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<\/div><\/p>\nSoftmax\u56de\u5f52\u8be6\u89e3<\/h1>\n\u8ba9\u6b65\u6bd4<\/h2>\n
\u7531\u4e8esoftmax\u56de\u5f52\u66f4\u591a\u7684\u662f\u903b\u8f91\u56de\u5f52\u7684\u591a\u5206\u7c7b\u5f62\u5f0f\uff0c\u6b64\u5904\u53ea\u7ed9\u51fasoftmax\u7684\u5b9a\u4e49\u53ca\u516c\u5f0f\u3002
\n \u8ba9\u6b65\u6bd4\u53ef\u4ee5\u7406\u89e3\u6210\u6709\u5229\u4e8e\u67d0\u4e00\u7279\u5b9a\u4e8b\u4ef6\u7684\u6982\u7387\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4e3a
\n$$
\n{\\frac{p}{1-p}}
\n$$
\n \u5728\u5df2\u77e5\u4e8c\u5206\u7c7b\u95ee\u9898\u7684\u60c5\u51b5\u4e0b\u6bcf\u4e2a\u5206\u7c7b\u7684\u6982\u7387\u5206\u522b\u4e3a$\\hat{y_i}$\u548c$1-\\hat{y_i}$\uff0c\u53ef\u4ee5\u5b9a\u4e49logit\u51fd\u6570\uff0c\u5373\u8ba9\u6b65\u6bd4\u7684\u5bf9\u6570\u5f62\u5f0f\uff08log-odds\uff09\u4e3a
\n$$
\n\\begin{align}
\n\\log{it}(\\hat{y_i}) & = \\log{\\frac{p(y=1|x,\\omega)}{p(y=0|x,\\omega)}} \\
\n& = \\log{\\frac{\\hat{y_i}}{1-\\hat{y_i}}} \\
\n& = \\log{\\frac{{\\frac{1}{1+e^{-\\omega^Tx}}}}{{\\frac{-\\omega^Tx}{1+e^{-\\omega^Tx}}}}} \\
\n& = \\omega^Tx
\n\\end{align}
\n$$
\n\u5176\u4e2d$\\log{it}(p)$\u51fd\u6570\u7b49\u4e8e\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u9664\u4ee5\u4e0d\u53d1\u751f\u7684\u6982\u7387\u53d6\u5bf9\u6570\uff0c\u5373\u8868\u793a\u7279\u5f81\u503c\u548c\u5bf9\u6570\u6982\u7387\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u3002<\/p>\n
\u4e0d\u540c\u7c7b\u4e4b\u95f4\u7684\u6982\u7387\u5206\u5e03<\/h2>\n
\u73b0\u5728\u5047\u8bbe\u6709\u4e00\u4e2a$k$\u5143\u5206\u7c7b\u6a21\u578b\uff0c\u5373\u6837\u672c\u7684\u8f93\u51fa\u503c\u4e3a$c_1,c_2,\\ldots,c_k$\uff0c\u5bf9\u4e8e\u67d0\u4e00\u4e2a\u5b9e\u4f8b\u9884\u6d4b\u4e3a$ci$\u6837\u672c\u7684\u6982\u7387\u603b\u548c\u4e3a$1$\uff0c\u5373 \u901a\u8fc7$p(y=k|x,\\omega)$\u5373\u53ef\u63a8\u51fa$p(y=j|x,\\omega)={\\frac{e^{{\\omegaj^T}x}}{1+\\sum<\/em>{t=1}^{k-1} e^{{\\omega_t^T}x}}} \\quad j=1,2,\\ldots,k-1$\uff0c\u56e0\u6b64\u53ef\u4ee5\u5f97\u5230$k$\u5143\u5206\u7c7b\u6a21\u578b\u7684$k$\u4e2a\u7c7b\u7684\u6982\u7387\u5206\u5e03\u4e3a <\/p> \u8054\u7cfb\u7ba1\u7406\u5458\u5fae\u4fe1tutu19192010\uff0c\u6ce8\u518c\u8d26\u53f7<\/p>\n<\/div>
