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

Regarding a point you raise in A: the sufficient statistic for the parameters $\beta$ of a logistic regression - that's a good question. There is no analytical solution to Logistic regression, only numerical. So, calling the final solution a function of the data $T(X,y)$ is a bit of a stretch to the definition of a function. It's true that you use the data in the fitting process - and technically you need only $X$ and $X^Ty$ - but there are other considerations: like the fitting algorithm (do you use Fisher-Scoring? Gradient Ascent? What threshold do you use? What learning step for the gradient?) Maybe others can answer if there is such a thing as a sufficient statistic for the logistic regression model.
Regarding point D: one of the main algorithms for Variational Inference (VI) / Variational Bayes - is called CAVI - Coordinate Ascent VI. It uses the "mean field" approximation (what John Madden called "Independent") $q(z)=\prod_j q_j(z_j)$. The CAVI update rule says that for each coordinate of the parameter, you update the variational distribution to be $q_j(z_j)\propto e^{\mathbb E_{-j}[\log p(z_j|z_{-j},x)]}$. This can be hard and require quite a lot of analytical work to find out what is this distribution, and what are it's parameters (take expectation on all other variational distributions, etc.). If you further assume that each "complete conditional" $p(z_j | x, z_{-j})$ is in the expo. family, the algorithm simplifies: the variational distributions have the same exponential family form as their complete-conditionals counterparts, and the update rule becomes simply updating the natural parameters of that family, $\theta_j$, by setting $\theta_j=\mathbb E_{-j}[\theta_{-j}]$ (where $\theta_{-j}$ will be a function of the conditioning set $x,z_{-j}$). For more information check sections 4.1 and 4.2 in "Variational Inference: A Review for Statisticians" here.
