
Wilson Score sorting algorithm, Wilson Score, is used for quality sorting. The data contains positive and negative reviews. Taking into account the **number of comments** and **praise rate**, the higher the score, the higher the quality.

$$
n = u + v \\
p = \frac{u}{v} \\
S = \frac{(p + \frac{z_{α}^{2}}{2n} - \frac{z_{α}^{2}}{2n}\sqrt{4n(1-p)p+z_{α}^{2}} )}{(1+\frac{z_{α}^{2}}{n})}
$$

$u$ represents the number of positive examples (good reviews), $v$ represents the number of negative examples (bad reviews), $n$ represents the total number of instances (total number of reviews), $p$ represents the positive rate, and $z$ is the score of the normal distribution. Number of bits (parameter), $S$ represents the final Wilson score. $z$ generally takes a value of 2, which is 95% confidence.

Here is the quantile table for the normal distribution:
![](https://upload-images.jianshu.io/upload_images/749674-35a2dc9beddd2083.png?imageMogr2/auto-orient/strip|imageView2/2/w/456/format/webp)

**Algorithm traits:**

1. Properties: The range of score $S$ is $[0,1)$, effect: normalized, suitable for sorting
2. Properties: When the number of positive examples $u$ is $0$, $p$ is $0$, and the score $S$ is $0$; effect: no positive comments, the score is the lowest;
3. Properties: When the number of negative examples $v$ is $0$, $p$ is $1$, degenerates to $1/(1 + z^2 / n)$, and the score $S$ is always less than $1$; effect: the score has permanent comparability;
4. Properties: When $p$ remains unchanged, the larger $n$ is, the reduction speed of the numerator is smaller than the reduction speed of the denominator, and the greater the score $S$, and vice versa; Effect: The positive rate $p$ is the same, the total number of instances is $n The more $, the more $S$ you score;
5. Properties: When $n$ tends to infinity, it degenerates to $p$, and the score $S$ is determined by $p$; Effect: When the total number of comments $n$ is more, the positive rate $p$ will bring the score $S The improvement of $ is more obvious;
6. Properties: When the quantile $z$ is larger, the total number of reviews $n$ is more important, and the praise rate $p$ is less important, and vice versa; Effect: The larger $z$ is, the total number of reviews $n$ is more important. The degree of discrimination is low; the smaller $z$ is, the more important the praise rate $p$ is;

[Click here to get source code](https://github.com/SpikeKing/MachineLearningTutorial/tree/master/wilson_score)

Python Implementation.

```python
def wilson_score(pos, total, p_z=2.):
    pos_rat = pos * 1. / total * 1.  # 正例比率
    score = (pos_rat + (np.square(p_z) / (2. * total))
             - ((p_z / (2. * total)) * np.sqrt(4. * total * (1. - pos_rat) * pos_rat + np.square(p_z)))) / \
            (1. + np.square(p_z) / total)
    return score
```

Distribution plot of Wilson score algorithm

![](https://upload-images.jianshu.io/upload_images/749674-3340e75c9438e31e.png?imageMogr2/auto-orient/strip|imageView2/2/w/1200/format/webp)

Let's try to put this theory into use with following example.

**Example:** Suppose Doctor A has 100 reviews, 1 negative review and 99 positive reviews. Doctor B has 2 reviews, both of which are good reviews. Which one should be ranked first?

When $z=2$, that is, 95% confidence level, doctor A's score is 0.9440, doctor B's score is 0.3333, and doctor A ranks in front.

---

PS: **Rating Level** Question: What should we do if we have a five-star evaluation system or a hundred percent evaluation system?

Just change the Wilson score formula from **Bernoulli distribution** to **normal distribution**.

$$
S = \frac{(μ + \frac{z_{α}^{2}}{2n} - \frac{z_{α}^{2}}{2n}\sqrt{4nσ^{2}+z_{α}^{2}} )}{(1+\frac{z_{α}^{2}}{n})} \\
μ ∈(0,1)
$$

**Note**: The mean and variance are both normalized values.

Python Implementation.

```python
def wilson_score_norm(mean, var, total, p_z=2.):
    # The mean variance needs to be normalized to fit the quantiles of the normal distribution
    score = (mean + (np.square(p_z) / (2. * total))
             - ((p_z / (2. * total)) * np.sqrt(4. * total * var + np.square(p_z)))) / \
            (1 + np.square(p_z) / total)
    return score
```

Example of normalization:

```python
def test_of_values():
    max = 5.  # Maximum evaluation value.
    min = 1.  # Minimum evaluation value.
    values = np.array([1., 2., 3., 4., 5.])  # Example

    norm_values = (values - min) / (max - min)  # Normalize
    total = norm_values.size
    mean = np.mean(norm_values)
    var = np.var(norm_values)
    return total, mean, var
```

---

PS: About the **z parameter**, that is, the normal quantile. The normal quantile affects the distribution of Wilson scores, and the value of the $z$ parameter is based on the magnitude of the number of samples. For example: the same 100 samples, 90 positive reviews, $z$ value is 2 or 6, the scores are very different, and the number of samples accommodated (or distinguished) by the system is also very different (the same is 0.82 points and 90% positive reviews, $z=2$ requires 100 samples, $z=6$ requires 1000 samples), **generally speaking, the larger the magnitude of the number of samples, the larger the value of z.**

```python
print 'score: %s' % wilson_score(90, 90 + 10, p_z=2.)
print 'score: %s' % wilson_score(90, 90 + 10, p_z=6.)
print 'score: %s' % wilson_score(900, 900 + 100, p_z=6.)

# take 2-100：score: 0.823802352689
# take 6-100：score: 0.606942322627
# take 6-1000：score: 0.828475631056
```

References:

- [Binomial proportion confidence interval](https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval)
- [Normal distribution](https://en.wikipedia.org/wiki/Normal_distribution)
- [How Not To Sort By Average Rating](http://www.evanmiller.org/how-not-to-sort-by-average-rating.html)
- [Relationship between Binomial and Normal Distributions](http://www.real-statistics.com/binomial-and-related-distributions/relationship-binomial-and-normal-distributions/)
- [Relevant Source Code](https://github.com/SpikeKing/MachineLearningTutorial/blob/master/wilson_score/wilson_score_model.py)


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The Wilson Score - Ranking Algorithm