One noisy observation could have come from several clean peaks. Move xt and watch the posterior weights decide how much each local conditional score contributes to the marginal score.
Fixed here: αt=0.85 and three equally likely clean origins x0∈{−3, 0, 3}.
The curves show relative conditional and marginal densities. Drag the red observation, or use the slider. Thin arrows are the conditional scores; their opacity follows the posterior weights. All arrows share one display scale at the current xt. When the chart is focused, the left and right arrow keys also move it.