Exploratory experiment: Desire as Regulariser
What is computed
Let \( p(\theta \mid D) \) be a Bayesian posterior given data \( D \), and let \( \pi_d(\theta) \) be a desire prior with \( D_{\mathrm{KL}}(\pi_d \| p) > 0 \). The runs form the tempered posterior
\[ p_d(\theta \mid D) \;\propto\; p(\theta \mid D) \cdot \pi_d(\theta)^\alpha, \quad 0 < \alpha < 1, \]and record its calibration error (gap between predicted and observed frequencies) against the untempered posterior and against a desire prior aligned with the evidence, at 20, 80 and 200 observations.
Mechanism under test
A prior that pulls the posterior away from the data acts as a shrinkage term, in the same way as Tikhonov regularisation. The question the runs explore is whether a partially contradicting prior does this more usefully than an aligned one at finite sample sizes, and whether any effect persists as the horizon grows.
Runs
| Observations | Aligned desire | Misaligned desire |
|---|---|---|
| 20 | Baseline | Lower calibration error in this run |
| 80 | Baseline | Lower calibration error in this run |
| 200 | Baseline | Lower calibration error in this run |
The per-horizon error values are on the experiment pages. The same construction is tried in three other settings; the numbers are reported as data on the linked pages, without a cross-domain conclusion.
| Setting | Construction | Page |
|---|---|---|
| Bayesian beliefs | Partially contradicting desire prior | Deep anima beliefs |
| Repeated games | Partially contradicting desire in the agents' payoffs | Multi-agent games |
| GANs | Discriminator as opposing objective | GAN training |
| Annealing | Counter-gradient desire on the learning-rate schedule | exp_skeptical_annealing.sx |
Relation to the conjectures
Conjecture 6.3 (annealed skepticism) and Conjecture 6.5 (sustained skepticism) are the two readings of these runs; both are listed as explored.
Experiment Files
exp_anima_deep.sx — belief interaction, consolidation, desire prior
exp_anima_correlated.sx — correlated belief streams with and without a learned interaction
exp_skeptical_annealing.sx — sustained, annealed and absent skeptic at several horizons