\n$$
\n\\sum<\/em>{i=1}^k p(y=i|x,\\omega) =1
\n$$
\n \u8be5$k$\u5143\u5206\u7c7b\u6a21\u578b\u4f9d\u636e\u8ba9\u6b65\u6bd4\u7684\u5bf9\u6570\u5f62\u5f0f\u53ef\u4ee5\u5f97\u5230
\n$$
\n\\begin{align}
\n& \\ln{\\frac{p(y=1|x,\\omega)}{p(y=k|x,\\omega)}} = {\\omega_1^T}x \\
\n& \\ln{\\frac{p(y=2|x,\\omega)}{p(y=k|x,\\omega)}} = {\\omega2^T}x \\
\n& \\cdots \\
\n& \\ln{\\frac{p(y=k-1|x,\\omega)}{p(y=k|x,\\omega)}} = {\\omega<\/em>{k-1}^T}x \\
\n& \\ln{\\frac{p(y=k|x,\\omega)}{p(y=k|x,\\omega)}} = {\\omega_{k}^T}x = 0 \\
\n\\end{align}
\n$$
\n\u901a\u8fc7\u5bf9\u4e0a\u8ff0\u516c\u5f0f\u5316\u7b80\u53ef\u5f97
\n$$
\n\\begin{align}
\n& {\\frac{p(y=1|x,\\omega)}{p(y=k|x,\\omega)}} = e^{{\\omega_1^T}x} \\
\n& {\\frac{p(y=2|x,\\omega)}{p(y=k|x,\\omega)}} = e^{{\\omega2^T}x} \\
\n& \\cdots \\
\n& {\\frac{p(y=k-1|x,\\omega)}{p(y=k|x,\\omega)}} = e^{{\\omega<\/em>{k-1}^T}x} \\
\n\\end{align}
\n$$
\n$$
\n\\begin{align}
\ne^{{\\omega_1^T}x}+e^{{\\omega1^T}x}+\\cdots+e^{{\\omega<\/em>{k-1}^T}x} & = \\sum_{i=1}^{k-1} e^{{\\omegai^T}x} \\
\n& = {\\frac{p(y=1|x,\\omega)}{p(y=k|x,\\omega)}} + {\\frac{p(y=2|x,\\omega)}{p(y=k|x,\\omega)}} + \\cdots + {\\frac{p(y=k-1|x,\\omega)}{p(y=k|x,\\omega)}} \\
\n& = {\\frac{p(y=1|x,\\omega)+p(y=2|x,\\omega)+\\cdots+p(y=k-1|x,\\omega)}{p(y=k|x,\\omega)}} \\
\n& = {\\frac{1-p(y=k|x,\\omega)}{p(y=k|x,\\omega)}} \\
\n\\end{align}
\n$$
\n\u65e2\u5f97$p(y=k|x,\\omega)={\\frac{1}{1+\\sum<\/em>{i=1}^{k-1} e^{{\\omega_i^T}x}}}$<\/p>\n
\n$$
\np(c=k|x,\\omega)=
\n\\begin{cases}
\n{\\frac{e^{{\\omegaj^T}x}}{1+\\sum<\/em>{t=1}^{k-1} e^{{\\omegat^T}x}}} \\quad j=1,2,\\ldots,k-1 \\quad if\u7c7b\u522b\u4e3a1,2,\\ldots,k-1 \\
\n{\\frac{1}{1+\\sum<\/em>{i=1}^{k-1} e^{{\\omega_i^T}x}}} \\quad if\u7c7b\u522b\u4e3ak \\
\n\\end{cases}
\n$$<\/p>\n\u76ee\u6807\u51fd\u6570<\/h2>\n
\u67e5\u770b\u66f4\u591a<\/h2>