The Probability of an Event under a Simplicity Prior

In this short post, I talk about a difficulty we encountered while working on a model of the fragility of value. You can read my post about that work here. The TL;DR is that the probability of a fixed event can be made arbitrarily large or small depending on the choice of simplicity prior. I'm sure this result was already known, but I found it unintuitive and thought it might be helpful to share. Background In our model of the alignment problem, AI agents go through idealized alignment training before being deployed to heavily optimize the world. The alignment training restricts the set of value functions that the agent can wind up with to those that satisfy a proxy condition; such value functions are called proxies. We also have definitions of catastrophic and partially catastrophic value functions that are bad to optimize for. One thing we’re interested in is the probability of training resulting in an agent with a catastrophic value function for a given proxy condition. A natural way to model a prior over learned functions is to use a simplicity prior — that is, a distribution over proxies where lower K-complexity proxies are more likely. However, the difficulty we encountered is that the choice of simplicity prior can make this probability arbitrarily large or small. Arbitrary Probability of Catastrophe Let mjx-container[jax="CHTML"] { line-height: 0; } mjx-container [space="1"] { margin-left: .111em; } mjx-container [space="2"] { margin-left: .167em; } mjx-container [space="3"] { margin-left: .222em; } mjx-container [space="4"] { margin-left: .278em; } mjx-container [space="5"] { margin-left: .333em; } mjx-container [rspace="1"] { margin-right: .111em; } mjx-container [rspace="2"] { margin-right: .167em; } mjx-container [rspace="3"] { margin-right: .222em; } mjx-container [rspace="4"] { margin-right: .278em; } mjx-container [rspace="5"] { margin-right: .333em; } mjx-container [size="s"] { font-size: 70.7%; } mjx-container [size="ss"] { font-size: 50%; } mjx-container [size="Tn"] { font-size: 60%; } mjx-container [size="sm"] { font-size: 85%; } mjx-container [size="lg"] { font-size: 120%; } mjx-container [size="Lg"] { font-size: 144%; } mjx-container [size="LG"] { font-size: 173%; } mjx-container [size="hg"] { font-size: 207%; } mjx-container [size="HG"] { font-size: 249%; } mjx-container [width="full"] { width: 100%; } mjx-box { display: inline-block; } mjx-block { display: block; } mjx-itable { display: inline-table; } mjx-row { display: table-row; } mjx-row > * { display: table-cell; } mjx-mtext { display: inline-block; text-align: left; } mjx-mstyle { display: inline-block; } mjx-merror { display: inline-block; color: red; background-color: yellow; } mjx-mphantom { visibility: hidden; } _::-webkit-full-page-media, _:future, :root mjx-container { will-change: opacity; } mjx-math { display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0; } mjx-container[jax="CHTML"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="CHTML"][display="true"][width="full"] { display: flex; } mjx-container[jax="CHTML"][display="true"] mjx-math { padding: 0; } mjx-container[jax="CHTML"][justify="left"] { text-align: left; } mjx-container[jax="CHTML"][justify="right"] { text-align: right; } mjx-mi { display: inline-block; text-align: left; } mjx-c { display: inline-block; } mjx-utext { display: inline-block; padding: .75em 0 .2em 0; } mjx-msub { display: inline-block; text-align: left; } mjx-mo { display: inline-block; text-align: left; } mjx-stretchy-h { display: inline-table; width: 100%; } mjx-stretchy-h > * { display: table-cell; width: 0; } mjx-stretchy-h > * > mjx-c { display: inline-block; transform: scalex(1.0000001); } mjx-stretchy-h > * > mjx-c::before { display: inline-block; width: initial; } mjx-stretchy-h > mjx-ext { /* IE */ overflow: hidden; /* others */ overflow: clip visible; width: 100%; } mjx-stretchy-h > mjx-ext > mjx-c::before { transform: scalex(500); } mjx-stretchy-h > mjx-ext > mjx-c { width: 0; } mjx-stretchy-h > mjx-beg > mjx-c { margin-right: -.1em; } mjx-stretchy-h > mjx-end > mjx-c { margin-left: -.1em; } mjx-stretchy-v { display: inline-block; } mjx-stretchy-v > * { display: block; } mjx-stretchy-v > mjx-beg { height: 0; } mjx-stretchy-v > mjx-end > mjx-c { display: block; } mjx-stretchy-v > * > mjx-c { transform: scaley(1.0000001); transform-origin: left center; overflow: hidden; } mjx-stretchy-v > mjx-ext { display: block; height: 100%; box-sizing: border-box; border: 0px solid