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arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = Object.assign({ h3: "h3", p: "p", table: "table", thead: "thead", tr: "tr", th: "th", tbody: "tbody", td: "td", strong: "strong", span: "span", math: "math", semantics: "semantics", mrow: "mrow", mo: "mo", msub: "msub", mi: "mi", msubsup: "msubsup", mn: "mn", annotation: "annotation", div: "div", mtext: "mtext", munderover: "munderover", mfrac: "mfrac", msqrt: "msqrt", msup: "msup", svg: "svg", path: "path", mover: "mover", mspace: "mspace", pre: "pre", code: "code", a: "a", img: "img", ul: "ul", li: "li", ol: "ol" }, _provideComponents(), props.components); return _jsxs(_Fragment, { children: [_jsx(_components.h3, { children: "はじめに" }), "\n", _jsx(_components.p, { children: "今回は、最尤推定量について掘り下げていこうと思います。" }), "\n", _jsx(_components.p, { children: "なお、筆者は統計学についてはかじった程度なので、間違い等あったら指摘していただけると助かります。" }), "\n", _jsx(_components.h3, { children: "最尤推定法とは" }), "\n", _jsx(_components.p, { children: "まず、最尤推定法について軽く説明します。" }), "\n", _jsx(_components.p, { children: "最尤推定法は確率分布のパラメータを求める方法の一種で、「ある確率分布(ただしパラメータが不明)があったとして、その確率分布から今あるデータが起こったという事象が起きているという状況から、不明なパラメータはなにであったか」を推定するものです。例を見てみましょう。" }), "\n", _jsx(_components.p, { children: "あるクラスの生徒の数学の点数は以下のようになっています。" }), "\n", _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsx(_components.th, { children: "A" }), _jsx(_components.th, { children: "B" }), _jsx(_components.th, { children: "C" }), _jsx(_components.th, { children: "D" }), _jsx(_components.th, { children: "E" }), _jsx(_components.th, { children: "F" }), _jsx(_components.th, { children: "G" }), _jsx(_components.th, { children: "H" }), _jsx(_components.th, { children: "I" }), _jsx(_components.th, { children: "J" })] }) }), _jsx(_components.tbody, { children: _jsxs(_components.tr, { children: [_jsx(_components.td, { children: "87" }), _jsx(_components.td, { children: "69" }), _jsx(_components.td, { children: "71" }), _jsx(_components.td, { children: "99" }), _jsx(_components.td, { children: "51" }), _jsx(_components.td, { children: "24" }), _jsx(_components.td, { children: "26" }), _jsx(_components.td, { children: "51" }), _jsx(_components.td, { children: "61" }), _jsx(_components.td, { children: "57" })] }) })] }), "\n", _jsx(_components.p, { children: "数学の点数は独立同分布で正規分布にしたがっていることがわかっているとき、正規分布のパラメータである平均と分散はいくつでしょうか？" }), "\n", _jsx(_components.p, { children: "最尤推定法は、**「パラメータをいろいろ変えて今あるデータである確率(連続型確率分布の場合は確率密度関数の値)を出してみて、最も高いものをパラメータの推定量としよう」**という考えのもと推定していきます。" }), "\n", _jsxs(_components.p, { children: ["当然パラメータは無限の可能性がありいちいち代入していられないので、パラメータを変数とみた関数を考え、これが最大となるパラメータを最尤推定量とします。このパラメータを変数とみた関数を", _jsx(_components.strong, { children: "尤度関数" }), "といいます。"] }), "\n", _jsxs(_components.p, { children: ["データの値を", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "{" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msubsup, { children: [_jsx(_components.mo, { stretchy: "false", children: "}" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\{x_i\\}_{i=1}^n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: 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_jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2587em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "、正規分布の平均、分散をそれぞれ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }) }), "と", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "σ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sigma" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" })] }) })] }) }), "とおくと、今回の尤度関数Lは以下のようになります。(積で書くことができるのは独立だからです。)"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "f" }), 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className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2777em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.4911em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", 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"pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["最大値を求めるには、パラメータで偏微分してあげて、0になる点を求めます。