical point estimator, given by the sample average (900). This difference is due to Bayes weighing the data via the Prior standard deviation and mean.
However, if any or all of the information regarding Bayes Prior (either the distribution or its parameters) which are used for deriving the Bayesian CI, is inaccurate or incorrect, then the CI obtained with such information will also be incorrect.
In such case, the Bayes point estimator (908.26) will be biased. In addition, the resulting CI, with its narrower half width and centered on this biased point estimator, will most likely not cover the true parameter.