Why is the exponential family so important in statistics? Why is the exponential family so important in statistics?
I was recently reading about the exponential family within statistics. As far as I understand, the exponential family refers to any probability distribution function that can be written in the following format (notice the "exponent" in this equation):

This includes common probability distribution functions such as the normal distribution, the gamma distribution, the Poisson distribution, etc. Probability distributions from the exponential family are often used as the "link function" in regression problems (e.g., in count data settings, the response variable can be related to the covariates through a Poisson distribution) - probability distribution functions that belong to the exponential family are often used due to their "desirable mathematical properties". For example, these properties are the following:

Why are these properties so important?
A) The first property is about "sufficient statistics". A "sufficient statistic" is a statistic that provides more information for any given data set/model parameter compared to any other statistic.
I am having trouble understanding why this is important. In the case of logistic regression, the logit link function is used (part of the exponential family) to link the response variable with the observed covariates. What exactly are the "statistics" in this case (e.g.. in a logistic regression model, do these "statistics" refer to the "mean" and "variance" of the beta-coefficients of the regression model)? What are the "fixed values" in this case?
B) Exponential families have conjugate priors.
In the Bayesian setting, a prior p(thetha | x) is called a conjugate prior if it is in the same family as the posterior distribution p(x | thetha). If a prior is a conjugate prior - this means that a closed form solution exists and numerical integration techniques (e.g., MCMC) are not required to sample the posterior distribution. Is this correct?
C) Is the third property essentially similar to the second property?
D) I don't understand the fourth property at all. Variational Bayes are an alternative to MCMC sampling techniques that approximate the posterior distribution with a simpler distribution - this can save computational time for high dimensional posterior distributions with big data. Does the fourth property mean that variational Bayes with conjugate priors in the exponential family have closed form solutions? So any Bayesian model that uses the exponential family does not require MCMC - is this correct?
References:

*

*Exponential family

*Sufficient statistic

*Conjugate prior

Since I do not see the result mentioned on that thread so far, let me mention an often neglected if significant issue, namely that, concerning question A, exponential families are closely linked with the notion of sufficiency due to the Pitman-Koopman-Darmois lemma:

Suppose $X_n$ , $n = 1 , 2 , 3 , \dots$ are independent identically
distributed random variables whose distribution is known to be in some
family of probability distributions with fixed support. Only if that
family is an exponential family there is a sufficient statistic
(possibly vector-valued) $\displaystyle T(X_{1},\dots ,X_{n})$ whose
number of scalar components does not increase as the sample size $n$
increases. Wikipedia

In short, there is no fixed-support family outside the exponential families that enjoys a fixed dimension sufficient statistic.
And while I am at it, point B is not a compelling argument as closed-form expressions are not a strong argument for defining a prior distribution that reflects one's prior beliefs. Furthermore, there exist exponential families whose conjugates are not manageable, witness the Beta distribution. Conjugates are thus better seen as handy approximations in the spirit of question D.
