🎲 NumPy Multinomial Distribution – Simulate Categorical Outcomes in Python

🧲 Introduction – Why Learn the Multinomial Distribution in NumPy?

The multinomial distribution is an extension of the binomial distribution. While the binomial deals with two outcomes (like success/failure), the multinomial handles multiple categories (like apple/orange/banana). It’s used in NLP, machine learning, A/B/n testing, surveys, and more—anywhere you want to simulate or analyze multiple outcomes per trial.

With np.random.multinomial(), NumPy allows you to generate realistic simulations for multi-class experiments.

🎯 By the end of this guide, you’ll:

• Generate samples using the multinomial distribution
• Understand n, pvals, and size parameters
• Simulate multi-category experiments
• Visualize and interpret output
• Learn real-world use cases like survey modeling and A/B/n testing

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🔢 Step 1: Generate Multinomial Samples with NumPy

import numpy as np

sample = np.random.multinomial(n=10, pvals=[0.2, 0.5, 0.3])
print(sample)

🔍 Explanation:

• n=10: Total number of trials
• pvals=[0.2, 0.5, 0.3]: Probabilities for each of 3 categories
  ✅ Output: [2 5 3] → 2 in category A, 5 in B, 3 in C
  📌 Each row always sums to n (10 here)

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📊 Step 2: Generate Multiple Samples with size

samples = np.random.multinomial(n=10, pvals=[0.2, 0.5, 0.3], size=5)
print(samples)

🔍 Explanation:

• Generates 5 rows of samples
• Each row represents one trial with 10 events distributed across categories
  ✅ Output example:

[[1 5 4]
 [2 4 4]
 [3 5 2]
 [2 6 2]
 [1 5 4]]

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📈 Step 3: Visualize Category Frequencies

import matplotlib.pyplot as plt
import seaborn as sns

data = np.random.multinomial(n=100, pvals=[0.3, 0.4, 0.3], size=1000)
category_totals = data.sum(axis=0)

categories = ['A', 'B', 'C']
sns.barplot(x=categories, y=category_totals, palette="Set2")
plt.title("Multinomial Category Totals over 1000 Trials")
plt.ylabel("Total Count")
plt.show()

🔍 Explanation:

• Simulates 1000 experiments of 100 trials each
• Aggregates total count per category
  ✅ Useful for comparing expected vs. observed proportions

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🎯 Step 4: Real-World Use Case – A/B/C Testing

clicks = np.random.multinomial(n=1000, pvals=[0.1, 0.5, 0.4])
print(f"Variant A: {clicks[0]} clicks, B: {clicks[1]}, C: {clicks[2]}")

🔍 Explanation:

• Simulates click results across 3 website variants
  ✅ Ideal for multi-variant performance comparison

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🧠 Step 5: NLP Use Case – Word Counts in Documents

vocab_probs = [0.1, 0.15, 0.25, 0.3, 0.2]  # Probabilities for 5 words
word_counts = np.random.multinomial(n=50, pvals=vocab_probs)
print("Word counts in synthetic document:", word_counts)

🔍 Explanation:

• Simulates 50-word document
• Each word chosen based on its probability
  ✅ Use in topic modeling, bag-of-words, or language modeling

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📐 Step 6: Check Distribution Properties

trials = np.random.multinomial(10, [0.2, 0.3, 0.5], size=10000)
mean_counts = trials.mean(axis=0)
print("Empirical mean counts:", mean_counts)

🔍 Explanation:

• Running many simulations approximates the expected values
• mean_counts ≈ [2, 3, 5] as per p-values and total trials

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🧮 Parameters Summary

Parameter	Description
n	Total number of trials per experiment
pvals	Probabilities of each category (must sum to 1)
size	Number of repetitions (experiments)

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⚠️ Common Mistakes to Avoid

Mistake	Fix
Probabilities not summing to 1	Ensure sum(pvals) == 1.0
Using floats expecting exact counts	Output is integer-only (count of occurrences)
Mismatched array dimensions	Use .sum(axis=0) to aggregate across samples
Confusing rows vs columns	Rows = experiments, columns = categories

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📌 Summary – Recap & Next Steps

The multinomial distribution lets you model multi-class outcomes, simulating data like survey results, A/B/n test results, or document word frequencies. With np.random.multinomial(), you can efficiently simulate, visualize, and analyze multi-outcome processes.

🔍 Key Takeaways:

• np.random.multinomial(n, pvals, size) simulates multi-category trials
• Output shape: (size, len(pvals))
• Each row sums to n
• Great for A/B/n testing, classification simulation, and survey modeling

⚙️ Real-world relevance: Used in digital marketing, natural language processing, quality control, and social science experiments.

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❓ FAQs – NumPy Multinomial Distribution

❓ Can I use decimal probabilities in pvals?
✅ Yes, as long as they sum to 1.0.

❓ Does multinomial() return floats or integers?
✅ Always returns integers — the counts per category.

❓ How do I simulate a single trial across 5 outcomes?
✅ Use:

np.random.multinomial(1, [0.1, 0.2, 0.3, 0.2, 0.2])

❓ What if my pvals don’t sum to 1?
❌ NumPy will raise an error or return incorrect results. Always normalize.

❓ Can I use multinomial for classification simulation?
✅ Yes, simulate class labels based on class probabilities.

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