Hello, I dont even know how to start the proof
This is true but only if I know that \(\displaystyle w = \frac{ µ_2 - µ_1 }{ ||µ_2-µ_1|| } \) if \(\displaystyle \beta = w^T \frac {µ_1 + µ_2} {2}\) from here (picture above in link) so that in my case w should be:
\(\displaystyle w = \frac {w_∆ - w_o} {||w_∆ - w_o||} = \frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T}\)
if I insert everything in \(\displaystyle w^Tx-\beta\):
\(\displaystyle (\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^Tx-(\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^T \frac {w_o + w_∆} {2} = \)
\(\displaystyle (\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)})x-(\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)}) \frac {w_o + w_∆} {2} \) and now I have the expected beta
But how to calculate this if I dont know what w is, I only have \(\displaystyle \beta = w^T \frac {w_o + w_∆} {2}\) and \(\displaystyle w = w_o - w_∆\) from the formula above
This is true but only if I know that \(\displaystyle w = \frac{ µ_2 - µ_1 }{ ||µ_2-µ_1|| } \) if \(\displaystyle \beta = w^T \frac {µ_1 + µ_2} {2}\) from here (picture above in link) so that in my case w should be:
\(\displaystyle w = \frac {w_∆ - w_o} {||w_∆ - w_o||} = \frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T}\)
if I insert everything in \(\displaystyle w^Tx-\beta\):
\(\displaystyle (\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^Tx-(\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^T \frac {w_o + w_∆} {2} = \)
\(\displaystyle (\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)})x-(\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)}) \frac {w_o + w_∆} {2} \) and now I have the expected beta
But how to calculate this if I dont know what w is, I only have \(\displaystyle \beta = w^T \frac {w_o + w_∆} {2}\) and \(\displaystyle w = w_o - w_∆\) from the formula above