Why not use the same distribution for the prior in Bayesian statistics? I am wondering why introductory books on statistics use a conjugate distribution family for the prior instead of using the same pdf of the one we are trying to infer the parameters?
For example, the binomial distribution has the Beta distribution as the prior. They are very similar except for a normalization constant due to the Beta distribution. Why not use another binomial distribution (which can be also uninformative) as the prior?
A case where I see the conjugate prior as necessary is the case when inferring $\Sigma$ of a multivariate Normal given $\mu$ which certainly needs a distribution with symmetrical paramters (e.g. Wishart/Inverse Wishart distribution).


For example, the binomial distribution has the Beta distribution as the prior. They are very similar except for a normalization constant due to the Beta distribution. Why not use another binomial distribution (which can be also uninformative) as the prior?

Actually, it's a great counterexample. The beta-binomial model is
$$\begin{align}
p &\sim \mathsf{Beta}(\alpha, \beta) \\
X &\sim \mathsf{Bin}(p, n)
\end{align}$$
where $X$ is a discrete random variable $X \in \{ 0, 1, 2, 3, \dots , n\}$, and the binomial distribution is parametrized by probability of success $p$ and the sample size $n$. We cannot use binomial distribution as a prior for $p$, because $p$ is continuous and bounded $p \in (0, 1)$. The same applies to many other distributions, for example, a prior for the Poisson's $\lambda$ parameter would be continuous and non-negative, unlike discrete Poisson distribution.
