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Perplexity
In information theory, perplexity is a measure of uncertainty in the value of a sample from a discrete probability distribution. The larger the perplexity, the less likely it is that an observer can guess the value which will be drawn from the distribution. Perplexity was originally introduced in 1977 in the context of speech recognition by Frederick Jelinek, Robert Leroy Mercer, Lalit R. Bahl, and James K. Baker.
The perplexity PP of a discrete probability distribution p is a concept widely used in information theory, machine learning, and statistical modeling. It is defined as
where x ranges over the events, where 0−0 is defined to be 1, and where the value of b does not affect the result; b can be chosen to be 2, 10, e, or any other positive value other than 1. In some contexts, this measure is also referred to as the (order-1 true) diversity.
The logarithm log PP(p) is the entropy of the distribution; it is expressed in bits if the base of the logarithm is 2, and it is expressed in nats if the natural logarithm is used.
Perplexity of a random variable X may be defined as the perplexity of the distribution over its possible values x. It can be thought of as a measure of uncertainty or "surprise" related to the outcomes.
For a probability distribution p where exactly k outcomes each have a probability of 1 / k and all other outcomes have a probability of zero, the perplexity of this distribution is simply k. This is because the distribution models a fair k-sided die, with each of the k outcomes being equally likely. In this context, the perplexity k indicates that there is as much uncertainty as there would be when rolling a fair k-sided die. Even if a random variable has more than k possible outcomes, the perplexity will still be k if the distribution is uniform over k outcomes and zero for the rest. Thus, a random variable with a perplexity of k can be described as being "k-ways perplexed," meaning it has the same level of uncertainty as a fair k-sided die.
Perplexity is sometimes used as a measure of the difficulty of a prediction problem. It is, however, generally not a straightforward representation of the relevant probability. For example, if you have two choices, one with probability 0.9, your chances of a correct guess using the optimal strategy are 90 percent. Yet, the perplexity is 2−0.9 log2 0.9 - 0.1 log2 0.1= 1.38. The inverse of the perplexity, 1/1.38 = 0.72, does not correspond to the 0.9 probability.
The perplexity is the exponentiation of the entropy, a more commonly encountered quantity. Entropy measures the expected or "average" number of bits required to encode the outcome of the random variable using an optimal variable-length code. It can also be regarded as the expected information gain from learning the outcome of the random variable, providing insight into the uncertainty and complexity of the underlying probability distribution.
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Perplexity
In information theory, perplexity is a measure of uncertainty in the value of a sample from a discrete probability distribution. The larger the perplexity, the less likely it is that an observer can guess the value which will be drawn from the distribution. Perplexity was originally introduced in 1977 in the context of speech recognition by Frederick Jelinek, Robert Leroy Mercer, Lalit R. Bahl, and James K. Baker.
The perplexity PP of a discrete probability distribution p is a concept widely used in information theory, machine learning, and statistical modeling. It is defined as
where x ranges over the events, where 0−0 is defined to be 1, and where the value of b does not affect the result; b can be chosen to be 2, 10, e, or any other positive value other than 1. In some contexts, this measure is also referred to as the (order-1 true) diversity.
The logarithm log PP(p) is the entropy of the distribution; it is expressed in bits if the base of the logarithm is 2, and it is expressed in nats if the natural logarithm is used.
Perplexity of a random variable X may be defined as the perplexity of the distribution over its possible values x. It can be thought of as a measure of uncertainty or "surprise" related to the outcomes.
For a probability distribution p where exactly k outcomes each have a probability of 1 / k and all other outcomes have a probability of zero, the perplexity of this distribution is simply k. This is because the distribution models a fair k-sided die, with each of the k outcomes being equally likely. In this context, the perplexity k indicates that there is as much uncertainty as there would be when rolling a fair k-sided die. Even if a random variable has more than k possible outcomes, the perplexity will still be k if the distribution is uniform over k outcomes and zero for the rest. Thus, a random variable with a perplexity of k can be described as being "k-ways perplexed," meaning it has the same level of uncertainty as a fair k-sided die.
Perplexity is sometimes used as a measure of the difficulty of a prediction problem. It is, however, generally not a straightforward representation of the relevant probability. For example, if you have two choices, one with probability 0.9, your chances of a correct guess using the optimal strategy are 90 percent. Yet, the perplexity is 2−0.9 log2 0.9 - 0.1 log2 0.1= 1.38. The inverse of the perplexity, 1/1.38 = 0.72, does not correspond to the 0.9 probability.
The perplexity is the exponentiation of the entropy, a more commonly encountered quantity. Entropy measures the expected or "average" number of bits required to encode the outcome of the random variable using an optimal variable-length code. It can also be regarded as the expected information gain from learning the outcome of the random variable, providing insight into the uncertainty and complexity of the underlying probability distribution.