When is using Loss Function for posterior distribution necessary? Let $X_i\sim Poisson(\theta)$ for $i = 1, 2, ..., n.$ Let $\theta \sim Gamma(\alpha, \beta)$ be the conjugate prior distribution.
It is easy to show that the posterior
$$
\theta|\textbf{x} \sim Gamma(n\bar{x} + \alpha,\beta+n),
$$
If we already know how the posterior distribution behaves, do we need to use loss function to estimate the parameter?
Can't we use
$$
\hat{\theta} = \frac{n\bar{x} + \hat{\alpha}}{\hat{\beta}+n}.
$$
the estimated mean from the posterior distribution as the estimator for $\theta$?

To auto-quote from The Bayesian Choice (2001),

Considering that the overall purpose of most inferential studies is to
provide the statistician (or a client) with a
decision, it seems reasonable to ask for an evaluation criterion of decision procedures that assesses the consequences of each decision
and depends on the parameters of the model, i.e., the true state of
the world (or of Nature).
These decisions can be of various kinds (...) and also include
assessing whether a new scientific theory is compatible with the
experimental evidence at hand.  If no evaluation criterion is
available, it is impossible to compare different decision procedures
and absurd solutions, such as proposing $\hat\theta = 3$ for any real
estimation problem or even more dramatically the answer one wants to
impose, can only be eliminated by ad-hoc reasoning. To avoid such
reasoning implies a reinforced axiomatization of the statistical
inferential framework, called decision theory.
Although almost everybody agrees on the need for such an evaluation
criterion, there is an important controversy running about the choice
of this evaluation criterion, since the consequences on the decision
are not innocuous. This difficulty even led some statisticians to
totally reject decision theory, on the basis that a practical
determination of the decision-maker evaluation criterion is utterly
impossible in most cases.
This criterion is usually called loss and is defined as follows,
where $\mathcal D$ denotes the set of possible decisions. $\mathcal D$
is called the {\it decision} space and most theoretical examples focus
on the case $\mathcal D = \Theta$, which represents the standard
estimation setting.
Definition 2.3 A loss function is any function $\mathrm{L}$ from $\Theta \times {\mathcal D}$ in $[0,+\infty)$.
This loss function is supposed to evaluate the penalty (or error)
$\mathrm L(\theta,d)$ associated with the decision $d$ when the
parameter takes the  value $\theta$. In a traditional setting of
parameter estimation, when $\mathcal D$ is $\Theta$ or $h(\Theta)$,
the loss function $\mathrm L(\theta,\delta)$ measures the error made
in evaluating $h(\theta)$ by $\delta$. Section 2.2 introduces a set of
so-called rationality axioms that ensures the existence of such a
function in a decision setting.

Further comments link to the dual nature of loss and prior stressed by Hermann Rubin (1987):

Many criticisms have been addressed on theoretical or psychological
grounds against the notion of  rationality of the decision-makers
(...) First, it seems illusory to assume that
individuals can compare all rewards, that is, that they can provide a
total ordering of $\mathcal P$ because their
discriminating abilities are necessarily limited, especially about
contiguous or extreme alternatives.  The transitivity assumption is also too strong, since
examples in sports or politics show that real-life orderings of
preferences often lead to nontransitivity, as illustrated by Condorcet and Simpson
paradoxes (...) More fundamentally, the
assumption that the ordering can be extended (...) has
been strongly attacked because it implies that a social
ordering can be derived from a set of individual
orderings and this is not possible in general (see Arrow (1951) or Blyth (1993)). However, while
recognizing this fact, Rubin (1987) notes that this
impossibility just implies that utility and prior are not separable,
not that an optimal (Bayesian) decision cannot be
obtained, and he gives a restricted set of axioms pertaining to this
purpose. In general, the criticisms above are obviously valuable, but
cannot stand against the absolute need of an axiomatic framework
validating decision-making under  uncertainty.