ここでは真の値と区別するため、推定する", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }) }), "と", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sigma^2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }) }), "を", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "μ" }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\mu}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t 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math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mover, { accent: "true", children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "^" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\sigma^2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0035em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "1.0035em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7401em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }), _jsxs(_components.span, { style: { top: "-3.3091em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.25em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) })] }) })] }) }), "とおきます。"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), 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_jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, 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_jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8804em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.01968em" }, children: "l" }), _jsx(_components.span, { className: "mord mathnormal", children: "o" }), _jsx(_components.span, { className: "mord mathnormal", children: "gL" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", 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_jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "これを解くと以下のようになります。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "μ" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "ˉ" })] }), _jsx(_components.mspace, { linebreak: "newline" }), _jsxs(_components.mover, { accent: "true", children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "ˉ" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\hat{\\mu} = \\frac{1}{n}\\sum_{i=1}^nx_i=\\bar{x} \\\\ \\hat{\\sigma^2} = \\frac{1}{n}\\sum_{i=1}^n(x_i-\\bar{x})^2 " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.2222em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9291em", verticalAlign: "-1.2777em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8723em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2777em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5678em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.5678em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.2222em" }, children: _jsx(_components.span, { className: "mord", children: "ˉ" }) })] })] }) }) }) })] }), _jsx(_components.span, { className: "mspace newline" }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0035em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "1.0035em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7401em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }), _jsxs(_components.span, { style: { top: "-3.3091em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.25em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }) }) }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9291em", verticalAlign: "-1.2777em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8723em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2777em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1141em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord accent", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsxs(_components.span, { className: "vlist", style: { height: "0.5678em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.2222em" }, children: _jsx(_components.span, { className: "mord", children: "ˉ" }) })] })] }) }) }) }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "したがって、このデータの平均と分散の最尤推定量は以下のように求めることができます。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\n\na = np.array([", _jsx(_components.span, { className: "hljs-number", children: "87" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "69" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "71" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "99" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "51" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "24" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "26" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "51" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "61" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "57" }), "])\nmu = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "(a) / ", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a)\nsigma2 = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "((a - mu)**", _jsx(_components.span, { className: "hljs-number", children: "2" }), ") / ", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a)\n"] }) }), "\n", _jsx(_components.p, { children: "結果は平均が59.6、分散が507.44と出ます。" }), "\n", _jsx(_components.h3, { children: "最尤推定量の性質" }), "\n", _jsx(_components.p, { children: "さて、とりあえずパラメータを推定することができました。