BMSE (Bayesian mean squared error)

Bayesian mean squared error (BMSE) is a measure of the quality of an estimator, which takes into account the uncertainty in the estimated quantity. It is derived from Bayesian statistics, which is a branch of statistics that deals with the probability of a hypothesis being true given the available evidence. BMSE can be used to evaluate the performance of various estimators in different settings.

In this article, we will explain BMSE in detail, including its definition, calculation, and interpretation. We will also discuss its advantages and limitations, and provide examples to illustrate its use.

Definition of BMSE

BMSE is a measure of the expected squared difference between the true value of a quantity and its estimated value, taking into account the uncertainty in the estimation. It is defined as the expected value of the squared difference between the true value and the estimator, where the expectation is taken over the posterior distribution of the parameter being estimated.

Mathematically, BMSE can be expressed as follows:

BMSE = E[(θ - ẑ)^2|D]

where θ is the true value of the parameter being estimated, ẑ is the estimator, D is the data, and E[.] denotes the expectation taken over the posterior distribution of θ given D. In other words, BMSE measures the average squared deviation between the true value and the estimator, where the average is taken over the uncertainty in the estimation.

Calculation of BMSE

The calculation of BMSE depends on the method used to estimate the posterior distribution of θ given the data D. In Bayesian statistics, the posterior distribution is obtained by applying Bayes' theorem:

p(θ|D) = p(D|θ) p(θ) / p(D)

where p(θ|D) is the posterior distribution, p(D|θ) is the likelihood function, p(θ) is the prior distribution, and p(D) is the marginal likelihood or evidence. The likelihood function describes the probability of observing the data D given the parameter θ, the prior distribution represents the prior belief about θ before observing the data, and the marginal likelihood is a normalization constant that ensures the posterior distribution integrates to one.

There are several methods to estimate the posterior distribution, such as Markov chain Monte Carlo (MCMC) methods, variational inference, and Laplace approximation. Once the posterior distribution is estimated, the BMSE can be calculated using the formula mentioned above.

Interpretation of BMSE

BMSE measures the expected squared deviation between the true value of a quantity and its estimated value, taking into account the uncertainty in the estimation. A lower BMSE indicates a better estimator, as it means that the average squared deviation between the true value and the estimator is smaller.

BMSE can be used to compare the performance of different estimators in different settings. For example, suppose we have two estimators, A and B, for estimating a parameter θ. If the BMSE of estimator A is lower than that of estimator B, we can conclude that estimator A is better than estimator B, as it has a smaller average squared deviation from the true value.

Advantages of BMSE

BMSE has several advantages over other measures of estimator quality, such as mean squared error (MSE) and bias-variance trade-off. First, BMSE takes into account the uncertainty in the estimation, which is not considered in MSE. MSE only measures the average squared deviation between the estimator and the true value, assuming that the estimator is unbiased and has minimum variance. However, in many real-world situations, the estimator is biased or has high variance, which can lead to poor performance.

Second, BMSE provides a more informative measure of estimator quality than bias-variance trade-off. Bias-variance trade-off decomposes the mean squared error of an estimator into two components: bias and variance. Bias measures the systematic error of the estimator, i.e., the difference between the expected value of the estimator and the true value of the parameter. Variance measures the random error of the estimator, i.e., the variability of the estimator across different samples.

While bias-variance trade-off is a useful concept, it does not take into account the uncertainty in the estimation, which can be large in some cases. BMSE, on the other hand, measures the expected squared deviation between the estimator and the true value, taking into account the uncertainty in the estimation. This makes it a more informative measure of estimator quality, especially in situations where the uncertainty is large.

Third, BMSE is a natural measure of estimator quality in Bayesian statistics, as it is based on the posterior distribution of the parameter being estimated. Bayesian statistics provides a coherent framework for modeling uncertainty and making decisions based on the available evidence. BMSE fits naturally within this framework, as it measures the expected squared deviation between the true value and the estimator, taking into account the uncertainty in the estimation.

Limitations of BMSE

While BMSE has several advantages, it also has some limitations. First, BMSE depends on the prior distribution of the parameter being estimated, which can have a large impact on the posterior distribution and the resulting BMSE. If the prior distribution is misspecified or inappropriate, the posterior distribution and the resulting BMSE may be biased or inaccurate.

