{"id":2104,"date":"2023-04-01T22:18:09","date_gmt":"2023-04-01T14:18:09","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2104"},"modified":"2023-04-06T13:55:54","modified_gmt":"2023-04-06T05:55:54","slug":"fundamentals-of-deep-learning-probability-theory-normal-distribution-gaussian-distribution","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/01\/fundamentals-of-deep-learning-probability-theory-normal-distribution-gaussian-distribution\/","title":{"rendered":"\u6df1\u5ea6\u5b66\u4e60\u57fa\u7840\uff1a\u6982\u7387\u8bba(1)_\u6b63\u6001\u5206\u5e03(\u9ad8\u65af\u5206\u5e03)"},"content":{"rendered":"<h2>\u6b63\u6001\u5206\u5e03<\/h2>\n<h3>\u5b9a\u4e49<\/h3>\n<p>$$<br \/>\nN(x; \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}<br \/>\n$$<\/p>\n<p><!-- more --><\/p>\n<p>\u5176\u4e2d$\\mu$\u662f\u5206\u5e03\u7684\u5747\u503c\uff0c\u5373$E(x) = \\mu$\uff1b$\\sigma$\u662f\u6807\u51c6\u5dee\uff0c$\\sigma^2$\u662f\u65b9\u5dee\uff0c\u5373$D(x) = \\sigma$\u3002<\/p>\n<h3>\u56fe\u50cf<\/h3>\n<p>\u5728\u6b63\u6001\u5206\u5e03\u4e2d\uff0c\u53c2\u6570$\\mu$\u662f\u7528\u6765\u63a7\u5236\u4e2d\u5fc3\u5cf0\u503c\uff0c\u800c$\\sigma^2$\u662f\u7528\u6765\u63a7\u5236\u5cf0\u7684\u9661\u5ced\u7a0b\u5ea6\uff0c\u5176\u4e2d$\\sigma^2$\u8d8a\u5c0f\uff0c\u5cf0\u8d8a\u9661\u5ced\u3002<\/p>\n<h3>\u66f4\u9ad8\u6548\u7684\u53c2\u6570\u5316\u5206\u5e03<\/h3>\n<p>\u4e3a\u4e86\u65b9\u4fbf\u5bf9\u4e0d\u540c\u53c2\u6570\u4e0b\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6c42\u503c\uff0c\u901a\u5e38\u4f7f\u7528\u53c2\u6570$\\beta = \\frac{1}{\\sigma^2}$\u6765\u66ff\u6362$\\sigma^2$\uff0c\u901a\u5e38$\\beta$\u79f0\u4e3a<strong>\u7cbe\u5ea6<\/strong>\uff0c\u90a3\u4e48\u6b63\u6001\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p>$$<br \/>\nN(x; \\mu, \\beta^{-1}) = \\frac{\\beta}{2\\pi}e^{-\\frac{\\beta (x-\\mu)^2}{2}}<br \/>\n$$<\/p>\n<h3>\u9ed8\u8ba4\u5206\u5e03\u901a\u5e38\u9009\u62e9\u6b63\u6001\u5206\u5e03\u7684\u539f\u56e0<\/h3>\n<p>\uff081\uff09\u4f9d<strong>\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406<\/strong>\uff0c\u5927\u91cf\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u7684\u548c\u670d\u4ece\u8fd1\u4f3c\u6b63\u6001\u5206\u5e03\u3002\u56e0\u6b64\uff0c\u5b9e\u9645\u4e2d\u5f88\u591a\u590d\u6742\u60c5\u51b5\u4e0b\u53ef\u4ee5\u88ab\u5efa\u6a21\u6210\u6b63\u6001\u5206\u5e03\u3002<br \/>\n\uff082\uff09\u5728\u5177\u6709\u76f8\u540c\u65b9\u5dee\u7684\u6240\u6709\u53ef\u80fd\u7684\u5206\u5e03\u4e2d\uff0c\u6b63\u6001\u5206\u5e03\u5177\u6709\u6700\u5927\u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u4e5f\u5c31\u662f<strong>\u71b5<\/strong>\u6700\u5927\u3002<\/p>\n<h2>\u591a\u7ef4\u6b63\u6001\u5206\u5e03<\/h2>\n<h3>\u5b9a\u4e49<\/h3>\n<p>\u6b63\u6001\u5206\u5e03\u53ef\u4ee5\u63a8\u5e7f\u5230${\\Bbb {R}}^n$\u7a7a\u95f4\uff0c\u5176\u5206\u5e03\u51fd\u6570\uff1a<\/p>\n<p>$$<br \/>\nN(x; \\mu, \\Sigma) = \\frac{1}{\\sqrt {(2\\pi)^n|\\Sigma|}}e^{-\\frac{1}{2}(x-\\mu)^T\\Sigma^{-1} \\, (x-\\mu)}<br \/>\n$$<\/p>\n<p>\u5176\u4e2d\u53c2\u6570$\\mu$\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u8868\u793a\u5206\u5e03\u7684\u5747\u503c\uff1b\u53c2\u6570$\\Sigma$\u662f\u4e00\u4e2a\u534f\u65b9\u5dee\u77e9\u9635\u3002<\/p>\n<h3>\u66f4\u9ad8\u6548\u7684\u53c2\u6570\u5316\u5206\u5e03<\/h3>\n<p>\u4ee4\u7cbe\u5ea6\u77e9\u9635$\\beta = \\Sigma^{-1}$\uff0c\u5206\u5e03\u51fd\u6570\u53ef\u4ee5\u8868\u793a\u4e3a<\/p>\n<p>$$<br \/>\nN(x; \\mu, \\Sigma) = \\sqrt {\\frac{|\\beta|}{(2\\pi)^n}}e^{-\\frac{1}{2}(x-\\mu)^T\\beta(x-\\mu)}<br \/>\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6b63\u6001\u5206\u5e03 \u5b9a\u4e49 $$ N(x; \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\si [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[527],"tags":[532],"class_list":["post-2104","post","type-post","status-publish","format-standard","hentry","category-ai","tag-532"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2104","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2104"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2104\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}