### abstract ###
AIMX	in this paper we derive the equations for loop corrected belief propagation on a continuous variable gaussian model
OWNX	using the exactness of the averages for belief propagation for gaussian models  a  different way of obtaining the covariances is found   based on belief propagation on cavity graphs
OWNX	we discuss the relation of this  loop correction algorithm to expectation propagation  algorithms for the case in which the model is no longer  gaussian  but slightly perturbed by nonlinear terms
### introduction ###
MISC	message passing techniques in graphical models allow for the computation of  approximate   marginal probabilities in a time interval scaling polynomially in the  model size
MISC	their discovery has consequently revolutionized several  fields of applications in the past years  of which error correcting codes and vision are probably the most prominent examples
MISC	in many cases  the corresponding graphs are loopy  implying either that the error resulting from the application of loopy belief propagation  bp  is negligible for the particular model  or it  can be tolerated for the particular purpose bp serves
MISC	in other cases  more sophisticated refinements of bp are necessary  taking into account  part of  the loop errors
MISC	finding the optimal treatment of these   loop errors    motivates an active field of research  in which  different solutions applying to different model classes are developed
MISC	for models involving many short loops   like on regular lattices  cvm type approaches work well  CITATION   or tree ep approaches  CITATION
MISC	the latter may also be  applied to correct for an incidental large loop
MISC	unifying frameworks like the region graphs of  CITATION   lead to general strategies for selecting the basic clusters underlying such approaches for general model classes
MISC	a recent analysis has shown that the local update equations of bp may be interpreted as the zero order term of an expansion in   cavity connected correlations  
MISC	these quantities are parameterizations of the   cavity distributions     i e   the  distribution over neighbor variables of a central variable which has been removed from the graph
MISC	the bethe approximation and bp are recovered when this  cavity distribution is assumed to factorize  whereas the first order correction to the local update equations is obtained when one takes into account the pair cumulants  CITATION
MISC	estimation of these pair cumulants is possible with extra runs of bp  allowing for new polynomial time algorithms  reducing errors to order    when applying algorithms of which running time scales with an extra factor  of    CITATION
MISC	although this scaling seems heavy  the large benefit of the approach is that it does not require selection of basic clusters or underlying  tree structures  since it takes into account the effect of all loops that contribute to nontrivial correlations in the cavity distribution at once
MISC	the above   loop correction    strategy is applicable in the class of models where a perturbative expansion around the bethe approximation makes sense  i e   in models with large loops and relatively weak interactions
MISC	the principal requirement  is that the magnitude of pair variable cumulants of cavity distributions is an order smaller than the  single variable cumulants  and third order cumulants are even smaller  etc
MISC	however  heuristics based on the strategy allow for other good algorithms  performing well outside these parameter regimes  CITATION
CONT	so far the approach has been developed for discrete variable models on a more abstract  CITATION  versus practical level  CITATION
AIMX	in this paper we apply the idea to graphical models for continuous variables
OWNX	we derive the loop corrected belief propagation equations  for simple tractable gaussian models   yielding a message passing scheme that  besides the correct average marginals  also yields the correct variances
OWNX	besides that we discuss some approaches potentially  applicable to cases in which extra function approximations are necessary   and the relation with expectation propagation
CONT	a by product of our loop corrected belief propagation equations is an algorithm that calculates exact covariance matrices for gaussian models like the one discussed in  CITATION   but without explicitly using linear response
