Difference between Mixed Logit model and hierarchical bayesian logit?

Question

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 ?

Answer 1

I would agree. I think the name "Hierarchical Bayes Logit" and its variants combine a model and an estimation technique which seems confusing when one starts reading about this topic.

A good starting point which discusses both ideas separately is Train (2009). Chapter 6 explains the mixed logit and how you can estimate with the traditional maximum simulated likelihood estimator.

Chapter 12 and 12.6 explain how you can use Hierarchical Bayes to estimate the parameters of a mixed logit while referencing the relevant publications from Allenby and Rossi.