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The conditioning theory of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-weighted least squares and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-weighted pseudoinverse problems is explored in this article. We begin by introducing three types of condition numbers for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-weighted pseudoinverse problem: normwise, mixed, and componentwise, along with their explicit expressions. Utilizing the derivative of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-weighted pseudoinverse problem, we then provide explicit condition number expressions for the solution of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-weighted least squares problem. To ensure reliable estimation of these condition numbers, we employ the small-sample statistical condition estimation method for all three algorithms. The article concludes with numerical examples that highlight the results obtained.