How to Avoid Idea Averaging
andrewxhill.com
andrewxhill.com
More specifically, it's an example of when a group of people will tend to agree on something that is the least objectionable on average. This means that it's less likely that an unconventional idea will be accepted.
This concept has been studied for a long time in the social psychology literature. For a survey of related topics: http://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?artic... (see specifically "Hidden Profiles and Common Knowledge").
[1]http://www.businessinsider.com/mitt-romney-intrade-bets-trad...
Of course, this also means you won't generate that once-in-a-million super great unconventional idea. But then again, I think that's the point. If you try to generate the great unconventional idea, the base rate of failure is so high that you should not have so much hubris as to believe you actually can do it, and most often you should admit that what you think is a great unconventional idea really probably can't do any better than whatever conventional wisdom would choose.
I think my view on this is not popular because everyone wants to believe in the idea underdog -- the idea that was spurned by conventional wisdom but proved itself regardless and made someone famous for executing upon it.
That's great and all, but in the same way that putting all of your money on a single stock is a bad strategy, failing to average across ideas embraced by conventional wisdom is also (often, but not always) a bad strategy.
On a different note though, there is a lot of value in your comment even connecting it back to the "wisdom of crowds" stuff. One of the issues with "wisdom of crowds" is that the aggregated average tends only to be better whenever there is a sufficient variance of opinions in the crowd. This way, there is not systematic bias in everyone's thinking, meaning that the average too will have that bias. The more variation, the more that each participant's particular way of being wrong will contribute to "canceling out" the errors, leaving the average in a good position.
To me it suggests that when unconventional ideas do hit it big, it is because they are highly related to some source of demonstrable bias in conventional wisdom, and by removing that bias, such ideas can actually improve upon whatever is the current state of conventional wisdom.
But most of the time, it seems we only hit upon these bias-based improvements by accident because tons and tons of people keep trying (and failing, often to their personal detriment) to commit to unconventional ideas that aren't articulated in terms of any demonstrable bias, and it ends up basically being an idea lottery to see who happens to hit the "jackpot" idea. That's what makes conventional wisdom appealing as an alternative, steady expected return with low risk, but it's also what means it will never lead you to the next big thing.
I would expect the "wisdom of the crowds" to have a different result if, when asking for each person how many peas are in a bottle, each person is first told of the current running average or median. I think it would achieve a better result if everyone made their own guess independent of the "crowd". Otherwise, the random bias that occurred from the first 20 people's guesses is going to percolate because everyone who is guessing after will have their guess anchored when they're told of the group opinion.
A lot of this is spelled out in the literature about why ensemble methods tend to be superior to individual methods in statistical inference, which has a lot of ties to why averages of non-expert opinions tend to be as accurate or even more accurate than small pools of expert opinions.
In the limit, when you reallocate credibility to a set of ideas in a manner that is fully derived from your prior beliefs and the evidence you have, it becomes exactly Bayesian reasoning. A posterior distribution is exactly an "idea average" where each idea (each potential outcome) gets as much (and no more) credibility as it deserves, according to the prior and the probability model at hand.
Here's somewhat of a popular account of it:
< https://www.washingtonpost.com/business/is-it-better-to-trus... >
There is also the recent stuff under the marketing buzzword "Wisdom of Crowds" e.g. < http://onlinelibrary.wiley.com/doi/10.1111/j.1551-6709.2011.... > or the Wikipedia article on it too (and many sources that point out potential problems and corner cases that are also important).
In general though, I think I have to disagree with the article's premise. Unless two distinct ideas truly are mutually exclusive (most things aren't), then it's better to have a diversified portfolio over the space of all the ideas than to put all your eggs in one basket, and averaging ideas is sort of a humble heuristic that seems to work in a lot of areas.
For example, you're chasing someone and you come to a fork in the road. You know they must have taken the left or the right branch, so your posterior distribution on their location has two modes. It also has a mean value, in the middle of the forest between the two branches, but unless you have specific evidence they've abandoned the road, this is a very low-probability state.
The type of averaging prescribed by Bayesian decision theory is averaging of utilities, not of beliefs or actions. You take the action having the highest expected utility, where the expectation is taken over your posterior distribution. Assuming you actually want to catch your suspect, the expected utility of following either the left or the right fork will be much higher than the utility of dropping the pursuit to poke around in the middle of the forest.
Further, you're simply wrong about Bayesianism being about averages of utilities. The best account of Bayesian probability as a mapping of plausibilities to the unit interval is in Jaynes' The Logic of Science, but there is also a brief account of it in David Mumford's essay The Dawning of the Age of Stochasticity and e.g. in the introduction of Bayesian estimation supersedes the t-test by Kruschke [1] (where he explicitly describes it in terms of reallocation of credibility). Further, a la Jaynes, I think the right way to understand probability at all is in a mind-projection fallacy sense of the term: it describes your state of ignorance about the uncertain item.
