Pseudocount
Encyclopedia
A pseudocount is an amount (not generally an integer, despite its name) added to the number of observed cases in order to change the expected probability
in a model of those data, when not known to be zero
. Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value. It may only be zero (or the possibility ignored) if impossible by definition, such as the possibility of a decimal digit of pi being a letter, or a physical possibility that would be rejected and so not counted, such as a computer printing a letter when a valid program for pi is run, or excluded and not counted because of no interest, such as if only interested in the zeros and ones. Generally, there is also a possibility that no value may be computable or observable in a finite time (see Turing's halting problem
). The relative values of pseudocounts represent the relative prior expected probabilities of their possibilities. The sum of the pseudocounts, which may be very large, represents the estimated weight of the prior knowledge compared with all the actual observations (one for each) when determining the expected probability.
In any observed data set or sample
there is the possibility, especially with low-probability event
s and/or small data sets, of a possible event not occurring. Its observed frequency is therefore zero, apparently implying a probability of zero. This is an oversimplification, which is inaccurate and often unhelpful, particularly in probability-based machine learning
techniques such as artificial neural network
s and hidden Markov model
s. By artificially adjusting the probability of rare (but not impossible) events so those probabilities are not exactly zero, we avoid the zero-frequency problem
. Also see Cromwell's rule
.
The simplest approach is to add one to each observed number of events including the zero-count possibilities. This is sometimes called Laplace's rule of succession
. It is a type of additive smoothing
.
More generally and accurately, the pseudocounts should be set in proportion to the prior estimate of their probabilities; equal only if there is no prior reason to prefer one over another — see the principle of indifference
. Adding one to the actual counts usually refers to the simplest binary choice situation. In the absence of any prior knowledge to the contrary, the sum of the pseudocounts should be two, however many possibilities there are, even if known to be unequal — see generalization to any number of possibilities. However, given appropriate prior knowledge, that sum should be increased in proportion to the expectation that the prior probabilities should be considered correct, despite evidence to the contrary — see further analysis.
A more complex approach is to estimate the probability
of the events from other factors and adjust accordingly.
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
in a model of those data, when not known to be zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value. It may only be zero (or the possibility ignored) if impossible by definition, such as the possibility of a decimal digit of pi being a letter, or a physical possibility that would be rejected and so not counted, such as a computer printing a letter when a valid program for pi is run, or excluded and not counted because of no interest, such as if only interested in the zeros and ones. Generally, there is also a possibility that no value may be computable or observable in a finite time (see Turing's halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
). The relative values of pseudocounts represent the relative prior expected probabilities of their possibilities. The sum of the pseudocounts, which may be very large, represents the estimated weight of the prior knowledge compared with all the actual observations (one for each) when determining the expected probability.
In any observed data set or sample
Sample (statistics)
In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size...
there is the possibility, especially with low-probability event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...
s and/or small data sets, of a possible event not occurring. Its observed frequency is therefore zero, apparently implying a probability of zero. This is an oversimplification, which is inaccurate and often unhelpful, particularly in probability-based machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...
techniques such as artificial neural network
Artificial neural network
An artificial neural network , usually called neural network , is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes...
s and hidden Markov model
Hidden Markov model
A hidden Markov model is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states. An HMM can be considered as the simplest dynamic Bayesian network. The mathematics behind the HMM was developed by L. E...
s. By artificially adjusting the probability of rare (but not impossible) events so those probabilities are not exactly zero, we avoid the zero-frequency problem
PPM compression algorithm
Prediction by partial matching is an adaptive statistical data compression technique based on context modeling and prediction. PPM models use a set of previous symbols in the uncompressed symbol stream to predict the next symbol in the stream....
. Also see Cromwell's rule
Cromwell's rule
Cromwell's rule, named by statistician Dennis Lindley, states that one should avoid using prior probabilities of 0 or 1, except when applied to statements that are logically true or false...
.
The simplest approach is to add one to each observed number of events including the zero-count possibilities. This is sometimes called Laplace's rule of succession
Rule of succession
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem....
. It is a type of additive smoothing
Additive smoothing
In statistics, additive smoothing, also called Laplace smoothing , or Lidstone smoothing, is a technique used to smooth categorical data...
.
More generally and accurately, the pseudocounts should be set in proportion to the prior estimate of their probabilities; equal only if there is no prior reason to prefer one over another — see the principle of indifference
Principle of indifference
The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
. Adding one to the actual counts usually refers to the simplest binary choice situation. In the absence of any prior knowledge to the contrary, the sum of the pseudocounts should be two, however many possibilities there are, even if known to be unequal — see generalization to any number of possibilities. However, given appropriate prior knowledge, that sum should be increased in proportion to the expectation that the prior probabilities should be considered correct, despite evidence to the contrary — see further analysis.
A more complex approach is to estimate the probability
Density estimation
In probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function...
of the events from other factors and adjust accordingly.