Reasoning Verification › Worked Example: Manual PRM Scoring

To ground the abstract framework in something concrete, let us walk through how a PRM would score a specific mathematical reasoning chain. Consider the following problem and a candidate solution.

Problem: "A store sells widgets for $8 each. They offer a 15% discount on orders of 10 or more. If I buy 12 widgets, how much do I pay in total?"

Candidate reasoning chain:

Step 1: The normal price for 12 widgets is 12 times $8 equals $96. Step 2: Since I am buying 12, which is at least 10, the 15% discount applies. Step 3: The discount amount is 15% of $96, which is 0.15 times 96 equals $14.40. Step 4: The final price is $96 minus $14.40 equals $81.60.

A PRM would process this chain as follows. The model reads the problem statement and each step in sequence, and at each step marker produces a correctness probability.

Step 1: The calculation is straightforward. 12 times 8 equals 96. This is arithmetically correct and logically appropriate (computing total before discount). A well-trained PRM assigns a high score, perhaps 0.97.

Step 2: This step checks the eligibility condition. The problem states "orders of 10 or more" and 12 is indeed at least 10. The logical inference is valid. A PRM assigns a high score, perhaps 0.96.

Step 3: The arithmetic is 0.15 times 96 equals 14.40. This is correct: 0.15 times 96 equals 0.1 times 96 plus 0.05 times 96 equals 9.6 plus 4.8 equals 14.4. The PRM assigns a high score, perhaps 0.95.

Step 4: The final calculation is 96 minus 14.40 equals 81.60. This is correct. The PRM assigns a high score, perhaps 0.96.

The chain score using the minimum aggregation is $\min(0.97, 0.96, 0.95, 0.96) = 0.95$. This chain would be selected over one with a low-scoring step.

Now consider an alternative candidate reasoning chain with an error:

Step 1: The normal price for 12 widgets is 12 times $8 equals $96. Step 2: The discount is 15%, so I pay 85% of the normal price. Step 3: 85% of $96 is 0.85 times 96 equals $81.60.

This is also correct and produces the same answer. A PRM trained on diverse examples might score this chain comparably or even higher, since it is more concise and each step is clearly valid. The PRM does not require a specific reasoning path, only that each step be locally correct.

Now consider a chain with an error:

Step 1: The normal price for 12 widgets is 12 times $8 equals $96. Step 2: Since I am buying 12, the 10% bulk discount applies. Step 3: The discount amount is 10% of $96, which is $9.60. Step 4: The final price is $96 minus $9.60 equals $86.40.

Step 2 contains a factual error: it states "10% bulk discount" when the problem specifies 15%. A PRM would assign a low score to this step because it misreads the discount percentage from the problem statement. The minimum chain score drops to the score for step 2, perhaps 0.08, and this chain would be ranked below the correct chains.

This example illustrates the PRM's core function: it can identify the specific step where a reasoning chain diverges from correctness, giving both a quality score and error localization that an ORM cannot.