Bayes estimation is a well-established method for information fusion based on a probabilistic setup. The parameter θ is to be estimated. The prior information is encoded by the density p(θ). In addition we have the observation y, which is related to the parameter θ by the likelihood function L(θ)=p(y|θ). It is the conditional probability density for y given some value θ and tells how likely some value θ is, if an observational value is given. Bayes' theorem provides the a posteriori density p(θ|y)=p(θ)p(y|θ)/p(y). A maximum a posteriori estimate then is the maximum of this density.
The example shown below assumes the prior density is a mixture of two Gaussians with the same standard deviation σ=0.1, the means μ1 and μ2, and the probabilities P1=P(μ= μ1) and P2=1-P1. The observation with (a usually high) probability Pin is an inlier with a standard deviation of σy. Otherwise it is assumed to be an outlier, randomly lying in the interval [0,3].
You can interactively change (1) the two mean values μ1 and μ2 by shifting the lower red points to the left or the right, (2) the prior probability P1 for the mean μ1 by shifting the left red point up or down, enforcing P1∈[0,1], (3) the probability Pin by shifting the left blue point up or down, again enforcing Pin∈[0,1], (4) the observational value y by shifting the lower blue point to the left or the right, and its standard deviation σy (being the distance to the ordinate axis) by shifting the lower left blue point to the right or left.
Depending on the chosen values, the applet then shows (1) the prior density p(θ) as red curve, (2) the likelihood L(θ)=p(y|θ) as dashed blue curve, and (3) the resulting posterior density p(θ|y) as the green curve. The initial configuration assumes the means are μ1=1.2 and μ2=1.6, the mean μ1 is slightly more probable than the mean μ2, namely P1=0.55, the inlier rate is Pin=90%, and the observational value is y=1.5. All values are shown with two digits.