## 2 Inference and the INFINITE Methodology
The main reason for this choice is that \(ln(Q)\) grows significantly more slowly than \(Q\), ensuring that the decay towards zero is gradual rather than abrupt. Given that \(Q\) is always a positive integer starting from 1, the optimal scenario occurs when \(Q=1\)  
Figure 1: Major components impacting inference and their corresponding parameters that may be recorded while executing a code-generating task. signifying the problem is resolved with a single query. As \(Q\) increases, the index should approach zero, reflecting the growing effort required to reach a conclusion. However, this decay should not be excessively rapid, as different problems exhibit varying complexity and ambiguity. To account for this variability, we introduce a problem-dependent multiplier \(m\) applied to \(ln(Q)\), which adjusts the decay rate. The parameter \(m\) is user-defined, must be a positive real number, and may take values less than 1 for highly ambiguous problems (e.g., open-ended research questions or real-world dynamic systems). In this manner, index \(C\) is formulated as  
\[C=\frac{1}{1+m\cdot ln(Q)} \tag{5}\]  
For the present work, parameter \(m\) is set to 1. To formulate an index for Accuracy, we must determine how closely the answer obtained by the framework aligns with the truth. In the case of numerical predictions in code-generating tasks, this can be quantified either through the R\({}^{2}\) score (for a series of predictions) or through some relative error, such as MAPE for single-value prediction or series of predictions. The R\({}^{2}\) or MAPEs of all predicted variables may be averaged to indicate the average accuracy. When the coefficient of determination is considered, if R\({}^{2}_{av}\) represents the averaged R\({}^{2}\) from all predictions, index \(A\) for Accuracy can be expressed as  
\[A=\text{R}^{2}_{av} \tag{6}\]  
If R\({}^{2}_{av}\) is less than or equal to 0, it is set to 0, and index A is 0 (indicating the poorest performance). When MAPE is considered, if (MAPE)\({}_{av}\) represents the averaged MAPE from all predictions, index \(A\) for Accuracy can be expressed as  
\[A=1-\frac{\text{(MAPE)}_{av}}{100} \tag{7}\]  
If (MAPE)\({}_{av}\) exceeds 100%, it is set to 100, and index A is 0 (indicating again the poorest performance). MAPE can become problematic when observed values are close to zero, leading to division by small numbers and resulting in artificially large or infinite errors. In contrast to MAPE, R\({}^{2}\) measures how well the model explains the variance in the data without suffering from division issues. If it is known a priori that observed values are not close to 0, then Equation (7) may be used, otherwise Equation (6) is suggested. Next, as we progress to the third step of the INFINITE methodology, index InI can be defined through a weighted average of the three indices \(E\), \(C\), and \(A\) as  
\[\text{InI}=w_{E}\cdot E+w_{C}\cdot C+w_{A}\cdot A \tag{8}\]  
where weights \(w_{E}\), \(w_{C}\) and \(w_{A}\) are assigned based on the significance of each of the three components as determined by the user. For the specific problem we are addressing, we carry out simple averaging and set all weights equal to \(\frac{1}{3}\). The proposed index is a unique tool that comprehensively evaluates three key components of inference: Efficiency, Consistency, and Accuracy. These components are quantified through distinct indices, incorporating factors such as the number of questions asked, the number of queries required to obtain a valid answer (including SB responses), average response time, and comparing results with the ground truth. By integrating these diverse measures, InI provides a holistic assessment of the LLM framework's performance, ensuring a balanced evaluation of its capability to answer the posed questions.