Haack's Foundherentism and Bayesian Inference

Susan Haack, in discussing her epistemological theory that she calls “Foundherentism”, proposes that

Because quality of evidence is multi-dimensional, we should not necessarily expect a linear ordering of degrees of justification … Nor … does it look realistic to aspire to anything as ambitious as a numerical scale of degrees of justification.

The three dimensions to evidential quality she proposes are supportiveness, independent security, and comprehensiveness:

How justified \(A\) is in believing that \(P\) … depends on

• how well the belief in question is supported by his experiential evidence and reasons [supportiveness];

• how justified his reasons are, independent of the belief in question [independent security];

• and how much of the relevant evidence his evidence includes [comprehensiveness].¹

I’ll argue here that Bayesian inference can be seen as a way of providing just such a linear ordering and numerical scale in some circumstances.

We can start by associating Haack’s terms with the jargon of Bayesian inference.

beliefs and evidence

The belief \(P\) might be identified with a statement about model parameters that can be assigned a posterior probability, e.g. “parameter \(x\) has a value less than \(0.5\) (though see further discussion below). \(A\)’s “experiential evidence” seems similar to the set of observables in a Bayesian model.

Haack uses a crossword analogy in which a \(P\) is a proposal for a given entry (“3 Down is ‘onomatopoeia’”) and the experiential evidence is provided by the clue for the given entry.

independent security

In the crossword analogy, the degree of independent security is determined by how confident \(A\) is in the other intersecting entries that are already filled in (and thus provide part of the evidence for a proposed value of the entry itself). This confidence is, in turn, based on the clues for those entries plus entries that intersect them, until eventually we reach bare clues.

In a Bayesian model of the whole crossword, I would normally consider all the clues to be observables and all the entries to be random variables. However, we could conceive of a model with a single entry as the parameter of interest and the corresponding clue as the observable. The intersecting entries could then be “nuisance” parameters to be marginalized over. Our confidence in the intersecting entries would be expressed as priors over their possible values.

Therefore, I propose that independent security is identified with the prior. Note that the model itself can be considered to be part of the prior: it can be conceived of as having been drawn from a larger space of all possible models and assigned a prior of 1 (with all other models assigned 0). Of course, this (and any) choice of prior can be debated, just as an assessment of the independent security of \(A\)’s belief that \(P\) can be debated.

supportiveness

Haack’s supportiveness seems closely related to the likelihood since it incorporates \(A\)’s “experiential evidence”. In the single-entry crossword example, the likelihood would be defined in terms of how well a candidate entry fit the clue (the “experiential evidence”).

Haack also wants supportiveness to include \(A\)’s “reasons,” which seem to be non-experiential evidence or prior beliefs. In that case we’d need to identify supportiveness with the posterior, which incorporates the prior on intersecting entries (the “independent security”). However, since we’d like a separate notion of “supportiveness”, I propose instead to stick with the likelihood, which only depends on the “experiential evidence” contained in the observables.

We are now free to identify the Bayesian posterior probability as our scalar measure of justification.

comprehensiveness

Comprehensiveness is addressed by model expansion in Bayesian inference. Let’s say \(A\) proposes a model that incorporates a set of observables \(y\). Based on the posterior of this model \(A\) assigns a particular degree of justification \(p(P | y)\) to belief \(P\). \(B\) criticizes \(A\) by suggesting that she overlooked some “relevant evidence” \(z\). How can \(A\) reassess her justification for believing \(P\)? Trivially, if \(z\) simply consists of more instances of the same type of observations as \(y\), then they can just be appended to \(y\) to form \(y'\) and the posterior recalculated with the same model \(p(P | y')\).

In the general case we might have to change not just the observables but the entire specification of the model to non-trivially incorporate new evidence. This points to the idea that the meaning of “belief” might be context dependent. Within the framework of a given model it might refer to statements like “parameter \(x < 0.5\)”, which can be assigned a posterior probability. When the model itself is called into question, however, the “belief in question” expands to encompass the whole model, prior, parameter-value complex. To assign a scalar degree of justification to this expanded belief, we must define an overarching model that includes all competing beliefs (sub-models) and all relevant evidence.

Bayesian inference also naturally incorporates Haack’s idea of “degrees of relevance.” Consider a (sub-)model that depends, in part, on a type of observation that we don’t posses. We have to marginalized over the missing data, which will generally lower the likelihood of the observed data. Therefore adding the missing data should increase the likelihood of the sub-model and tend to increase its “justification” (posterior probability), while leaving unchanged the posterior of a sub-model that is indifferent to the extra evidence.

However, if the data is “irrelevant” – if it adds no information to the model and therefore fails to increase the likelihood of the observed data under the model – it will not increase the likelihood. ²

There are certainly other approaches to model comparison besides model expansion. Model comparison has a long and contentious history in Bayesian and non-Bayesian contexts, which I won’t discuss.

My argument here is that it is possible to use the Bayesian posterior probability as the numerical scale Haack referred to with skepticism, not that it is always possible to agree on the inputs to the process. In other words, Bayesian inference gives us a procedure for combining these dimensions, even if there’s still room for argument about each dimension.

Notes