The conditional query fallacy: Applying Bayesian inference from discrete mathematics perspective

Preamble

    The Tilled Field,
Joan Miró
(Wikipedia)

One of the core concepts in data sciences is conditional probabilities, $p(x|y)$ appear as logical description of many of the tasks, such as formulating regression or as a core concept in Bayesian Inference. However, there is operationally no special meaning of a conditional or joint probabilities as their arguments are no more than a compositional event statements. This raise a question: Is there any fundamental relationship between Bayesian Inference and discrete mathematics that is practically relevant to us as practitioners? Since, both topics are based on discrete statements returning a Boolean variables. Unfortunately, the answer to this question is a rabbit hole and probably even an open research.  There is no clearly established connections between discrete mathematics fundamentals and Bayesian Inference.  

Statement mappings as definition of probability 

Statement is a logical description of some events, or set of events. Let's have a semi-formal description of such statements.

Definition: A mathematical or logical statement formed with boolean relationships $\mathscr{R}$ (conjunctions) among set of events $\mathscr{E}$,  so a statement $\mathbb{S}$ is formed with at least a tuple of $\langle \mathscr{R},  \mathscr{E} \rangle$. 

Relationships can be any binary operator and events could explain anything perceptional, i.e., a discretised existence. This is the core discrete mathematics and almost all problems in this domain formed in this setting from defining functions to graph theory. A probability is no exception and definition naturally follows, as so called statement mapping.

Definition: A probability $\mathbb{P}$ is a statement mapping, $\mathbb{P}: \mathbb{S} \rightarrow [0,1]$. 

The interpretation of this definitions that a logical statement is always True if probability is 1 and always False if it is 0. However, having conditionals based on this is not that clear cut.

Conditional Query Fallacy 

A non-commutative statement may imply, reversing the order of statements should not yield to the same filtered set on the data for Bayesian Inference. However, Bayes' theorem would have a fallacy for statement mappings for conditionals in this sense. 

Definition: The conditional query fallacy is defined as one can not update belief in probability, because reversing order of statements in conditional probabilities halts Bayes' update,  i.e., back to back query results into the same dataset for inference.

At first glance, this appears as a Bayes' rule does not support commutative property, practically posterior being equal to likelihood.  However, this fallacy appears to be a notational misdirection. Inference on the filtered dataset back to back constituting a conditional fallacy i.e., when a query language is used to filter data to get A|B and B|A yielding to the same dataset regardless of filtering order. 

Although, in inference with data, likelihood is actually not a conditional probability, strictly speaking and not a filtering operation. It is merely a measure of update rule. We compute likelihood by multiplying values obtained by i.i.d. samples inserted into conjugate prior, a distribution is involved. Hence, the likelihood computationally is not really a reversal of conditional as in $P(A|B)$ written as reversed, $P(B|A)$.  

Outlook

In computing conditional probabilities for Bayesian Inference, our primary assumption is that conditional probabilities; likelihood and posterior are not identical. Discrete mathematics only allows Bayesian updates, if time evolution is explicitly stated with non-commutative statements for conditionals.

Going back to our initial question, indeed there is a deep connection between the fundamentals of discrete mathematics and Bayesian belief update on events as logical statements. The fallacy sounds a trivial error in judgement but (un)fortunately goes into philosophical definitions of probability that simultaneous tracking of time and sample space is not encoded in any of the notations explicitly, making statement filtering definition of probability a bit shaky.

Glossary of concepts

Statement Mapping A given set of mathematical statements mapped into a domain of numbers.

Probability A statement mapping, where domain is $\mathscr{D} = [0,1]$.

Conditional query fallacy Differently put than the above definition. Thinking that two conditional probabilities as reversed statements of each other in Bayesian inference, yields to the same dataset regardless of time-ordering of the queries.