Second, BMSE assumes that the estimator is optimal with respect to the loss function being used. In Bayesian statistics, the loss function is typically chosen to reflect the goals and preferences of the decision maker. If the chosen loss function is inappropriate or does not reflect the decision maker's preferences, the resulting BMSE may not be a good measure of estimator quality.

Finally, BMSE may be computationally expensive to calculate, especially for complex models and large datasets. Estimating the posterior distribution of the parameter may require advanced computational methods, such as MCMC, which can be time-consuming and require a large amount of computational resources.

Examples of BMSE

To illustrate the use of BMSE, we provide two examples below.

Example 1: Linear regression

Suppose we have a dataset consisting of pairs of input-output values (x_i,y_i), i=1,...,N, and we want to fit a linear regression model of the form:

y_i = a + b x_i + ε_i,

where ε_i are independent and identically distributed Gaussian errors with mean zero and variance σ^2. We assume that the parameters a, b, and σ^2 are unknown and need to be estimated.

We can use Bayesian linear regression to estimate the posterior distribution of the parameters, given the data. The likelihood function for the data is:

p(y|a,b,σ^2,x) = N(y|a+b x, σ^2),

where N(μ, σ^2) denotes the Gaussian probability density function with mean μ and variance σ^2. We assume that the prior distributions for a, b, and σ^2 are:

p(a) = N(a|0,1), p(b) = N(b|0,1), p(σ^2) = IG(σ^2|1,1),

where N(μ, σ^2) denotes the Gaussian probability density function with mean μ and variance σ^2, and IG(α, β) denotes the inverse Gamma probability density function with shape parameter α and scale parameter β.

Using Bayes' theorem, we can obtain the posterior distribution of the parameters as:

p(a,b,σ^2|y,x) = N(a|a_N, σ^2/b_N) N(b|b_N, σ^2/a_N) IG(σ^2|α_N,β_N),

where a_N, b_N, α_N, and β_N are the posterior parameters that depend on the prior parameters and the data. The BMSE for the parameter a is then given by:

BMSE(a) = E[(a - a_N)^2] + Var(a),

where E[.] denotes the expected value and Var(.) denotes the variance.

Similarly, we can calculate the BMSE for the parameters b and σ^2. The total BMSE, which measures the expected squared deviation between the true values and the estimated values, is given by:

BMSE_total = BMSE(a) + BMSE(b) + BMSE(σ^2).

Example 2: Bayesian logistic regression

Suppose we have a binary classification problem, where we want to predict the probability of a binary outcome y (0 or 1) based on a set of input features x_1, ..., x_p. We can use Bayesian logistic regression to estimate the posterior distribution of the parameters, given the data.

The logistic regression model assumes that the log-odds of the probability of y=1 are linearly related to the input features:

logit(p(y=1|x)) = β_0 + β_1 x_1 + ... + β_p x_p,

where logit(p) = log(p/(1-p)) is the log-odds transformation, and β_0, ..., β_p are the parameters that need to be estimated.

We assume that the prior distributions for the parameters β are Gaussian:

p(β) = N(β|0, Σ),

where Σ is a diagonal matrix with diagonal entries equal to a small value, such as 0.01.

Using Bayes' theorem, we can obtain the posterior distribution of the parameters as:

p(β|y,x) = N(β|μ_N, Σ_N),

where μ_N and Σ_N are the posterior parameters that depend on the prior parameters and the data.

The BMSE for the parameters β is then given by:

BMSE(β) = E[(β - μ_N)^2] + Tr(Σ_N),

where Tr(.) denotes the trace of a matrix.

The total BMSE, which measures the expected squared deviation between the true values and the estimated values, is given by:

BMSE_total = BMSE(β_0) + BMSE(β_1) + ... + BMSE(β_p).

Conclusion

In summary, BMSE is a useful measure of estimator quality in Bayesian statistics, as it takes into account the uncertainty in the estimation and provides a natural measure of the expected squared deviation between the estimator and the true value. BMSE has several advantages over other measures of estimator quality, such as bias-variance decomposition, especially in situations where the uncertainty is large. However, BMSE also has some limitations, such as its dependence on the prior distribution and the loss function being used, and its computational cost. Overall, BMSE provides a useful tool for evaluating and comparing estimators in Bayesian statistics.