#### [MADS] Law of Total Variance

This simple law, despite my trial of full text search, was not showing in ADS. As discussed in systematic errors, astronomers, like physicists, show their error components in two additive terms; statistical error + systematic error. To explain such decomposition and to make error analysis statistically rigorous, the law of total variance (LTV) seems indispensable.

V[X] = V[E[X|Y]] + E[V[X|Y]]

(X and Y are random variables, and X indicates observed data. In addition, V and E stands for variance and expectation, respectively. Instead of X, f(X_1,…,X_n) can be plugged in to represent a best fit. In other words, a best fit is a solution of the chi-square minimization which is a function of data). For Bayesian, the uncertainty of theta, parameter of interest, is

V[theta]=V[E[theta|Y]] + E[V[theta|Y]]

Suppose Y is related to systematics. E[theta|Y] is a function of Y so that V[E[theta|Y] indicates systematic error. V[theta|Y] is statistical error given Y which reflects the fact that unless the parameters of interest and systematics are independent, statistical error cannot be quantified into a single factor next to a best fit. If parameter of interest, theta is independent of Y and Y is fixed, then the uncertainty in theta is solely come from statistical uncertainty (Let’s not consider “model uncertainty” for the time being).

In conjunction of astronomers’ systematic error and statistical error decomposition or representing uncertainties in quadrature (error2total = error2 stat+error2sys), statisticians use mean square error (MSE) as total error, in which variance matches statistical error, and bias^2 does systematic error.

MSE = Variance+ bias^2

Now it comes to a question, is systematic error bias? Those methods based on quadratures or parameterization of systematics for marginalization consider systematic error as bias although no account explicitly said so. According to the law of total variance unless it’s orthogonal/independent, quadrature is not proper way to handle systematic uncertainties prevailing in all instruments. Generally parameters (data) and systematics are nonlinearly correlated and hard to factorize (instrument specific empirical studies exist to offer correction factors due to systematics; however, such factors work only on specific cases and the process of defining correction factors is hard to be generalized). Because of the varying nature of systematics over the parameter space, instead of MSE

$MISE = \int [\hat f(x)- f(x)]^2 f(x) dx$

or mean integrated square error might be of use. The estimator of f(x), or \hat f(x) is either parametrically or nonparametrically estimable while incorporating systematics and correlation structures with statistical errors as a function of a certain domain x. MISE can be viewed as a robust version of chi-square methods but details have not been explored to account for the following identity.

$MISE=\int [E\hat f(x) -f(x)]^2 dx + \int Var \hat f(x) dx$

This equation may or may not look simple. Perhaps, the expansion of the above identify could explain more on the error decomposition.

$MISE(\hat f(x)) = \int E(\hat f(x)-f(x))^2 dx$
$=\int MSE_x (\hat f) dx = \int (\hat f(x)- f(x))^2 f(x) dx$
$= \int [E \hat f(x)- f(x)]^2dx +\int Var \hat f(x) dx$
integreated squared bias + the integrated variance (overall systematic error + overall statistical error)

Furthermore, it robustly characterizes uncertainties from systematics, i.e. calibration uncertainties in data analysis. Note that estimating f(x) or \hat f(x) reflects complex structures in uncertainty analysis; whereas, the chi square minimization estimates f(x) via piecewise straight horizontal lines, assumes the homogeneous error in each piece (bin), forces statistical and systematic errors to be orthogonal, and as a consequence, inflates the size of error or produces biased best fits.

Either LTV, MSE, or MISE, even if we do not know the true model f(x) — if unknown, assessing statistical analysis results such as confidence levels/intervals may not be feasible; the reason that chi-square methods offer best fits and its N sigma error bars is that it assume the true model is Gaussian, or N(f(x),\sigma^2), or E(Y|X)=f(X)+\epsilon, V(\epsilon)=\sigma^2 where f(x) is a source model. On the other hand, Monte Carlo simulations, resampling methods like bootstrap, or posterior predictive probability (ppp) allows to infer the truth from which one can evaluate the p-value to indicate one’s confidence on the result from fitting analysis in a nonparametric fashion. — setting up proper models for \hat f(x) or \theta|Y would help assessing the total error in a more realistic manner than the chi-square minimization, additive errors, gaussian quadrature, or subjective expertise on systematics. The underlying notions and related theoretical statistics methodologies of LTV, MSE, or MISE could clarify the questions like how to quantify systematic errors and how systematic uncertainties are related to statistical uncertainties. Well, nothing will make me and astronomers happier if those errors are independent and additive. Even more exuberant, systematic uncertainty can be factorized.