Credible intervals

Credible interval In a Bayesian analysis, the area under the posterior distribution. Represents the degree of belief, including all past and current information,... 

Similar to a confidence interval, except that a credibility interval represents the degree of belief regarding the true value of a statistical parameter. 

The Bayesian would stress, however, that the more practical concept is that of the credible interval. This would give an interval such that (say) the treatment effect lay between these limits with 95% probability. Of course, such an interval would have to be subjective and it would not be the same for me as for you. In many cases, however, if a so-called uninformative prior is used then for many applications the credible interval corresponds closely or even exactly to the confidence interval, so that two different interpretations of the same interval (or limits) can be given. 

Credible interval. A Bayesian term. An interval based on a posterior probability distribution which has been calculated by updating an uninformative prior using the likelihood according to Bayes theorem. 

Note the interpretation of a in these models is very useful in decision making. Specifically, when fhe posterior probability that a is significantly > 0 is high, it indicates an AE that may indicate a dose-response relationship. Further evaluation of fhe potential dose-response relationship would then be warranted. On the other hand, if the posterior probability that a is significantly < 0 is high, it indicates that the particular AE decreases with decreasing dose levels. If the 95% credible interval covers 0, there is no indication of a potential dose-related AE rate. Thus, the Bayesian model allows direct evaluation of potential dose-response relationship via parameter a. 

From the posterior distribution we can deduce for example, a (1 — a). 100% credibility interval for fi and for 6. We can also compute the predictive distribution for anew measurement X, using the law of total probability ... 

This analysis provides a basis for comparing different jars. The basis is estimates of 6, credibility intervals for 6, and predictive distributions and predictions of X. 

Following the Bayesian uncertainty assessment of the expected downtimes, we report the results through credibility intervals and predictive distributions. The predictive capacity is one of the key features from Bayesian analysis, and enables us to integrate variation in data and epistemic imcertainties about unknown parameters. 

If desired, we may now express our uncertainty about this estimate by a 90% credibility interval. For details, e.g., see Rausand and Floyland (2004). 

We consider the first update (/ = 1). By the approach in step 3, we estimate the rate of DU-failures by either Xdu.i or Xdu.i- In the following, assume that we have chosen to use the Bayesian estimate Xdu.i-(If we use the empirical estimate, we get the same formulas). Next, determine the 90% confidence (or credibility) interval for Xdu.i We then calculate the ratio XDU.0/Xdu.i - This ratio indicates the fractional change in failure rate and thus the allowed change of the test interval. By using eq. (1), an updated test interval t can now be calculated as ... 

Moreover, as a consequence of working with conditional probabilities that force them to change their point of view back and forth fi om data to model, Bayesians are less prone to fall in love with their model, whichhelps to step back, discuss hypotheses and entertain the cycle of statistical analysis (Box, 1980). The possibility to take into account expert knowledge, the more natural way of interpreting probabilities, credibility intervals, statistical tests are other interesting features in practical applications. In spite ofthat, it cannot be denied that Bayesian setting is not very common in the industrial practice, at least less common than the frequentist approach. 

Uncertainty the model-based accident risk lies in a range of credible values for the risk of the real operation (e.g. a 95% credibility interval). 

For all of the parameters in the DCPN-based model and risk decomposition, a credibility interval could be determined. This gives insight into which parameter values are reasonably certain, and for which more data might need to be collected. 

DRM Method The results of the previous steps provide point estimates and credibility intervals for the probabilities of safety events in various conditions. Comparison of these results with safety criteria provides insight in risk acceptability and risk margins. Furthermore, these results can be used to identify safety bottlenecks (aspects of the operation that contribute to unacceptable risk levels) and they provide a basis to determine safety requirements and safety objectives. 

The uncertainty directly can be transferred to other measures of interest, like the Mean Time To Failure (MTTF). For example using the collection of the parameters as above, the 95% credibility interval of MTTF is [47 192] years. This interval... 

The second type of inference is where we find an interval of possible values that has a specific probability of containing the true parameter value. In the Bayesian approach, we have the posterior distribution of the parameter given the data. Hence we can calculate an interval that has the specified posterior probability of containing the random parameter 6. These are called credible intervals. 

When we want to find a (1 - a) x 100% credible interval for 9 from the posterior we are looking for an interval 9i,9u) such that the posterior probability... 

Example 4 (continued) We find the equal tail area 95% credible interval using the numerical posterior density. The lower limit is the solution of... 

We find the 95% credible interval is (—1.888, 5.299). The density with the credible interval is shown in Figure 3.4. 

Figure 3.4 The 95% credible interval for the mean with equal tail areas.

The null hypothesis is true only at the single point = do so it is called a point null hypothesis. Its negation, the alternative hypothesis is two-sided. This would be called a two-sided hypothesis test. Under the Bayesian approach with a continuous prior, the posterior will also be continuous, so the posterior probability of the point null hypothesis will always be 0. So we can t test a two-sided hypothesis by calculating the posterior probability of the null hypothesis. Clearly we can t test the "truth" of the null hypothesis. Our choice of a continuous prior means we do not believe the null hypothesis can be literally true. Instead we calculate a (1 — a) x 100% credible interval for the parameter 0. If the null value do lies in the credible interval, then we cannot reject the null hypothesis and do remains a credible value. Note we are not testing the "truth" of the null hypothesis, but rather its "credibility."... 

The random sample from the posterior can be used to calculate an equal-tail credible interval. If we had the exact posterior, we would find the value 0i and such that P d < i) = f and P 9 > = f respectively. Since we are using the random... 

Figure 3.8 The histogram of the sample from the posterior distribution showing the 95% credible interval.

Example 4 (continued) We calculate the equal tail 95% credible interval using the random sample from the unsealed posterior having shape given by... 

It is (-1.88407, 5.26849). The histogram of the random sample with the 95% credible interval are given in Figure 3.8. This is the sample analog of Figure 3.3. 

Smith and Roberts (1992) point out we can explore the posterior using exploratory data analysis techniques on the random sample from the posterior. This is statistics at the most basic level. However, the data analysis techniques are used on the sample from the posterior, not on the observed data. Inferences, including point estimates, credible intervals, and hypothesis tests, can be made from a random sample from the posterior. They are the sample analogs to the procedures we would use to make that inference from the exact numerical posterior. These inferences are approximations, since they come from a random sample the posterior. These approximations can be made as accurate as we need by taking a large enough sample size. 

Equal tail (1 — a) x 100% credible interval is an interval 0i, 6u) such that... 

There is (1 - a) posterior probability that 9 lies in the interval 9i,0u)-We can test the two sided hypothesis Hq 0 = 0o Hi 0 0q zt the a level of significance by observing whether or not the null value lies inside the (1 - q) x 100% credible interval for 0. If it does not we reject the null hypothesis at the level a. If it does, we cannot reject the null hypothesis and conclude remains a credible value. 

Equal tail (1 - a) x 100% credible interval for 0 is the interval (0i,0u) where the proportion of the posterior sample less than 9i is f and the proportion of the posterior sample is greater than is. We can test a two-sided hypothesis at the a level of significance by seeing whether the null value 9q lies inside or outside the (1 — a) x 100% credible interval for6 . 

Finding normal prior for Intercopt. The scientist would have some prior belief about the overall proportion of "success" to be expected for an "average" observation. We will find the normal bo, Sg) prior for /3q that matches the scientist s prior 95% credible interval. Suppose that, beforehand, the scientist believes with 95% probability that the proportion of "successes will be between I and u. The prior mean bo and standard deviation So will be found by solving the two simultaneous equations... 

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