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

None of the properties that OP mentions were important when the most popular exponential family distributions were discovered or were put in use. That is not to say that the properties are irrelevant or not important. These are all interesting and useful features, but these are not the reasons for popularity of the distributions. Distributions such as Gaussian and Poisson became popular because they are ubiquitous in nature, and also approximate very well processes in engineering and the industry.
Poisson.
Here's one example: Poisson distribution. It and its related distributions, such as exponential distribution, are observed in radioactive decay phenomenon. Not only it is simply convenient to approximate the observed data with them, but they arise from the theory of radioactive decay in their exact form, and the observations are consistent with equations beyond any doubt with arbitrary precision limited only by the precision of our instrumentation. This has been discovered in early 20th century and ever increasing precision of measurements has not produced any deviation from the exact equations. At this point it's simply a fact that Poisson distribution explains the number of decaying nuclei in a given period of time, and that time between two events is from exponential distribution.
Gaussian. Gaussian distribution is observed in numerous natural phenomena, e.g. Brownian motion. I gave an exotic example in my comment to the question: Heisenberg's uncertainty principle (UP) from quantum mechanics (QM).
The first thing to note here is that physicists so far failed to find any deviation from QM finding. The precision with which it's confirmed to work is mind boggling and unprecedented in any field of study in sciences. So, you can safely accept UP as a fact of nature.
In its simple form it states that the uncertainties about the measurement of the moment and coordinate of a particle simultaneously has a lower limit:$$\Delta p\Delta x\ge h/4\pi$$
If you measure the coordinate very precisely, then the moment measurement will be very imprecise etc.
But when does the inequality become equality $\Delta p\Delta x= h/4\pi$? For a Gaussian wave packet!
We don't always have a choice. In the above examples we do not pick Poisson or Gaussian because they have some nice statistical properties. We use them because we must use them. That's how nature works, and if we want to describe the observed phenomena then we don't have choice but to use these distributions regardless of their convenience or elegance.