It's not about utilities of actions -- utilities can be modeled with probabilities if the utility of certain actions is uncertain, but that is different than the base concept of probability being about which outcomes are more valued.
Idea averaging is about taking multiple ideas and merging them into one. The Bayesian posterior is merely assigning probabilities to each individual idea. I.e., idea averaging is "knife + spoon = spoon with sharp edge". Bayesian posterior is "50% knife is best, 50% spoon is best => evidence => 5% knife is best, 95% spoon is best".
The latter is what the article is saying we should do; don't build a new idea incorporating everyone's contributions, instead just figure out which individual idea is the best and do it.
> idea averaging is "knife + spoon = spoon with sharp edge". Bayesian posterior is "50% knife is best, 50% spoon is best => evidence => 5% knife is best, 95% spoon is best".
There's a lot wrong with the above quote from your comment. First, you're confounding two very different things. One thing is whether two ideas are mutually exclusive (e.g. the product is fundamentally non-functional if it is a spoon with sharp edge, so blending the two ideas is not in the set of feasible solutions). I already granted in my first comment that genuine mutual exclusivity can be a reason to avoid idea averaging -- but it's just overstated: true mutual exclusivity like that doesn't happen as much as people argue it does.
Secondly, and more importantly, you could break up your budget to pursue the knife solution and the spoon solution independently. What fraction of your budget should you allocate to the knife part? That is, what if 'knife' and 'spoon' are two distinct models that you are considering, and instead of doing crappy model selection to choose merely one of them, you want to produce a model that is a superposition of the two -- what metaprior probability should you choose that the 'knife' model will be the one that succeeds, versus the 'spoon' model?
Maybe something more concrete would be helpful, just think of a Gaussian Mixture Model. You have many different models and you're not choosing just one. You are blending them (and people call this model averaging) by using whatever each different component model's prior probability is to govern what the overall outcome is.
Maybe we are just talking past one another, but I really think the way you keep emphasizing the idea "posterior is a distribution" makes me think you continue to misunderstand me.
Another way to put it is that you are averaging over a space of distributions -- sort of like the Dirichlet distribution. The thing you end up with, the thing that is the result of computing an average, is itself a probability distribution, that comes as an average over a bunch of other probability distributions.
This is why I don't agree with your claim, "The posterior is not remotely like idea averaging." It's exactly like averaging when you're talking about model averaging and I think model averaging is a very useful and good analogue to idea averaging in cases where there is not genuine mutual exclusivity between ideas.
To make predictions about the world, you compute an nexpected value (average) over Models. The explicit assumption of both GM and BMA is that only one gaussian or only one model is correct - you just don't know which one, and therefore need to average over your uncertainty to take into account all possibilities.
Idea averaging (as described in the article) is about averaging over world states not models.
This is so wrong I don't even know where to start. In fact, this is basically the notion of frequentism! One of the very most fundamental ideas of Bayesian reasoning is that there is no one true set of parameters nor is there one true model. There is only your state of knowledge about the space of all possible parameter sets or all possible models. I'm very surprised to see you, of all people, claiming this. Even a cursory Google search of BMA fundamentals disconfirms what you are saying, e.g. [1]
[1] http://www.stat.colostate.edu/~jah/papers/statsci.pdf
> Madigan and Raftery (1994) note that averaging over all the models in this fashion provides better average predictive ability, as measured by a logarithmic scoring rule, than using any single model Mj, conditional on M. Considerable empirical evidence now exists to support this theoretical claim...
and so on.
Frequentism doesn't even allow you to represent your belief with a probability distribution.
One of the very most fundamental ideas of Bayesian reasoning is that there is no one true set of parameters nor is there one true model. There is only your state of knowledge about the space of all possible parameter sets or all possible models.
Bayesian reasoning says there is one true set of parameters/model, you just don't know which one it is. The posterior distribution allows you to represent relative degrees of belief and figure out which model/parameter is more likely to be true.
Assuming you gather enough data, your posterior distribution will eventually approximate a delta function centered around that one true model. This is also what happens when you do BMA or gaussian mixtures.
The fact that model averaging provides better predictive ability doesn't contradict what I said.
What happens when your data is actually distributed according to a mixture of gaussians (c.f. Pearson's crabs).
The generative model (first choose gaussian, then draw from it) is the specific model. As you gather a sufficiently large sample from the population P, eventually you'll converge to a delta-function on the space of M.
