Manuel Baltieri<p>After a long collaboration with <span class="h-card" translate="no"><a href="https://mathstodon.xyz/@martinbiehl" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@<span>martinbiehl</span></a></span>, <span class="h-card" translate="no"><a href="https://mathstodon.xyz/@mc" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@<span>mc</span></a></span> and <span class="h-card" translate="no"><a href="https://mathstodon.xyz/@Nathaniel" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@<span>Nathaniel</span></a></span> I’m excited to share the first of (hopefully) many outputs:<br>“A Bayesian Interpretation of the Internal Model Principle”<br><a href="https://arxiv.org/abs/2503.00511" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="">arxiv.org/abs/2503.00511</span><span class="invisible"></span></a>.</p><p>This work combines ideas from control theory, applied <a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>categorytheory</span></a> and <a href="https://mathstodon.xyz/tags/Bayesian" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Bayesian</span></a> reasoning, with ramifications for <a href="https://mathstodon.xyz/tags/cognitive" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>cognitive</span></a> science, <a href="https://mathstodon.xyz/tags/AI" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>AI</span></a>/#ML, <a href="https://mathstodon.xyz/tags/ALife" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>ALife</span></a> and biology to be further explored in the future.</p><p>In these fields, we come across ideas of “models”, “internal models”, “world models”, etc. but it is hard to find formal definitions, and when one does, they usually aren’t general enough to cover all the aspects these different fields consider important.</p><p>In this work, we focus on two specific definitions of models, and show their connections. One is inspired by work in control theory, and one comes from Bayesian inference/filtering for cognitive science, AI and ALife, and is formalised with Markov categories.</p><p>In the first part, we review and reformulate the “internal model principle” from control theory (at least, one of its versions) in a more modern language heavily inspired by categorical systems theory (<a href="https://www.davidjaz.com/Papers/DynamicalBook.pdf" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">davidjaz.com/Papers/DynamicalB</span><span class="invisible">ook.pdf</span></a>, <a href="https://github.com/mattecapu/categorical-systems-theory/blob/master/main.pdf" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">github.com/mattecapu/categoric</span><span class="invisible">al-systems-theory/blob/master/main.pdf</span></a>).</p>