UNPKG

mathmlben

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Visualization for goldstandard

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<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper F left-parenthesis StartLayout 1st Row a comma 1 minus a 2nd Row c EndLayout semicolon z right-parenthesis equals left-parenthesis 1 minus z right-parenthesis Superscript c minus 1 Baseline upper F left-parenthesis StartLayout 1st Row one-half left-parenthesis c minus a right-parenthesis comma one-half left-parenthesis a plus c minus 1 right-parenthesis 2nd Row c EndLayout semicolon 4 z left-parenthesis 1 minus z right-parenthesis right-parenthesis"> <semantics> <mrow data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="88" data-semantic-children="85,87" data-semantic-content="18"> <mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-id="85" data-semantic-children="0,17" data-semantic-content="84,0" data-semantic-parent="88"> <mi data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-id="0" data-semantic-parent="85" data-semantic-operator="appl">F</mi> <mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-id="84" data-semantic-parent="85" data-semantic-added="true" data-semantic-operator="appl"></mo> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="17" data-semantic-children="15" data-semantic-content="1,16" data-semantic-parent="85"> <mo data-semantic-type="fence" data-semantic-role="open" data-semantic-id="1" data-semantic-parent="17" data-semantic-operator="fenced">(</mo> <mrow data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-id="15" data-semantic-children="12,13,14" data-semantic-content="13" data-semantic-parent="17"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0" data-semantic-type="multiline" data-semantic-role="binomial" data-semantic-id="12" data-semantic-children="10,11" data-semantic-parent="15"> <mrow data-semantic-added="true" data-semantic-type="line" data-semantic-role="binomial" data-semantic-id="10" data-semantic-children="8" data-semantic-parent="12"> <mrow data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-id="8" data-semantic-children="2,3,7" data-semantic-content="3" data-semantic-parent="10"> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="2" data-semantic-parent="8">a</mi> <mo data-semantic-type="punctuation" data-semantic-role="comma" data-semantic-id="3" data-semantic-parent="8" data-semantic-operator="punctuated">,</mo> <mrow data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="7" data-semantic-children="4,6" data-semantic-content="5" data-semantic-parent="8"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="4" data-semantic-parent="7">1</mn> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="5" 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data-semantic-parent="17" data-semantic-operator="fenced">)</mo> </mrow> </mrow> <mo data-semantic-type="relation" data-semantic-role="equality" data-semantic-id="18" data-semantic-parent="88" data-semantic-operator="relseq,=">=</mo> <mrow data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-id="87" data-semantic-children="28,83" data-semantic-content="86" data-semantic-parent="88"> <msup data-semantic-type="superscript" data-semantic-role="leftright" data-semantic-id="28" data-semantic-children="81,27" data-semantic-parent="87" data-semantic-fencepointer="23"> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="81" data-semantic-children="80" data-semantic-content="19,23" data-semantic-parent="28"> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="19" data-semantic-parent="81">(</mo> <mrow data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="80" data-semantic-children="20,22" data-semantic-content="21" data-semantic-parent="81"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="20" data-semantic-parent="80">1</mn> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="21" data-semantic-parent="80" data-semantic-operator="infixop,−"><!-- − --> </mo> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="22" data-semantic-parent="80">z</mi> </mrow> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="23" data-semantic-parent="81">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD" data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="27" data-semantic-children="24,26" data-semantic-content="25" data-semantic-parent="28"> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="24" data-semantic-parent="27">c</mi> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="25" data-semantic-parent="27" data-semantic-operator="infixop,−"><!-- − --> </mo> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="26" data-semantic-parent="27">1</mn> </mrow> </msup> <mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="86" data-semantic-parent="87" data-semantic-added="true" data-semantic-operator="infixop,⁢"></mo> <mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-id="83" data-semantic-children="29,79" data-semantic-content="82,29" data-semantic-parent="87"> <mi data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-id="29" data-semantic-parent="83" data-semantic-operator="appl">F</mi> <mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-id="82" data-semantic-parent="83" data-semantic-added="true" data-semantic-operator="appl"></mo> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="79" data-semantic-children="77" data-semantic-content="30,78" data-semantic-parent="83"> <mo data-semantic-type="fence" data-semantic-role="open" data-semantic-id="30" data-semantic-parent="79" data-semantic-operator="fenced">(</mo> <mrow data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-id="77" data-semantic-children="63,64,76" data-semantic-content="64" data-semantic-parent="79"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0" data-semantic-type="multiline" data-semantic-role="binomial" data-semantic-id="63" data-semantic-children="61,62" data-semantic-parent="77"> <mrow data-semantic-added="true" data-semantic-type="line" data-semantic-role="binomial" data-semantic-id="61" data-semantic-children="59" data-semantic-parent="63"> <mrow data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-id="59" data-semantic-children="56,39,58" data-semantic-content="39" data-semantic-parent="61"> <mrow