Entropy and Information Density of Text
When you look at natural language text, you might wonder how much information is actually carried by each token. This is where the concept of entropy comes in. In information theory, entropy measures the average amount of information produced by a source of data — in this case, a stream of text tokens. The average bits per token tells you how many binary digits, on average, are needed to represent each token, assuming you encode tokens as efficiently as possible. If a token is highly predictable, it carries less information and can be represented with fewer bits. Conversely, less predictable tokens require more bits. This means that the information density of text — the amount of information packed into each token — depends on how surprising or uncertain each token is. The length of a token, or how many bits are needed to encode it, is thus directly linked to how predictable it is: the less predictable, the longer (in bits) its encoding.
In the context of text, entropy quantifies the unpredictability or randomness of token occurrence. High entropy means token appearances are less predictable, while low entropy means they are more regular. Entropy is significant for tokenization because it sets a lower bound on how efficiently text can be compressed: you cannot, on average, encode tokens using fewer bits than the entropy of their distribution.
1234567891011121314151617181920import numpy as np # Sample text text = "the cat sat on the mat and the cat ate the rat" # Tokenize by splitting on spaces tokens = text.split() # Count frequencies token_counts = {} for token in tokens: token_counts[token] = token_counts.get(token, 0) + 1 # Compute probabilities total_tokens = len(tokens) probs = np.array([count / total_tokens for count in token_counts.values()]) # Compute empirical entropy (in bits) entropy = -np.sum(probs * np.log2(probs)) print(f"Empirical entropy: {entropy:.3f} bits per token")
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When you look at natural language text, you might wonder how much information is actually carried by each token. This is where the concept of entropy comes in. In information theory, entropy measures the average amount of information produced by a source of data — in this case, a stream of text tokens. The average bits per token tells you how many binary digits, on average, are needed to represent each token, assuming you encode tokens as efficiently as possible. If a token is highly predictable, it carries less information and can be represented with fewer bits. Conversely, less predictable tokens require more bits. This means that the information density of text — the amount of information packed into each token — depends on how surprising or uncertain each token is. The length of a token, or how many bits are needed to encode it, is thus directly linked to how predictable it is: the less predictable, the longer (in bits) its encoding.
In the context of text, entropy quantifies the unpredictability or randomness of token occurrence. High entropy means token appearances are less predictable, while low entropy means they are more regular. Entropy is significant for tokenization because it sets a lower bound on how efficiently text can be compressed: you cannot, on average, encode tokens using fewer bits than the entropy of their distribution.
1234567891011121314151617181920import numpy as np # Sample text text = "the cat sat on the mat and the cat ate the rat" # Tokenize by splitting on spaces tokens = text.split() # Count frequencies token_counts = {} for token in tokens: token_counts[token] = token_counts.get(token, 0) + 1 # Compute probabilities total_tokens = len(tokens) probs = np.array([count / total_tokens for count in token_counts.values()]) # Compute empirical entropy (in bits) entropy = -np.sum(probs * np.log2(probs)) print(f"Empirical entropy: {entropy:.3f} bits per token")
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