Model degrees of freedom ($\df$) is a fundamental concept in statistics because it quantifies the flexibility of a fitting procedure and is indispensable in model selection. The $\df$ is often intuitively equated with the number of independent variables in the fitting procedure. But for adaptive regressions that perform variable selection (e.g., the best subset regressions), the model $\df$ is larger than the number of selected variables. The excess part has been defined as the \emph{search degrees of freedom} ($\sdf$) to account for model selection. However, this definition is limited since it does not consider fitting procedures in augmented space, such as splines and regression trees; and it does not use the same fitting procedure for $\sdf$ and $\df$. For example, the lasso's $\sdf$ is defined through the \emph{relaxed} lasso's $\df$ instead of the lasso's $\df$. Here we propose a \emph{modified search degrees of freedom} ($\msdf$) to directly account for the cost of searching in the original or augmented space. Since many fitting procedures can be characterized by a linear operator, we define the search cost as the effort to determine such a linear operator. When we construct a linear operator for the lasso via the iterative ridge regression, $\msdf$ offers a new perspective for its search cost. For some complex procedures such as the multivariate adaptive regression splines (MARS), the search cost needs to be pre-determined to serve as a tuning parameter for the procedure itself, but it might be inaccurate. To investigate the inaccurate pre-determined search cost, we develop two concepts, \emph{nominal} $\df$ and \emph{actual} $\df$, and formulate a property named \emph{self-consistency} when there is no gap between the \emph{nominal} $\df$ and the \emph{actual} $\df$.

翻译：模型自由度（$\df$）是统计学中的基础概念，因它量化了拟合过程的灵活性，在模型选择中不可或缺。$\df$常被直观地等同于拟合过程中自变量的数量。但对于执行变量选择的自适应回归（如最优子集回归），模型$\df$大于所选变量的数量。超出部分被定义为搜索自由度（$\sdf$）以解释模型选择。然而，此定义存在局限：它未考虑增广空间中的拟合过程（如样条和回归树），且$\sdf$与$\df$使用了不同的拟合过程。例如，lasso的$\sdf$是通过放松型lasso（relaxed lasso）的$\df$而非lasso本身的$\df$定义的。本文提出修正搜索自由度（$\msdf$），直接量化原始空间或增广空间中的搜索代价。由于许多拟合过程可由线性算子表征，我们将搜索代价定义为确定此类线性算子所需的努力。当通过迭代岭回归为lasso构建线性算子时，$\msdf$为其搜索代价提供了新视角。对于多元自适应回归样条（MARS）等复杂过程，搜索代价需预先设定以作为该过程本身的调优参数，但可能不精确。为探究不精确的预设搜索代价，我们引入名义自由度（nominal $\df$）与实际自由度（actual $\df$）两个概念，并在名义$\df$与实际$\df$无差异时提出自洽性（self-consistency）这一性质。