Two sources may be used to check each other over time. One of these sources may be your own senses, which show whether the event that the other source predicted occurred or not. The observation of an event is really another signal about the event. It is a noisy signal because your own eyes may lie (optical illusions, deepfakes).
First, one source sends a signal about the event, then the second source sends. You will never know whether the event actually occurred, but the second source is the aggregate of all the future information you receive about the event, so may be very accurate. The second source may send many signals in sequence about the event, yielding more info about the first source over time. Then the process repeats about a second event, a third, etc. This is how belief about the trustworthiness of a source is built.
You cannot learn the true accuracy of a source, because the truth is unavailable to your senses, so you cannot compare a source’s signals to the truth. You can only learn the consistency of different sources of sensory information. Knowing the correlation between various sensory sources is both necessary and sufficient for decision making, because your objective function (utility or payoff) is your perception of successfully achieving your goals. If your senses are deceived so you believe you have achieved what you sought, but actually have not, then you get the feeling of success, but if your senses are deceived to tell you you have failed, then you do not feel success even if you actually succeeded. The problem with deception arises purely from the positive correlation between the deceit and the perception of deceit. If deceit increases the probability that you later perceive you have been deceived and are unhappy about that perception, then deceit may reduce your overall utility despite initially increasing it temporarily. If you never suspect the deception, then your happiness is as if the deception was the truth.
Your senses send signals to your brain. We can interpret these signals as information about which hypothetical state of the world has occurred – we posit that there exists a world which may be in different states with various probabilities and that there is a correlation between the signals and these states. Based on the information, you update the probabilities of the states and choose a course of action. Actions result in probability distributions over different future sensations, which may be modelled as a different sensation in each state of the world, which have probabilities attached. (Later we may remove the states of the world from the model and talk about a function from past perceptions and actions into future perceptions. The past is only accessible through memory. Memory is a current perception, so we may also remove time from the model.)
You prefer some future sensations to others. These need not be sensory pleasures. These could be perceptions of having improved the world through great toil. You would prefer to choose an action that results in preferable sensations in the future. Which action this is depends on the state of the world.
To estimate the best action (the one yielding the most preferred sensations), you use past sensory signals. The interpretation of these signals depends on the assumed or learned correlation between the signals and the state. The assumption may be instinctive from birth. The learning is really about how sensations at a point in time are correlated with the combination of sensations and actions before that point. An assumption that the correlation is stable over time enables you to use past correlation to predict future correlation. This assumption in turn may be instinctive or learned.
The events most are interested in distinguishing are of the form “action A results in the most preferred sensations”, “action B causes the most preferred sensations”, “action A yields the least preferred sensations”. Any event that is useful to know is of a similar form by Blackwell’s theorem: information is useful if and only if it changes decisions.
The usefulness of a signal source depends on how consistent the signals it gives about the action-sensation links (events) are with your future perceptions. These future perceptions are the signals from the second source – your senses – against which the first source is checked. The signals of the second source have the form “memory of action A and a preferred sensation at present”. Optimal learning about the usefulness of the first source uses Bayes’ rule and a prior probability distribution on the correlations between the first source and the second. The events of interest in this case are the levels of correlation. A signal about these levels is whether the first source gave a signal that coincided with later sensory information.
If the first source recommended a “best action” that later yielded a preferred sensation, then this increases the probability of high positive correlation between the first source and the second on average. If the recommended action was followed by a negative sensation, then this raises the probability of a negative correlation between the sources. Any known correlation is useful information, because it helps predict the utility consequences of actions.
Counterfactuals should be mentioned as a side note. Even if an action A resulted in a preferred sensation, a different action B might have led to an even better sensation in the counterfactual universe where B was chosen instead. Of course, B might equally well have led to a worse sensation. Counterfactuals require a model to evaluate – what the output would have been after a different input depends on the assumed causal chain from inputs to outputs.
Whether two sources are separate or copies is also a learnable event.