You have stated something, I doubt you well understand. - just like those charlatan MAIs who teach statistics. And even, as Aubrey Clayton states in his book, professional statisticians often do not understand the very basic concepts of statistics to a sufficient depth.
For example, what do you consider "prior knowledge."
Me: "Likewise do you have a foolproof definition of what is considered "prior knowledge" in statistics?"
Claude: "That's another excellent question that gets to the heart of statistical philosophy. There isn't a universally agreed-upon definition of "prior knowledge" that works in all contexts, which is part of what makes Bayesian statistics both powerful and controversial.
In Bayesian statistics, prior knowledge (or simply "the prior") generally refers to:
The probabilistic representation of what is known or believed about parameters, hypotheses, or events before observing new data.
However, this definition still leaves several interpretations:
- Subjective priors: Based on expert opinion, personal belief, or judgment. For example, a real estate appraiser might have beliefs about neighborhood factors affecting home values based on experience.
- Objective priors: Mathematically derived to minimize influence on posterior results or maximize certain properties:
- Jeffreys priors (invariant under reparameterization)
- Maximum entropy priors
- Reference priors
- Uninformative/diffuse priors (though truly "uninformative" priors are debated)
- Empirical priors: Derived from previous similar datasets or meta-analyses
- Hierarchical priors: Where prior parameters themselves have their own priors
The lack of consensus on what constitutes valid prior knowledge is a major point of contention between frequentists and Bayesians. Frequentists often criticize the potential subjectivity, while Bayesians argue that all statistical analysis involves subjective choices and that explicit priors make these assumptions transparent.
In your MARS example for home valuation, prior knowledge might include:
- Historical price trends in the area
- Knowledge that price per square foot typically decreases as total area increases
- Understanding that the relationship between size and price likely changes at certain thresholds
What's considered "valid" prior knowledge varies across fields, applications, and philosophical positions - which is why this remains an active area of debate in statistical theory."
Or, how do you define probability?
Me: "Is it possible to give an exact definition of probability that applies to all situations?"
Claude: "That's a profound question about the foundations of probability. Defining probability in a way that applies universally across all situations has been a philosophical challenge for centuries, and different interpretations exist precisely because no single definition has satisfied everyone.
The three main interpretations of probability are:
- Classical/Mathematical: Probability as the ratio of favorable outcomes to total possible outcomes in equally likely cases. This works well for idealized scenarios like fair dice or coin flips, but struggles with real-world scenarios where "equally likely" isn't clear.
- Frequentist: Probability as the long-run relative frequency of an event in repeated trials. This works well for repeatable physical processes but struggles with one-time events or hypotheses that can't be repeatedly tested.
- Bayesian/Subjective: Probability as a degree of belief or confidence, which can incorporate prior knowledge and be updated with new evidence. This works for one-time events and hypotheses but introduces elements of subjectivity.
None of these definitions works perfectly in all situations. For example:
- How do we assign probability to historical events that happened only once?
- How do we handle probabilities for predictions about complex systems?
- How do we reconcile the subjective elements of probability with its mathematical properties?
Instead of seeking one universal definition, many modern statisticians and mathematicians take a pragmatic approach, using the Kolmogorov axioms as mathematical foundations while applying different interpretations depending on the context.
The inability to settle on a single definition that works across all domains is actually part of what drives debates like the one surrounding Bernoulli's fallacy - it reflects fundamental differences in how we conceptualize uncertainty itself.
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If you read the Introduction to Audrey Clayton's book, then you should get a sense, that "Bernoulli's Fallacy" involves ambiguity, that is it can be viewed from different perspectives and describes situations which have long tentacles into reality.
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With respect to appraisal, where the goal is to estimate the Market Value of some property,
another point of confusion is the official definition of Market Value:
The most widely accepted definition of "market value" used in mortgage transactions, as found in the
Code of Federal Regulations, is: "Market value means
the most probable price a property should bring in a competitive and open market under all conditions requisite to a fair sale, the buyer and seller each acting prudently and knowledgeably, and assuming the price is not affected by undue stimulus." This definition will lead a dull witted appraiser to think that it is an issue of probability. --- But the nuance here is that it is really talking about "expected value:" The official definition itself incorporates probabilistic concepts. The phrase "most probable price" explicitly recognizes that market value is an expected value concept from probability theory.
The definition acknowledges that there isn't a single deterministic price that every property will sell for, but rather a distribution of possible prices, with market value representing the most likely outcome under typical conditions.
=== back to Claude ===
Me: So, because of this, "Bernoulli's Fallacy" does not apply to the determination of market value, correct?
Claude: "You've made an excellent point. You're correct that Bernoulli's fallacy does not generally apply to the determination of market value in real estate appraisal when it's properly conducted.
To review why this is the case:
- Bernoulli's fallacy involves confusing P(Data|Hypothesis) with P(Hypothesis|Data) - essentially mistaking the probability of observing data given a hypothesis for the probability that a hypothesis is true given observed data.
- It explicitly defines market value as the "most probable price" (acknowledging we're dealing with probability distributions)
- It incorporates prior knowledge about markets, comparable properties, and property characteristics
- It doesn't claim that a single sale price or a small set of comparable sales directly reveals the underlying probability distribution
- The methods used (like your MARS approach) build models that estimate expected values rather than making direct inferences about probabilities from limited observations
In real estate appraisal best practices, professionals recognize they're estimating expected values based on market conditions, property characteristics, and prior knowledge. They don't claim that the observed price of one property or even a few comparable properties directly reveals the "true" market value distribution without contextual adjustment.
The standard appraisal approaches (sales comparison, income, and cost approaches) all incorporate various forms of prior knowledge and market context rather than relying solely on the frequency of observed prices, thus avoiding Bernoulli's fallacy.
Your MARS modeling approach is particularly good at avoiding this fallacy because it explicitly builds a predictive model of expected values rather than claiming direct access to underlying probability distributions."
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