data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-id="56" data-semantic-children="33,51" data-semantic-content="55" data-semantic-parent="59"> <mrow class="MJX-TeXAtom-ORD"> <mfrac data-semantic-type="fraction" data-semantic-role="vulgar" data-semantic-id="33" data-semantic-children="31,32" data-semantic-parent="56"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="31" data-semantic-parent="33">1</mn> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="32" data-semantic-parent="33">2</mn> </mfrac> </mrow> <mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="55" data-semantic-parent="56" data-semantic-added="true" data-semantic-operator="infixop,⁢"></mo> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="51" data-semantic-children="50" data-semantic-content="34,38" data-semantic-parent="56"> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="34" data-semantic-parent="51" data-semantic-operator="fenced">(</mo> <mrow data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="50" data-semantic-children="35,37" data-semantic-content="36" data-semantic-parent="51"> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="35" data-semantic-parent="50">c</mi> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="36" data-semantic-parent="50" data-semantic-operator="infixop,−"><!-- − --> </mo> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="37" data-semantic-parent="50">a</mi> </mrow> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="38" data-semantic-parent="51" data-semantic-operator="fenced">)</mo> </mrow> </mrow> <mo data-semantic-type="punctuation" data-semantic-role="comma" data-semantic-id="39" data-semantic-parent="59" data-semantic-operator="punctuated">,</mo> <mrow data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-id="58" data-semantic-children="42,54" data-semantic-content="57" data-semantic-parent="59"> <mrow class="MJX-TeXAtom-ORD"> <mfrac data-semantic-type="fraction" data-semantic-role="vulgar" data-semantic-id="42" data-semantic-children="40,41" data-semantic-parent="58"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="40" data-semantic-parent="42">1</mn> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="41" data-semantic-parent="42">2</mn> </mfrac> </mrow> <mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="57" data-semantic-parent="58" data-semantic-added="true" data-semantic-operator="infixop,⁢"></mo> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="54" data-semantic-children="53" data-semantic-content="43,49" data-semantic-parent="58"> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="43" data-semantic-parent="54" data-semantic-operator="fenced">(</mo> <mrow data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="53" data-semantic-children="52,48" data-semantic-content="47" data-semantic-parent="54"> <mrow data-semantic-type="infixop" data-semantic-role="addition" data-semantic-id="52" data-semantic-children="44,46" data-semantic-content="45" data-semantic-parent="53"> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="44" data-semantic-parent="52">a</mi> <mo data-semantic-type="operator" data-semantic-role="addition" data-semantic-id="45" data-semantic-parent="52" data-semantic-operator="infixop,+">+</mo> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="46" data-semantic-parent="52">c</mi> </mrow> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="47" data-semantic-parent="53" data-semantic-operator="infixop,−"><!-- − --> </mo> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="48" data-semantic-parent="53">1</mn> </mrow> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="49" data-semantic-parent="54" data-semantic-operator="fenced">)</mo> </mrow> </mrow> </mrow> </mrow> <mrow data-semantic-added="true" data-semantic-type="line" data-semantic-role="binomial" data-semantic-id="62" data-semantic-children="60" data-semantic-parent="63"> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="60" data-semantic-parent="62">c</mi> </mrow> </mfrac> </mrow> <mo data-semantic-type="punctuation" data-semantic-role="unknown" data-semantic-id="64" data-semantic-parent="77" data-semantic-operator="punctuated">;</mo> <mrow data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-id="76" data-semantic-children="65,66,73" data-semantic-content="74,75" data-semantic-parent="77"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="65" data-semantic-parent="76">4</mn> <mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="74" data-semantic-parent="76" data-semantic-added="true" data-semantic-operator="infixop,⁢"></mo> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="66" data-semantic-parent="76">z</mi> <mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="75" data-semantic-parent="76" data-semantic-added="true" data-semantic-operator="infixop,⁢"></mo> <mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="73" data-semantic-children="72" data-semantic-content="67,71" data-semantic-parent="76"> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="67" data-semantic-parent="73" data-semantic-operator="fenced">(</mo> <mrow data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-id="72" data-semantic-children="68,70" data-semantic-content="69" data-semantic-parent="73"> <mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-id="68" data-semantic-parent="72">1</mn> <mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="69" data-semantic-parent="72" data-semantic-operator="infixop,−"><!-- − --> </mo> <mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-id="70" data-semantic-parent="72">z</mi> </mrow> <mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="71" data-semantic-parent="73" data-semantic-operator="fenced">)</mo> </mrow> </mrow> </mrow> <mo data-semantic-type="fence" data-semantic-role="close" data-semantic-id="78" data-semantic-parent="79" data-semantic-operator="fenced">)</mo> </mrow> </mrow> </mrow> </mrow> <annotation encoding="application/x-tex">F\left({a,1-a \atop c};z\right)=(1-z)^{c-1}F\left({{\frac {1}{2}}(c-a),{\frac {1}{2}}(a+c-1) \atop c};4z(1-z)\right)</annotation> </semantics> </math>