The exponential family of distributions is a broad class of probability distributions that can be written in a specific form. It includes many well-known distributions such as the normal, binomial, Poisson, and gamma distributions. For a multivariate exponential family, the probability density function (PDF) is typically expressed as:

$f(x|\theta) = h(x) \exp \left( \eta(\theta)^{T} T(x) - A(\theta) \right)$

Where:

• $x$ is the random variable (which could be multivariate),
• $\theta$ is the natural parameter,
• $\eta(\theta)$ is the canonical parameter,
• $T(x)$ is the sufficient statistic,
• $A(\theta)$ is the log-partition function (normalizing constant),
• $h(x)$ is the base measure.

1. Expected Value of $d_j(X)$:

In this context, $d_j(X)$ could represent one of the sufficient statistics in the multivariate case. The expected value of $d_j(X)$, denoted $\mathbb{E}[d_j(X)]$, is:

$\mathbb{E}[d_j(X)] = \frac{\partial A(\theta)}{\partial \eta_j}$

This means that the expected value of a function $d_j(X)$ in the exponential family can be derived from the log-partition function $A(\theta)$.

2. Variance of $d_j(X)$:

The variance of $d_j(X)$ can be computed as the second derivative of the log-partition function $A(\theta)$ with respect to the parameter $\eta_j$:

$\text{Var}(d_j(X)) = \frac{\partial^2 A(\theta)}{\partial \eta_j^2}$

This shows that the variance is related to the curvature of the log-partition function.

3. Covariance of $d_j(X)$:

The covariance between two sufficient statistics $d_j(X)$ and $d_k(X)$ can be expressed as the mixed partial derivative of the log-partition function:

$\text{Cov}(d_j(X), d_k(X)) = \frac{\partial^2 A(\theta)}{\partial \eta_j \partial \eta_k}$

In a multivariate exponential family, this covariance matrix plays an important role in characterizing the joint distribution of the sufficient statistics.

Summary of Key Relationships:

1. $\mathbb{E}[d_j(X)] = \frac{\partial A(\theta)}{\partial \eta_j}$
2. $\text{Var}(d_j(X)) = \frac{\partial^2 A(\theta)}{\partial \eta_j^2}$
3. $\text{Cov}(d_j(X), d_k(X)) = \frac{\partial^2 A(\theta)}{\partial \eta_j \partial \eta_k}$

These derivatives of the log-partition function $A(\theta)$ provide essential information about the mean, variance, and covariance structure of the distribution in the exponential family.

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Would you like more details on how these concepts apply in specific distributions, or do you have further questions on this? Here are some additional questions you might explore:

1. How do these concepts apply to the Gaussian distribution in the exponential family?
2. What is the role of the Fisher information matrix in the context of covariance?
3. Can you derive these properties for a multivariate Poisson distribution?
4. How is the log-partition function related to moment-generating functions?
5. What are the implications of these results in Bayesian statistics?

Tip: Understanding how the log-partition function $A(\theta)$ governs both the mean and variance can give you insights into how distributions behave under parameter changes.