# Bayesian update of a confidence interval

How does one update a confidence interval using Bayes rule?

Say, for example, an experiment shows that the mean lies in [A, B] with 95% confidence. Later, a colleague says they ran a similar experiment and found that the mean lies in [C, D] with 95% confidence (or any other CI).

How does one "merge" the two data under Bayes?

If you would know this you can calculate the means simply by: $$mean_1 = (A+B)/2$$ $$mean_2 = (C+D)/2$$ $$std_1 = (A-B)/2/1.96*\sqrt(n_1)$$ $$std_2 = (A-B)/2/1.96*\sqrt(n_2)$$
Now the new mean needs to be normalized according to observations $$mean = (mean_1*obs_1 + mean_2*obs_2)/(obs_1 + obs_2)$$ The standard deviation can now be calculated from the number of samples, the $$std = \sqrt{\frac{n_1std_1^2 + n_2std_2^2 + n1(mean_1-mean)^2 + n_2(mean_2-mean)^2} {n_1+n_2}}$$ For quick derivation (for example): Is it possible to find the combined standard deviation? And the interval $$A = mean - 1.96 * std / \sqrt(n1+n2)$$ $$B = mean + 1.96 * std / \sqrt(n1+n2)$$