Difference between Mixed Logit model and hierarchical bayesian logit?

I'm studying the discrete choice analysis; The utility of person $$i$$ for alternative $$k$$ is: $$U_{ik} = \beta_kx_{ik} + \epsilon_{ik}$$ where $$\beta_k$$ is the parameter of interest and with $$\epsilon_{ik} \sim$$ extreme value.

For the mixed logit model, this specification is generalized by allowing $$\beta_k$$ to be random (follow some distribution $$f(\beta_k)$$). The utility of person $$i$$ for alternative $$k \in [K]$$ in the mixed logit model:

$$U_{ik} = \beta_{ik}x_{ik} + \epsilon_{ik}$$ with $$\epsilon_{ik} \sim$$ extreme value.

Now we can compute the probability person $$i$$ choose $$k$$ as follows:

$$\mathbb{P}(i \text{ choose } k) = \int L_i(\beta_k)f(\beta_k)d\beta_k$$

where $$L_i(\beta_k) = \frac{\text{exp}(U_{ik})}{\sum_{k'} \text{exp}(U_{ik'})}$$.

We can assume that $$\boldsymbol{\beta} \sim p(\boldsymbol{\beta} | \boldsymbol{\mu}, \Sigma)$$ where $$\boldsymbol{\beta} = (\beta_1, \dots, \beta_K) \in \mathbb{R}^K$$. And again $$\boldsymbol{\mu}, \boldsymbol{\Sigma}$$ has their own distributions and let their prior to be $$\boldsymbol{\mu} \sim p(\boldsymbol{\mu}), \boldsymbol{\Sigma} \sim p(\boldsymbol{\Sigma})$$. (I used bolded to represent the vector of size $$K$$)

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Now; I feel this is exactly the hierarchical bayesian logit model.The hierarchical structure is as follow

(1) Draw from prior, $$\boldsymbol{\mu} \sim p(\boldsymbol{\mu}), \boldsymbol{\Sigma} \sim p(\boldsymbol{\Sigma})$$.

(2) Draw $$\boldsymbol{\beta} \sim p(\boldsymbol{\beta} | \boldsymbol{\mu},\boldsymbol{\Sigma}) = \text{Normal}(\boldsymbol{\mu},\boldsymbol{\Sigma} )$$

(3 Posterior distribution would be :

$$p(\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\Sigma} | X) \propto L(\boldsymbol{\beta})p(\boldsymbol{\beta} | \boldsymbol{\mu},\boldsymbol{\Sigma})p(\boldsymbol{\mu})p(\boldsymbol{\Sigma})$$

To compute the probability formulated above

By specifying the likelihood as above,

(1) ran MCMC to get many of the samples to Monte Carlo estimate : $$\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\Sigma} \sim p(\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\Sigma} | X)$$

$$\mathbb{P}(i \text{ choose } k) = \int_{\boldsymbol{\mu}} \int_{\boldsymbol{\Sigma} } \int_{\beta_k} L_i(\beta_k)p(\boldsymbol{\beta},\boldsymbol{\mu},\boldsymbol{\Sigma} | X)$$

So; mixed logit is basically multinomial logit when you allow $$\beta$$ has its own distribution; and we can hence, build up hierarchical from there. Am I correct ?