New working paper, trying to channel some @SayWhatYouFound energy with the title: “Directional Motives and Different Priors are Observationally Equivalent”

Link:
andrewtlittle.com/wp-content/upl… Image
The general idea claimed in the title shows up in a lot of work, the ambition here is to make it precise and general.

A subject learns about a fact (or facts) w, by observing a signal (or signals) s, with no restrictions on the prior and signal structure.
Both before and after the signal, the subject has a motivated belief which balances accuracy and directional motives.

Accuracy measured by the Kullback-Leibler divergence from the Bayesian belief, directional motives just mean liking some states more than others. Image
The motivated beliefs take a simple form, where things the subject wants to believe get weighted up, and then renormalized to be a proper probability distribution.

It looks kinda like Bayesian updating! ImageImage
So, directional motives lead one to update like a subject who has observed an additional “signal”, resulting in a different prior before observing s. Call a Bayesian who observed such a signal the “Fully Bayesian Equivalent”.
The subject with motivated beliefs and the FBE respond to any signal (or sequence of signals) in an identical fashion. This is the sense in which directional motives and different priors are observationally equivalent.
Two immediate implications: (1) different groups (say, partisans) responding to info in different ways is not meaningful evidence of directional motives. It doesn't matter how clever one is in designing the signal structure, this just *never* works.
On the flipside, (2) coming up with a “Bayesian rationalization” for a pattern of updating is not meaningful evidence for a *lack* of directional motives. Being Bayesian just means going correctly from prior to posterior, but directional motives affect the prior too!
The rest of the paper digs deeper into particular empirical designs. The upshot is to detect directional motives we need to manipulate (or find natural variance in) directional motives, while keeping information constant.
Any comments appreciated! It’s now sort of in-between a short paper and regular length paper, so I’ll probably need to compress to the core or expand on the empirical discussion.

(end, for now...)

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