In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.
Contents
- Definition
- Minimum mean square error estimation
- Posterior mean
- Bayes estimators for conjugate priors
- Alternative risk functions
- Posterior median and other quantiles
- Posterior mode
- Generalized Bayes estimators
- Example
- Empirical Bayes estimators
- Admissibility
- Asymptotic efficiency
- Practical example of Bayes estimators
- References
Definition
Suppose an unknown parameter
If the prior is improper then an estimator which minimizes the posterior expected loss for each
Minimum mean square error estimation
The most common risk function used for Bayesian estimation is the mean square error (MSE), also called squared error risk. The MSE is defined by
where the expectation is taken over the joint distribution of
Posterior mean
Using the MSE as risk, the Bayes estimate of the unknown parameter is simply the mean of the posterior distribution,
This is known as the minimum mean square error (MMSE) estimator. The Bayes risk, in this case, is the posterior variance.
Bayes estimators for conjugate priors
If there is no inherent reason to prefer one prior probability distribution over another, a conjugate prior is sometimes chosen for simplicity. A conjugate prior is defined as a prior distribution belonging to some parametric family, for which the resulting posterior distribution also belongs to the same family. This is an important property, since the Bayes estimator, as well as its statistical properties (variance, confidence interval, etc.), can all be derived from the posterior distribution.
Conjugate priors are especially useful for sequential estimation, where the posterior of the current measurement is used as the prior in the next measurement. In sequential estimation, unless a conjugate prior is used, the posterior distribution typically becomes more complex with each added measurement, and the Bayes estimator cannot usually be calculated without resorting to numerical methods.
Following are some examples of conjugate priors.
Alternative risk functions
Risk functions are chosen depending on how one measures the distance between the estimate and the unknown parameter. The MSE is the most common risk function in use, primarily due to its simplicity. However, alternative risk functions are also occasionally used. The following are several examples of such alternatives. We denote the posterior generalized distribution function by
Posterior median and other quantiles
Posterior mode
Other loss functions can be conceived, although the mean squared error is the most widely used and validated. Other loss functions are used in statistics, particularly in robust statistics.
Generalized Bayes estimators
The prior distribution
However, occasionally this can be a restrictive requirement. For example, there is no distribution (covering the set, R, of all real numbers) for which every real number is equally likely. Yet, in some sense, such a "distribution" seems like a natural choice for a non-informative prior, i.e., a prior distribution which does not imply a preference for any particular value of the unknown parameter. One can still define a function
Such measures
The use of an improper prior means that the Bayes risk is undefined (since the prior is not a probability distribution and we cannot take an expectation under it). As a consequence, it is no longer meaningful to speak of a Bayes estimator that minimizes the Bayes risk. Nevertheless, in many cases, one can define the posterior distribution
This is a definition, and not an application of Bayes' theorem, since Bayes' theorem can only be applied when all distributions are proper. However, it is not uncommon for the resulting "posterior" to be a valid probability distribution. In this case, the posterior expected loss
is typically well-defined and finite. Recall that, for a proper prior, the Bayes estimator minimizes the posterior expected loss. When the prior is improper, an estimator which minimizes the posterior expected loss is referred to as a generalized Bayes estimator.
Example
A typical example is estimation of a location parameter with a loss function of the type
It is common to use the improper prior
so the posterior expected loss equals
The generalized Bayes estimator is the value
In this case it can be shown that the generalized Bayes estimator has the form
This is identical to (1), except that
Empirical Bayes estimators
A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator. Empirical Bayes methods enable the use of auxiliary empirical data, from observations of related parameters, in the development of a Bayes estimator. This is done under the assumption that the estimated parameters are obtained from a common prior. For example, if independent observations of different parameters are performed, then the estimation performance of a particular parameter can sometimes be improved by using data from other observations.
There are parametric and non-parametric approaches to empirical Bayes estimation. Parametric empirical Bayes is usually preferable since it is more applicable and more accurate on small amounts of data.
Example
The following is a simple example of parametric empirical Bayes estimation. Given past observations
First, we estimate the mean
Next, we use the relation
where
Finally, we obtain the estimated moments of the prior,
For example, if
Admissibility
Bayes rules having finite Bayes risk are typically admissible. The following are some specific examples of admissibility theorems.
By contrast, generalized Bayes rules often have undefined Bayes risk in the case of improper priors. These rules are often inadmissible and the verification of their admissibility can be difficult. For example, the generalized Bayes estimator of a location parameter θ based on Gaussian samples (described in the "Generalized Bayes estimator" section above) is inadmissible for
Asymptotic efficiency
Let θ be an unknown random variable, and suppose that
To this end, it is customary to regard θ as a deterministic parameter whose true value is
where I(θ0) is the fisher information of θ0. It follows that the Bayes estimator δn under MSE is asymptotically efficient.
Another estimator which is asymptotically normal and efficient is the maximum likelihood estimator (MLE). The relations between the maximum likelihood and Bayes estimators can be shown in the following simple example.
Consider the estimator of θ based on binomial sample x~b(θ,n) where θ denotes the probability for success. Assuming θ is distributed according to the conjugate prior, which in this case is the Beta distribution B(a,b), the posterior distribution is known to be B(a+x,b+n-x). Thus, the Bayes estimator under MSE is
The MLE in this case is x/n and so we get,
The last equation implies that, for n → ∞, the Bayes estimator (in the described problem) is close to the MLE.
On the other hand, when n is small, the prior information is still relevant to the decision problem and affects the estimate. To see the relative weight of the prior information, assume that a=b; in this case each measurement brings in 1 new bit of information; the formula above shows that the prior information has the same weight as a+b bits of the new information. In applications, one often knows very little about fine details of the prior distribution; in particular, there is no reason to assume that it coincides with B(a,b) exactly. In such a case, one possible interpretation of this calculation is: "there is a non-pathological prior distribution with the mean value 0.5 and the standard deviation d which gives the weight of prior information equal to 1/(4d2)-1 bits of new information."
Another example of the same phenomena is the case when the prior estimate and a measurement are normally distributed. If the prior is centered at B with deviation Σ, and the measurement is centered at b with deviation σ, then the posterior is centered at
For example, if Σ=σ/2, then the deviation of 4 measurements combined together matches the deviation of the prior (assuming that errors of measurements are independent). And the weights α,β in the formula for posterior match this: the weight of the prior is 4 times the weight of the measurement. Combining this prior with n measurements with average v results in the posterior centered at
Compare to the example of binomial distribution: there the prior has the weight of (σ/Σ)²−1 measurements. One can see that the exact weight does depend on the details of the distribution, but when σ≫Σ, the difference becomes small.
Practical example of Bayes estimators
The Internet Movie Database uses a formula for calculating and comparing the ratings of films by its users, including their Top Rated 250 Titles which is claimed to give "a true Bayesian estimate". The following Bayesian formula was initially used to calculate a weighted average score for the Top 250, though the formula has since changed:
where:
Note that W is just the weighted arithmetic mean of R and C with weight vector (v, m). As the number of ratings surpasses m, the confidence of the average rating surpasses the confidence of the prior knowledge, and the weighted bayesian rating (W) approaches a straight average (R). The closer v (the number of ratings for the film) is to zero, the closer W gets to C, where W is the weighted rating and C is the average rating of all films. So, in simpler terms, films with very few ratings/votes will have a rating weighted towards the average across all films, while films with many ratings/votes will have a rating weighted towards its average rating.
IMDb's approach ensures that a film with only a few hundred ratings, all at 10, would not rank above "the Godfather", for example, with a 9.2 average from over 500,000 ratings.