The Conservation of Expected Evidence and the Narratives
After the arrest of a woman accused of witchcraft, if she was afraid, it would prove her guilt; if she was not afraid, it would also prove her guilt
Friedrich Spee von Langenfeld was a German Jesuit priest who heard the confessions of condemned witches. He criticized the inquisition's criminal process and witch hunts. In 1631, he wrote a book titled "Cautio Criminalis" ("Prudence in Criminal Cases"), where he sharply described the decision tree for condemning witches. According to Spee, the process essentially cornered the accused in such a way that any reaction could be interpreted as guilt.
In the book, the Jesuit examines the paradoxical standards applied during witch trials, demonstrating how any aspect of a woman's actions or life decisions could be manipulated to "confirm" her guilt in witchcraft. The process would unfold as follows:
If the accused witch had led a dishonorable life, she was deemed guilty; conversely, a virtuous life was also seen as evidence of guilt, as it was believed witches would feign virtue to deceive others. Upon arrest, if a woman showed fear, it was taken as a sign of guilt; if she showed no fear, this too was evidence, suggesting witches would pretend innocence. Faced with accusations, she had two choices: flee, which would be seen as an admission of guilt, or stay, which was interpreted as the devil's influence to claim her soul.
Whether living virtuously or not, displaying fear or calmness when accused, or choosing to flee or stay upon confrontation, all actions were interpreted as proof of guilt. As evident, the system was rigged to leave the accused without recourse. Spee, having served as confessor to numerous accused witches, had insight into every facet of the accusatory process.
Consequently, anything the accused said or did was twisted into evidence against her. Typically, only one perspective was presented in these cases, and it was never the accused witches'. This approach underscores their dire situation and highlights both the illogical and profoundly unjust nature of the witch trials.
This passage also highlights the importance of scientists documenting their predictions before experiments through hypotheses or research questions, to uphold integrity and adhere to scientific and probabilistic principles. The concept that "the absence of evidence is evidence of absence" is an illustration of the broader probabilistic principle known as the Conservation of Expected Evidence, emphasizing the limitations imposed by Probability Theory on what can be conclusively determined from experimental data.
The Conservation of Expected Evidence, also known as the law of total expectation or the law of iterated expectations, asserts that for every expectation of finding evidence, there must be an equally weighted expectation of discovering counter-evidence. This principle affects both the direction in which expectations are updated and the extent of that update.
Essentially, the anticipation of encountering evidence, prior to its actual observation, should not alter pre-existing beliefs. The principle encourages a balanced view of evidence and counter-evidence, emphasizing objectivity in belief formation and adjustment.
Let's formally analyze this concept:
Consider a hypothesis H and an evidence (or observation) E. The prior probability of the hypothesis is P(H); the posterior probability is P(H|E) or P(H|¬E), depending on whether we observe E or non-E (evidence or counter-evidence). Here, we read P(H|E) as the probability of H given E. Similarly, we read P(H|¬E) as the probability of H given non-E. The probability of observing E is P(E), and the probability of observing non-E is P(¬E). Thus, the expected value of the posterior probability of the hypothesis is:
P(H|E) * P(E) + P(H|¬E) * P(¬E)
The prior probability of the hypothesis can similarly be categorized, leading us to the following conclusion:
P(H) = P(H,E) + P(H,¬E)
P(H) = P(H|E) * P(E) + P(H|¬E) * P(¬E)
Therefore, the expected value of the posterior probability matches the prior probability. This demonstrates that merely anticipating evidence does not alter pre-existing beliefs.
Conversely, if we anticipate the probability of a hypothesis will be altered upon encountering evidence, the magnitude of this alteration, given the evidence is affirmative, can be summarized as follows:
D1 = P(H|E) – P(H)
If the evidence is negative, we have:
D2 = P(H|¬E) – P(H)
The expected magnitude of change, upon receiving positive evidence, mirrors the inverse of the expected change when encountering counter-evidence. To formalize this concept, we proceed as follows:
D1 * P(E) = -D2 * P(¬E)
Anticipating a high likelihood of encountering weak evidence in one direction necessitates a corresponding low expectation of encountering strong evidence in the opposite direction.
Suppose you are a scientist highly confident in your theory, expecting results that align with your hypothesis. In such a case, positive outcomes slightly boost your belief. However, if the results contradict your expectations, this significant deviation prompts a critical reassessment of your beliefs.
Revisiting Spee's logic, we can distill his argument into a critique of contradictory reasoning: if leading a "good and proper life" is considered evidence of witchcraft, then logically, a "bad and improper life" should indicate innocence. Similarly, if the absence of divine proof is deemed a test of faith, then biblical miracles would paradoxically suggest divine non-existence. This highlights the flawed logic in applying inconsistent standards to prove a point.
Consider how the revised passage makes you feel. Notice any underlying discomfort or the subtle tension it may invoke. This reaction is significant and worth paying attention to, as it underscores the narrative's potentially strained logic or the uneasy implications of its arguments.
For a true Bayesian, seeking evidence solely to confirm a theory is a fallacy. There's no strategy or plan that can justifiably increase one's confidence in a predetermined hypothesis. The emphasis should always be on testing theories with evidence, rather than attempting to confirm them.
This insight should significantly reduce your stress. There's no need to fret over interpreting every potential experimental outcome to bolster your theory. Nor must you devise ways to make each piece of evidence support your theory, knowing well that for every piece of expected evidence, there's an equally likely counter-evidence.
Attempting to undermine the evidence against an "abnormal" observation inevitably leads to a diminished support for a "normal" observation by a precisely equal and opposite degree. Essentially, it's a zero-sum game.
No matter the cleverness, argumentation, or strategy employed, Probability Theory ensures that manipulative tactics cannot skew the logical outcome in a favored direction.
So, my friends, remain calm and allow the evidence to reveal itself in its own time. Attempting to manipulate evidence to fit a desired narrative is futile; neither logic nor Probability Theory permits the conversion of falsehood into truth, irrespective of external validation or acceptance. Regardless of the narrative devised.