Discussion

The beta parameter from earlier

Sec. 14.3 had used a factor $\beta$ in front of the utilities, and it was said that smaller $\beta$ leads to a more random choice, while larger $\beta$ leads to a stronger preference for the best options. What happened to this $\beta$ in the theoretical treatment of this chapter?

In fact, the $\beta$ from Sec. 14.3 is related to the width parameter $\mu$ showing up in some equations of this chapter. It is however not systematically treated by this text. The reason for this is that in the maximum likelihood estimation, it does not show up as a separate variable anyway. But what is the reason for this now?

What happens here is that the maximum likelihood estimation automatically includes the meaning of the prefactor $\beta$ or $\mu$ into the other $\beta_i$. So if the theoretical form says

$$p_X \propto e^{\mu \, V_X}$$

and

$$V_X = \sum_k \beta_k \, x_{X,k} \ ,$$

then the maximum likelihood estimation in practice estimates the products

$$\tilde \beta_k := \mu \, \beta_k \ .$$

The consequence of this is that, if a set of attributes is not useful to predict the choice, then all estimated $\tilde \beta_k$ will be small, leading to quasi-random choice.

[[also: which assumptions were made? Also see in ``improvements'']]