Micro Precision

1. Precision Recap (Binary Case)

• Precision = Of all predicted positives, how many are actually positive?

$Precision = \frac{TP}{TP + FP}$

Where:

• TP = True Positives
• FP = False Positives

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2. Multiclass Extension

For K classes, there are two main strategies: macro vs micro.

• Macro Precision: Compute precision per class (OvR), then average them equally.
• Micro Precision: Pool all classes together and compute precision globally.

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3. Definition of Micro Precision

$Precision_{micro} = \frac{\sum_{i=1}^{K} TP_i}{\sum_{i=1}^{K} (TP_i + FP_i)}$

• Instead of averaging per-class scores, we sum up counts across classes first, then compute precision.
• Equivalent to treating the multiclass problem as one big binary problem: “correct vs incorrect”.

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4. Key Characteristics

• Micro Precision = Micro Recall = Micro F1 in multiclass classification (because they’re all based on the same global TP, FP, FN counts).
• Dominated by majority classes: large classes contribute more to the total TP and FP.
• More useful when class imbalance exists and you want to measure overall system performance, not per-class fairness.

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5. Example

Suppose we have 3 classes (A, B, C).

Confusion matrix results:

• Class A: TP = 40, FP = 10
• Class B: TP = 30, FP = 20
• Class C: TP = 10, FP = 20

Macro Precision:

• Precision(A) = 40 / (40+10) = 0.80
• Precision(B) = 30 / (30+20) = 0.60
• Precision(C) = 10 / (10+20) = 0.33
• Macro Precision = (0.80 + 0.60 + 0.33) / 3 = 0.58

Micro Precision:

$\frac{40+30+10}{(40+10)+(30+20)+(10+20)} = \frac{80}{130} \approx 0.615$

Notice: Micro Precision (0.615) is closer to the large classes’ contribution.

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Summary

• Micro Precision = global precision across all classes.
• Counts TP and FP across classes before computing precision.
• Favours large classes, good for overall system evaluation.
• Macro Precision treats all classes equally.