可交换性不是分层模型的基本特征（至少在观察级别上不是）。它基本上是标准文献中“独立同分布”的贝叶斯类似物。它只是描述您对当前情况的了解的一种方式。这就是“洗牌”不会改变你的问题。我想这样想的一种方法是考虑给你的情况$x_{j}=5$但你没有被告知价值$j$. 如果学那个$x_{j}=5$会导致您怀疑$j$比其他人多，那么序列是不可交换的。如果它什么也没告诉你$j$，则序列是可交换的。请注意，可交换性是“在信息中”而不是“在现实中”——这取决于你所知道的。

虽然就观察到的变量而言，可交换性并不是必不可少的，但如果没有一些可交换性的概念，可能很难拟合任何模型，因为没有可交换性，您基本上没有理由将观察结果汇总在一起。所以我的猜测是，如果你在模型的某个地方没有可交换性，你的推论会弱得多。例如，考虑$x_{i}\sim N(\mu_{i},\sigma_{i})$为了$i=1,\dots,N$. 如果$x_{i}$是完全可交换的，那么这意味着$\mu_{i}=\mu$和$\sigma_{i}=\sigma$. 如果$x_{i}$有条件地交换给定$\mu_{i}$那么这意味着$\sigma_{i}=\sigma$. 如果$x_{i}$有条件地交换给定$\sigma_{i}$那么这意味着$\mu_{i}=\mu$. 但请注意，在这两种“有条件可交换”情况中的任何一种情况下，与第一种情况相比，推理的质量都会降低，因为有一个额外的$N$引入问题的参数。如果我们没有可交换性，那么我们基本上有$N$无关的问题。

基本上可交换性意味着我们可以做出推断$x_{i}\to \text{parameters}\to x_{j}$对于任何$i$和$j$可部分交换

“基本”太模糊了。但是压制技术细节，如果序列$X=\{X_i\}$是可交换的$X_i$给定一些未观察到的参数是条件独立的$\Theta$具有概率分布$\pi$. That is, $p(X) = \int p(X_i|\Theta)d\pi(\Theta)$. $\Theta$ needn't be univariate or even finite dimensional and may be further represented as a mixture, etc.

Exchangability is essential in the sense that these conditional independence relationships allow us to fit models we almost certainly couldn't otherwise.

It isn't! I'm no expert here, but i'll give my two cents. In general when you have a hierarchical model, say

$y|\Theta_{1} \sim \text{N}(X\Theta_{1},\sigma^2)$

$\Theta_{1}|\Theta_{2} \sim\text{N}(W\Theta_{2},\sigma^2)$

We make conditional independence assumptions, i.e., conditional on $\Theta_{2}$, the $\Theta_{1}$ are exchangeable. If the second level is not exchangeable, than you can incluce another level that makes it exchangeable. But even in the case that you can't make an assumption of exchaganbelity, the model may still be a good fit to your data at the first level.

Last, but not least, exchangeability is important only if you wanna think in terms of De Finetti's representation theorem. You might just think that priors are regularization tools that help you to fit your model. In this case, the exchangeability assumption is as good as it is your model fit to the data. In other words, if you think of Bayesian hierarchical model as way to get abetter fit to your data, then exchangeability is not essential in any sense.