How to explain Bayesian's summary? I ran Bayesian logistic regression as below.
n=1000
set.seed(1)
df <- data.frame(smoke = sample(c(0,1), n, TRUE),
                   cancer = sample(c(0, 1), n, TRUE),  #y
                   diabetes = sample(c(0, 1), n, TRUE),
                   depression = sample(c(0, 1), n, TRUE),
                   weight = sample(40:75, n, TRUE)) 
str(df)
df$smoke <- as.factor(df$smoke)
df$cancer <- as.factor(df$cancer)
df$diabetes <- as.factor(df$diabetes)
df$depression <- as.factor(df$depression)

fit_rstanarm <- stan_glm(cancer ~ .,data = df,family = binomial(link="logit"))
summary(fit_rstanarm)


There are no convergence issues, and the mean in Estimates shows the same values as the Estimate in Coefficients when I ran logistic regression. Does this mean I can say depression is an important factor to cause cancer? I only need to find which x variables are important to y, but I do not completely understand how to explain this result.
 A: 
Does this mean I can say depression is an important factor to cause cancer?

For different reasons: no!
First: you could derive that depression and cancer have something to do with each other but you can not tell whether one of these is the cause and the other the result or whether both depend on an unnamed third cause. Depression has an estimator and the 10% to 90% credible interval does not include zero.
Second: the coefficient of depression is negative. So more depression goes along with less cancer!
Third: you will need to define what you consider "important" before calling something important. It looks like a depression reduces the risk of cancer by about .2 on the log odds scale. Is that important in your eyes?
Fourth: You know how the data was constructed as depression = sample(c(0, 1), n, TRUE) so its all random! You can learn to be more cautios with statements. After all, you looked at 80% credible intervals and you looked at 5 of those. For a more critical view consult
plot(fit_rstanarm)
summary(fit_rstanarm, probs = c( .025, .05, .5, .95, .975))