So in Pearson's crab example, given enough data, you'll eventually either converge to the single model of 2 gaussian crab species, or you'll converge to a 3 gaussian crab species model, or you'll converge to a weibull single species model (assuming you put all 3 of those models into your prior).
You seem to have read the post quite differently, in a way that causes it to seem totally wrong.
I actually do think it's just a fact that "average" in both colloquial and mathematical usage means something like "to collapse multiple values to a single typical value" (even Bayesian model averaging collapses a distribution over probabilities into a single probability). But even if it were genuinely ambiguous what andrewxhill meant, you generally get a lot more out of a post by choosing the reading that allows it to be insightful over the reading the causes it to be nonsensical.
> even Bayesian model averaging collapses a distribution over probabilities into a single probability
as I mentioned in the child comment to the other reply to this comment, model averaging creates an average value, but the type of that value is a distribution. That is, you have a distribution over distributions (each coming from a different model) and the effect of averaging does not reduce you down to a point estimate, rather it reduces you down to just one distribution.
This is why it's totally fair to say a posterior can be an average. It is the average of a bunch of other distributions. I think if I had said it that way in my first comment, it would have removed some confusion.
But it is important, because the criticism that "a posterior isn't an average" is very wrong. A posterior most certainly can be the average of some other stuff, if that other stuff was itself a bunch of distributions -- and that's exactly what I am trying to talk about.
But to your other point, about the 'sensible' vs. 'not sensible' readings, I mostly agree. However, the problem is who gets to decide when two ideas fall into the "knife-vs-spoon-clearly-exclusive" category, or when it's more gray than that, and the choice is not so black and white, and there is not a need to over-commit to just one approach?
The reason the OP post strikes me as problematic is that it seems like a matter of opinion, or in the worst case a matter of bureaucratic/dictatorial mandate, as to when ideas are of the type that can be averaged vs. when they are not.
I'd generally like people to be more humble about it and tend to believe that conventional wisdom and model averaging are better, at least as a first heuristic, than deeply committing to just one thing. That way there might be less urgency to rush into the claim that some debate is "knife-vs-spoon".
I sort of see the whole "knife-vs-spoon" thing as a kind of Godwin's law of brainstorming. Once you invoke the "knife-vs-spoon-so-we-can't-average" claim, it's like game over and all useful intellectual discussion dies and everyone just either picks Team Knife or Team Spoon and then the political battles start. Unless it's really mutually exclusive, I'd rather that doesn't happen.
Product design ideas don't average together in this way. If one person wants to design a knife and another wants to design a spoon, and you average them together, you get a spoon with a sharp blade on one side, or maybe the blade is on the handle of the spoon, but either way, bad idea!! Or best case maybe you end up with a swiss army knife, in which case it's harder to clean, etc. Adding the knife feature sacrifices the simplicity of the spoon, which is extremely valuable.
I think you're pointing out a straw-man though. There are lots of cases where two ideas are not totally mutually exclusive, especially in design. And there are lots of times when the budget will allow you to explore building both designs in parallel and "investing in both" (thus diversifying your investment by not allocating the whole budget to only one project).
I agree when there is real mutual exclusivity you have to confidently choose one way to invest. I just think this is rarer than stated, and that lots of times you aren't actually forced to make that type of bifurcating decision, and so in general you should be eager to compromise more and more humble about not knowing which of the ideas will work, thus more willing to average over them, or divide up the budget and try multiple things.
I've become the "product" guy in the startup I'm in, and would love to learn more about helping a group innovate and push forward.
I was putting together a "best read of 2015" type post but never finished it. There were a couple on my list that I think are related and maybe you'd find interesting.
This one from HN a few weeks back, http://justinkan.com/three-stories
Higher level company organization stuff I thought these two books gave at times contradictory and at times very complimentary thoughts on what made teams work. In reality, neither were exactly about making teams work though :)
Playing to Win http://www.amazon.com/Playing-Win-Strategy-Really-Works/dp/1... is on creating strategies
Reinventing organizations http://www.amazon.com/Reinventing-Organizations-Frederic-Lal... is about holacracy but talks a lot to people's motivations in a company.
Hopefully I have a couple more interesting things to say on andrewxhill.com too, so keep an eye on it!
- "What should we put on our homepage?"
- A bunch of people have different ideas.
- "I know, we can put them all up with a carousel!"
Even Apple has succumbed to his, which is perhaps not surprising. Carousels often come out of collaborative cultures (everyone wins!), which has been a point of emphasis for Tim Cook.
Avoiding carousels requires that a single person is willing to make a bunch of colleagues unhappy by picking only one thing at a time to feature.