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Before you observe evidence, your expected posterior probability must equal your prior probability — averaged over all possible observations, weighted by their likelihood. This means you cannot rationally expect to be convinced of something you do not already believe: if you expect the evidence to support hypothesis H, then you should already believe H more strongly. Conservation of expected evidence is a powerful diagnostic for motivated reasoning: if you expect to find evidence confirming your current belief no matter what happens, something has gone wrong with your reasoning. Genuine inquiry must admit the possibility that evidence will push you in either direction.
Work through a concrete example: before flipping a coin you suspect is biased, write down how each possible sequence of outcomes would update your belief. Verify that the probability-weighted average of all possible posteriors equals your prior. Then apply this to real beliefs: before reading a study, ask what results would update you in each direction and by how much.
Conservation of expected evidence is a direct consequence of Bayes' theorem that serves as one of the most powerful diagnostic tools for detecting broken reasoning. The principle states: before you observe any evidence, the probability-weighted average of all your possible posterior beliefs must equal your current prior. Mathematically, this is just the law of total probability applied to belief updating, but its practical implications are profound.
Consider a concrete example. You suspect a coin might be biased toward heads. Before flipping it, you think through what each outcome would do to your belief. If heads comes up, you will update toward "biased." If tails comes up, you will update toward "fair." Now here is the key constraint: the amount you would update toward "biased" on heads, weighted by the probability of heads, must exactly equal the amount you would update toward "fair" on tails, weighted by the probability of tails. The expected net change in your belief is zero. This is not because evidence is useless -- once you actually flip the coin and see the result, you will genuinely update in one direction. The point is that you cannot predict in advance which direction you will update, on net, without that prediction itself being evidence you should have already incorporated.
This principle becomes a powerful diagnostic for motivated reasoning. Suppose a researcher says: "If the clinical trial shows positive results, I'll accept them. But if it shows negative results, I'll conclude the methodology was probably flawed." This person has pre-committed to updating upward on positive results and refusing to update downward on negative results. The probability-weighted average of their anticipated posteriors is higher than their prior -- which is mathematically impossible for correct Bayesian reasoning. They have violated conservation of expected evidence, and the violation reveals that they are not genuinely open to what the evidence might show. Their bottom line was already written.
The practical test you can apply to your own reasoning is simple: before examining evidence on any question, ask what results would push you in each direction, and by how much. If you find that every possible outcome would confirm your current belief -- or that disconfirming outcomes would always be explained away -- conservation of expected evidence is violated, and you are not reasoning honestly. Genuine inquiry must admit the possibility that the evidence will push you somewhere you did not want to go. This is uncomfortable, but it is the price of having beliefs that actually track reality rather than beliefs that merely feel good.
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