What can we conclude from a Bayesian credible interval? I learned that the credible interval does not have the frequentist property, but recently I read the following statements that derived from the credible interval/region: 

Point (0,0) is on the edge line of the 98% credible region for the joint
  posterior density. The test for overall treatment effects is significant with p-value 0.02.

in this thesis (Page 88) and 

We also depict the upper and lower 2.5% posterior quantiles in the figure.
  From these posterior inferences, we can further identify differentially expressed proteins.
  For example, if we require the 2.5% quantile above zero or the 97.5% quantile
  below zero, there are 19 up-regulated and 7 down-regulated proteins.

in this paper (Section 4).
Are they proper? Or what is the proper way to conclude from a credible interval? Any input will be greatly appreciated.
 A: Confidence intervals can be used equivalently to hypothesis tests, but highest density intervals are not the same as confidence intervals. Let's start with what $p$-value is by quoting Cohen (1994)

What we want to know is "Given this data what is the probability that
  $H_0$ is true?" But as most of us know, what it $p$-value tells us
  is "Given that $H_0$ is true, what is the probability of this (or more
  extreme) data?" These are not the same (...)

So $p$-value tells us what is the $P(D|H_0)$. In Bayesian approach we want to learn directly (rather than indirectly) about probability of some parameter given the data that we have $P(\theta|D)$ by employing the Bayes theorem and using priors for $\theta$
$$ \underbrace{P(\theta|D)}_\text{posterior} \propto \underbrace{P(D|\theta)}_\text{likelihood} \times \underbrace{P(\theta)}_\text{prior} $$
So if 95% confidence interval does not include the null value(s), than you can reject your null hypothesis: your data is more extreme than you would expect given your hypothesis. On the other hand, if in Bayesian setting your 95% highest density interval does not include null value(s), than you can conclude that probability of observing such value(s) is less than 95%.
Kruschke (2010) can be quoted for comparison of both approaches

The primary goal of NHST [Null Hypothesis Significance Testing] is determining whether a particular "null"
  value of a parameter can be rejected. One can also ask what range of
  parameter values would not be rejected. This range of non-rejectable
  parameter values is called the confidence interval.
  (...) The
  confidence interval tells us something about the probability of
  extreme unobserved data values that we might have gotten if we
  repeated the experiment (...)
A concept in Bayesian inference, that is somewhat analogous to the
  NHST confidence interval, is the highest density interval (HDI), (...) The 95% HDI consists of those
  values of $\theta$ that have at least some minimal level of posterior
  believability, such that the total probability of all such $\theta$ values is 95%.
  (...)  The NHST
  confidence interval, on the other hand, has no direct relationship
  with what we want to know; there's no clear relationship between the
  probability of rejecting the value $\theta$ and the believability of
  $\theta$.

Posterior probability can be used and is used for testing hypothesis, but you have to remember that it provides answer for a different question than $p$-values.
See also: What is the connection between credible regions and Bayesian hypothesis tests? and Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

Cohen, J. (1994). The earth is round (p<.05). American Psychologist, 49, 997-1003.
Kruschke, J.K. (2010). Doing Bayesian Data Analysis: A Tutorial with R and BUGS. Academic Press / Elsevier.