機械学習の多くの手法では、最尤推定法によってパラメータを推定することが多いです。しかし、このパラメータの推定量、本当に正しいんでしょうか。今回のブログでは、これを掘り下げていきます。" }), "\n", _jsx(_components.h3, { children: "最尤推定量は不偏推定量であるとは限らない" }), "\n", _jsxs(_components.p, { children: ["いきなりネガティブですが、", _jsx(_components.strong, { children: "最尤推定量は不偏推定量であるとは限りません。" })] }), "\n", _jsxs(_components.p, { children: ["不偏推定量とは、「推定されたパラメータの(標本を変えることによる)平均は真のパラメータと等しい」という性質です。不偏推定量については", _jsx(_components.a, { href: "./blog2.html", children: "以前のブログ" }), "でも書いているので参考にしてください。"] }), "\n", _jsxs(_components.p, { children: ["今回の場合、正規分布のパラメータの最尤推定量は、", _jsx(_components.strong, { children: "平均は不偏推定量になりますが、分散は不偏推定量になりません。" }), "(統計学を学んでいるならば、最尤推定量の分散と不偏分散は異なることに気づくでしょう。)"] }), "\n", _jsx(_components.p, { children: "実際に異なる標本で1000回正規分布の最尤推定量を出してその平均を出してみましょう。今回は平均50、分散900(標準偏差30)の正規分布から10データ取得しています。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " pandas ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " pd \n\nmu_lists, sigma2_lists = [], []\n\n", _jsx(_components.span, { className: "hljs-keyword", children: "for" }), " i ", _jsx(_components.span, { className: "hljs-keyword", children: "in" }), " ", _jsx(_components.span, { className: "hljs-built_in", children: "range" }), "(", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "):\n np.random.seed(i) \n a = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "50" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "30" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "10" }), ") \n mu = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "(a)/", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a) \n sigma2 = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "((a-mu)**", _jsx(_components.span, { className: "hljs-number", children: "2" }), ")/", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a) \n mu_lists.append(mu) \n sigma2_lists.append(sigma2) \n \nparam_df = pd.DataFrame(", _jsx(_components.span, { className: "hljs-built_in", children: "zip" }), "(mu_lists, sigma2_lists), columns=[", _jsx(_components.span, { className: "hljs-string", children: "\"mu\"" }), ",", _jsx(_components.span, { className: "hljs-string", children: "\"sigma2\"" }), "]) \nparam_df.mean()\n"] }) }), "\n", _jsx(_components.p, { children: "結果は、muは49.788432、sigma2は815.737927と出ました。muの値は真の値(50)に近いですが、sigma2(900)は少し遠いですね。" }), "\n", _jsx(_components.p, { children: "ヒストグラムは以下のようになっています。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " matplotlib.pyplot ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " plt\n\nplt.rcParams[", _jsx(_components.span, { className: "hljs-string", children: "\"figure.figsize\"" }), "] = ", _jsx(_components.span, { className: "hljs-number", children: "15" }), ",", _jsx(_components.span, { className: "hljs-number", children: "5" }), "\nparam_df.hist()\n"] }) }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog23/pic1.png", alt: "ヒストグラム1", title: "ヒストグラム1" }) }), "\n", _jsx(_components.p, { children: "特にsigma2はばらつきが大きいですね。" }), "\n", _jsx(_components.p, { children: "ちなみに、不偏分散は次の通りです。(最尤推定量の分散にn/(n-1)をかけると不偏分散になります。)" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: ["(param_df[", _jsx(_components.span, { className: "hljs-string", children: "\"sigma2\"" }), "] * ", _jsx(_components.span, { className: "hljs-number", children: "10" }), " / ", _jsx(_components.span, { className: "hljs-number", children: "9" }), ").mean()\n"] }) }), "\n", _jsx(_components.p, { children: "結果は906.3754743659838となり、最尤推定量より近い結果になります。" }), "\n", _jsx(_components.p, { children: "つまり、最尤推定量は必ずしも(不偏推定量的な意味で)いい推定量とは言えないですね。" }), "\n", _jsx(_components.h3, { children: "最尤推定量は(ある正則条件のもと)一致推定量であり、漸近正規性をもつ" }), "\n", _jsxs(_components.p, { children: ["じゃあ最尤推定量って統計的に意味のないものなのかと言われると、それは違います。", _jsx(_components.strong, { children: "ある正則条件のもと、最尤推定量は一致推定量であり、漸近正規性を持ちます。" })] }), "\n", _jsx(_components.p, { children: "これらは漸近理論と呼ばれるもので、データ数nが大きい時になりたつ性質です。" }), "\n", _jsx(_components.p, { children: "細かくいうと、一致推定量はデータ数nを大きくしたときに真のパラメータに近づく、漸近正規性はデータ数nを大きくしたときに確率分布は正規分布に従う性質を言っています。実際に確かめてみましょう。" }), "\n", _jsx(_components.p, { children: "まずは、先ほどの最尤推定量を求めるためのデータ数を10000に増やしてヒストグラムを書いてみます。