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220 lines
6.6 KiB
Plaintext
220 lines
6.6 KiB
Plaintext
#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
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\lyxformat 345
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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\language english
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\quotes_language english
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\author ""
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\author ""
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\end_header
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\begin_body
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\begin_layout Standard
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Notes on mlogit.
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\end_layout
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\begin_layout Standard
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Assume that
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\begin_inset Formula $J=3$
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\end_inset
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, so that there are
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\begin_inset Formula $2$
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\end_inset
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vectors of parameters for
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\begin_inset Formula $J-1$
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\end_inset
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.
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For now the parameters are passed around as
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\begin_inset Formula \[
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\left[\beta_{1}^{\prime}\beta_{2}^{\prime}\right]\]
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\end_inset
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\end_layout
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\begin_layout Standard
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So if
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\begin_inset Formula $K=3$
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\end_inset
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(including the constant), then the matrix of parameters is
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\begin_inset Formula \[
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\left[\begin{array}{cc}
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b_{10} & b_{20}\\
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b_{11} & b_{21}\\
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b_{12} & b_{22}\end{array}\right]^{\prime}\]
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\end_inset
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\end_layout
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\begin_layout Standard
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(changed to rows and added prime above, so this all changes and the score
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is also just transposed and flattend along the zero axis.) This is flattened
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along the zero axis for the sake of the solvers.
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So that it is passed internally as
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\begin_inset Formula \[
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\left[\begin{array}{cccccc}
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b_{10} & b_{20} & b_{11} & b_{21} & b_{12} & b_{22}\end{array}\right]\]
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\end_inset
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Now the matrix of score vectors is
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\begin_inset Formula \[
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\left[\begin{array}{cc}
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\frac{\partial\ln L}{\partial b_{10}} & \frac{\partial\ln L}{\partial b_{20}}\\
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\frac{\partial\ln L}{\partial b_{11}} & \frac{\partial\ln L}{\partial b_{21}}\\
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\frac{\partial\ln L}{\partial b_{12}} & \frac{\partial\ln L}{\partial b_{22}}\end{array}\right]\]
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\end_inset
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\end_layout
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\begin_layout Standard
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In Dhrymes notation, this would be column vectors
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\begin_inset Formula $\left(\partial\ln L/\partial\beta_{j}\right)^{\prime}\text{ for }j=1,2$
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\end_inset
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in our example.
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So, our Jacobian is actually transposed vis-a-vis the more traditional
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notation.
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So that the solvers can handle this, though, it gets flattened but the
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score gets flattened along the first axis to make things easier, which
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is going to make things tricky.
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Now, in traditional notation, the Hessian would be
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\begin_inset Formula \[
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\frac{\partial^{2}\ln L}{\partial\beta_{j}\partial\beta_{j}}=\frac{\partial}{\partial\beta_{j}}\vec{\left[\left(\frac{\partial\ln L}{\partial\beta_{j}}\right)\right]}\]
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\end_inset
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\end_layout
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\begin_layout Standard
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where
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\begin_inset Formula $\vec{}$
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\end_inset
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denotes a vectorized matrix, i.e., for a
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\begin_inset Formula $n\times m$
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\end_inset
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matrix
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\begin_inset Formula $X$
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\end_inset
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,
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\begin_inset Formula $\vec{X}=\left(x_{\cdot1}^{\prime},x_{\cdot2}^{\prime},...,x_{\cdot m}^{\prime}\right)^{\prime}$
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\end_inset
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such that
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\begin_inset Formula $x_{\cdot1}$
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\end_inset
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is the first
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\begin_inset Formula $n$
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\end_inset
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elements of column 1 of
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\begin_inset Formula $X$
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\end_inset
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.
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This matrix is
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\begin_inset Formula $mn\times n$
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\end_inset
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.
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In our case
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\begin_inset Formula $\ln L$
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\end_inset
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is a scalar so
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\begin_inset Formula $m=1$
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\end_inset
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, so each second derivative is
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\begin_inset Formula $n\times n$
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\end_inset
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and
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\begin_inset Formula $n=K=3$
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\end_inset
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in our example.
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Given our score
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\begin_inset Quotes eld
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\end_inset
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matrix,
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\begin_inset Quotes erd
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\end_inset
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our Hessian will look like
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\begin_inset Formula \[
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H=\left[\begin{array}{cc}
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\frac{\partial\ln L}{\partial\beta_{1}\partial\beta_{1}} & \frac{\partial\ln L}{\partial\beta_{1}\partial\beta_{2}}\\
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\frac{\partial\ln L}{\partial\beta_{2}\partial\beta_{1}} & \frac{\partial\ln L}{\partial\beta_{2}\partial\beta_{2}}\end{array}\right]\]
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\end_inset
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\begin_inset Formula \[
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H=\left[\begin{array}{cccccc}
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\frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{10}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{11}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{12}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{20}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{21}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{22}}\\
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\frac{\partial\ln L}{\partial b_{11}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{22}}\\
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\frac{\partial\ln L}{\partial b_{12}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{22}}\\
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\frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{10}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{11}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{12}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{20}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{21}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{22}}\\
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\frac{\partial\ln L}{\partial b_{21}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{22}}\\
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\frac{\partial\ln L}{\partial b_{22}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{22}}\end{array}\right]\]
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\end_inset
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\end_layout
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\begin_layout Standard
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But since our Jacobian is a row vector that alternate
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\end_layout
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\end_body
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\end_document
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