Notes and further reading

• Fallacy is one computes $P(A|B)=P(B|A)$, while filtering results into identical datasets. Correction would be that, one needs to use different sample sizes for reverse statement or compute joints and marginals separately on their own filtered datasets. Use the first filtering sample size in computing the probability not the total.
• Here, discrete mathematics we refer to appears within arguments of probability. The discussion of discrete parameter estimations are a different topic. Gelman discusses this, here.
• Conjunction Fallacy
• Probability Interpretations
• Bayesian rabbit holes: Decoding conditional probability with non-commutative algebra M. Süzen (2022)
• Holes in Bayesian Statistics Gelman-Yao (2021) : This is a beautifully written article. Specially, proposal that context dependence should be used instead of subjective
• John Allen Paulos’s  Innumeracy: Mathematical Illiteracy and Its Consequences, 

• Random House (1988).  Interestingly Paullos mentions fallacies regarding the conditional probabilities. 

  An oldy book on numerical fallacies but goodie.    

Bayesian rabbit holes: Decoding conditional probability with non-commutative algebra

Preamble

    The White Rabbit
(Wikipedia)

A novice analyst or even experienced (data) scientist would have thought that the bar notation $|$ in representing conditional probability carries some different operational mathematics. Primarily when written in explicit distribution functions $p(x|y)$. Similar approach applies to joint probabilities such as $p(x, y)$ too. One could see a mixture of these, such as $p(x, y | z)$. In this short exposition, we clarify that none of these identifications within arguments of probability do have any different resulting operational meaning. 

Arguments in probabilities: Boolean statement and filtering 

Arguments in any probability are mathematical statements of discrete mathematics that correspond to events in the experimental setting. These are statements declaring some facts with a boolean outcome. These statements are queries to a data set. Such as, if the temperature is above $30$ degrees, $T > 30$. Temperature $T$ is a random variable. Unfortunately, the term random variable is often used differently in many textbooks. It is defined as a mapping rather than as a single variable. The bar $|$ in conditional probability $p(x|y)$, implies statement $x$ given that statement $y$ has already occurred, i.e., if. This interpretation implies that $y$ first occurred before $x$, but it doesn't imply that they are causally linked. The condition plays a role in filtering, a where clause in query languages. $p(x|y)$ boils down to $p_{y}(x)$, where the first statement $y$ is applied to the dataset before computing the probability on the remaining statement $x$.

In the case of joint probabilities $p(x, y)$, events co-occur, i.e., AND statement. In summary, anything in the argument of $p$ is written as a mathematical statement. In the case of assigning a distribution or a functional form to $p$, there is no particular role for conditionals or joints; the modelling approach sets an appropriate structure.

Conditioning does not imply casual direction: do-Calculus do

A filtering interpretation of conditional $p(x|y)$ does not imply causal direction, but $do$ operator does, $p(x|do(y))$. 

Non-commutative algebra: When frequentist are equivalent to Bayesian

Most of the simple filtering operations would result in identical results if reversed. $p(x|y) = p(y|x)$, prior being equal to posterior. This remark implies we can't apply Bayesian learning with commutative statements. We need non-commutative statements; as a result, one can do Bayesian learning with the newly arriving data, i.e., the arrival of new subjective evidence. The reason seems to be due to the frequentist nature of filtering.

Outlook 

Even though we provided some revelations on decoding the operational meaning of conditional probabilities, we suggested that any conditional, joint or any combination of these within the argument of probabilities has no operational purpose other than pre-processing steps. However, the philosophical and practical implications of probabilistic reasoning are always counterintuitive. Probabilistic reasoning is a complex problem computationally. From a causal inference perspective, we are better equipped to tackle these issues with do-Bayesian analysis.  

Further reading

• Discrete Mathematics, Oscar Levin
• Causal Inference in Statistics, A Primer, Judea Pearl, Madelyn Glymour, Nicholas P. Jewell
• Indicative Conditionals Stanford Encyclopedia of Philosophy 
• Bayesian Epistemology Stanford Encyclopedia of Philosophy 

Please Cite as:

 @misc{suezen22brh, 

     title = {Bayesian rabbit holes: Decoding conditional probability with non-commutative algebra}, 

     howpublished = {\url{https://science-memo.blogspot.com/2022/07/bayesian-conditional-noncommutative.html}}, 

     author = {Mehmet Süzen},

     year = {2022}

}