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " pandas ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " pd\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " matplotlib.pyplot ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " plt\n\nmu_lists, sigma2_lists = [], []\n\n", _jsx(_components.span, { className: "hljs-keyword", children: "for" }), " i ", _jsx(_components.span, { className: "hljs-keyword", children: "in" }), " ", _jsx(_components.span, { className: "hljs-built_in", children: "range" }), "(", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "):\n np.random.seed(i) \n a = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "50" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "30" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "10000" }), ") \n mu = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "(a) / ", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a) \n sigma2 = ", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "((a-mu)**", _jsx(_components.span, { className: "hljs-number", children: "2" }), ") / ", _jsx(_components.span, { className: "hljs-built_in", children: "len" }), "(a) \n mu_lists.append(mu) \n sigma2_lists.append(sigma2) \n \nparam_df = pd.DataFrame(", _jsx(_components.span, { className: "hljs-built_in", children: "zip" }), "(mu_lists, sigma2_lists), columns=[", _jsx(_components.span, { className: "hljs-string", children: "\"mu\"" }), ", ", _jsx(_components.span, { className: "hljs-string", children: "\"sigma2\"" }), "])\n \nplt.rcParams[", _jsx(_components.span, { className: "hljs-string", children: "\"figure.figsize\"" }), "] = ", _jsx(_components.span, { className: "hljs-number", children: "15" }), ",", _jsx(_components.span, { className: "hljs-number", children: "5" }), " \nparam_df.hist()\n"] }) }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog23/pic2.png", alt: "ヒストグラム2", title: "ヒストグラム2" }) }), "\n", _jsx(_components.p, { children: "漸近正規性より、ヒストグラムは正規分布になります。また、ここで注目したいのはsigmaです。平均が900あたりになっていますね。" }), "\n", _jsx(_components.p, { children: "データ数を大きくすれば、最尤推定量は真のパラメータに近くなるのです。" }), "\n", _jsx(_components.p, { children: "さらに、平均は49から51の間にほとんどあり、分散も860から940の間にあります。" }), "\n", _jsx(_components.p, { children: "n数を増やすことでさらに間が縮まり、やがては真のパラメータに確率収束するようになります。これが一致推定量の性質になります。" }), "\n", _jsx(_components.p, { children: "つまり、最尤推定量はデータ数が多いときには性質のいい統計量になることがわかります。" }), "\n", _jsx(_components.h3, { children: "注意！" }), "\n", _jsx(_components.p, { children: "「ある正則条件のもと」と書きましたが、複雑なモデルであると満たすことが困難になってきます。" }), "\n", _jsx(_components.p, { children: "正則条件の詳細については参考文献1を参照してください。" }), "\n", _jsx(_components.p, { children: "正規分布などのよく知られている分布であれば正則条件を満たしますが、 混合ガウス分布、潜在ディリクレモデル、隠れマルコフモデル、ニューラルネットワークモデルなどの複雑なモデルでは正則条件に当てはまらなくなってきます。(実際に正則条件が成り立つかは確認していませんが、多分成り立たないと思います。これもいつかブログ書いてみたいです。)このような複雑なモデルで統計的性質をみる場合は工夫が必要です" }), "\n", _jsxs(_components.p, { children: ["今回の記事で何が言いたかったかというと、", _jsx(_components.strong, { children: "常に最尤推定法によるパラメータ推定はいい性質を持っているとは限らないので、データ数やモデルの複雑さ等を考えて最尤推定法が使えたらいいな" }), "、ということです。"] }), "\n", _jsx(_components.h3, { children: "まとめ" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "最尤推定法とは「ある確率分布から今あるデータが起こったという事象が起きている状況から、パラメータはなにであったか」を推定する手法である。" }), "\n", _jsx(_components.li, { children: "最尤推定法では、「パラメータをいろいろ変えて今あるデータである確率(連続型確率分布の場合は確率密度関数の値)を出してみて、最も高いものをパラメータの推定量としよう」という考えのもと、推定を行う。" }), "\n", _jsx(_components.li, { children: "最尤推定量は不偏推定量とは限らない。" }), "\n", _jsx(_components.li, { children: "最尤推定量は(ある正則条件のもと)一致推定量であり、漸近正規性をもつ。" }), "\n"] }), "\n", _jsx(_components.h3, { children: "参考文献" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "野田一雄・宮岡悦良 著 数理統計学の基礎 p224-p231" }), "\n", _jsx(_components.li, { children: "Annette J. Dobson 著 田中豊・森川敏彦・山中竹春・冨田誠 訳 一般化線形モデル入門 原著 第２版" }), "\n"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = Object.assign({}, _provideComponents(), props.components); return MDXLayout ? _jsx(MDXLayout, Object.assign({}, props, { children: _jsx(_createMdxContent, props) })) : _createMdxContent(props); } return { default: MDXContent }; a:["$","div",null,{"className":"container mx-auto px-4 py-8 max-w-7xl","children":["$","div",null,{"className":"flex flex-col md:flex-row gap-8","children":[["$","article",null,{"className":"w-full md:w-[70%] bg-white shadow-lg rounded-lg p-8","children":[["$","header",null,{"className":"mb-8 border-b pb-4","children":[["$","div",null,{"className":"flex items-center gap-2 mb-2","children":[["$","$L6",null,{"href":"/","className":"text-gray-500 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