How to Visualize Bayesianism

5gwern

1David Udell

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They seem hard(er) to visualize moment to moment with respect to updates, imo. One of the neat things about lines is that the renormalization step of an observational update isn't cognitively demanding, whereas translating areas of disks-with-bites-taken-out-of-them-by-an-inconsistent-observation into new disks with those same areas is cognitively demanding.

But different Bayesian visualizations will work best for different people.

Major spoilers forplanecrash (Book 2)and for Eliezer'sMasculine Mongoose #3.## How Bayesians Lie; How to Lie to Bayesians

## Weighted Possible Worlds and their Correlated Observations

Here's how I like to think about Bayesian priors and updates. Imagine the panoply of possible worlds. Now imagine only the subset of that panoply that looks, first-personally, like the world you've seen so far. You're eliminating all the worlds that don't have you as an observer, and all the worlds where you-as-an-observer exist but made different observations than you recall seeing.

You now have this overlay of possible worlds on top of your view of the world.

Weighteach possible world in the overlay by its relative likelihood: let worlds that are very probable be heavy, and worlds that are deeply implausible be light. Don't worry too much about justifying those weights right now; thewhole pointof Bayesian updates is that your prior will quickly update to something reasonable. Just try and get a feel for your best gut judgements of possible world plausibility and encode those gut judgements as relative weights.One way to visualize weight is as length. Let each possible world in your overlay be a line segment, in addition to its overlay across your visual field. When a possible world says that an event is 60% likely, that possible world is wagering 60% of its current weight on that event occurring and 40% against that event occurring. If the possible world is represented by a line segment, then 610=35 of the line segment is now colored blue for the event occurring and 410=25 of the segment is red for the event not occurring. If the event occurs, you live in the blue subset of the panoply -- keep only the blue lengths. If the event doesn't occur, keep only the red lengths. Your relative weighting of possible worlds is the relative length of the surviving line segments.

Another way of visualizing weight, which is a little harder for me, is as first-personal vividness of a possible world in your overlay. Flit back and forth between the possible worlds you might inhabit. The prior probability of each is its brightness or vividness. See what each of them wagers will occur next. Discard the subset inconsistent with your observations. The relative brightness of each remaining first-person viewpoint in your overlay is that viewpoint's credence in your newly updated prior.

The possible worlds that bet relatively heavily on the observations you end up making will be the worlds that end up weighing the most in your new prior.