We do ten experiments. A scientist observes the results, constructs a theory consistent with them, and uses it to predict the results of the next ten. We do them and the results fit his predictions. A second scientist now constructs a theory consistent with the results of all twenty experiments.

The two theories give different predictions for the next experiment. Which do we believe [more]? Why?

My intuition is that we should believe the first theory more. Making a successful and surprising prediction makes a theory more plausible. The second theory also makes a successful "prediction" of the second set of results but it is not surprising (since those results were already factored into that theory). So I think the second theory has greater risk of over-fitting.
But somehow I can't manage to formalize this reasoning.
Also, I find it strange that this apparently simple problem was not addressed in the books I read on probability theory.

There were a few different attempts on this thread, but none seems quite successful.
I see how other factors such as the skills of the scientists and the priors of both theories (Occam's Razor) could come into play.
But I hope that a formal analysis would isolate the "prediction vs explanation" factor.

This issue is recurrent as many basic economic questions which need constant explaining. For instance, it came up recently in the presidential debate and last week in discussions of Apple possibly setting up factories in the US.

Now consider that the factory may be in a shipping yard and that the workers may be dock workers. The technology is trade.

Just like any factory, it requires engineers (to design the production line), managers (to coordinate the flow), workers (to handle and assemble stuff), resources (as inputs), technology (processes, information systems, shipping infrastructure) and time (to process the inputs into outputs).
From the outside, both look the same, they only differ in what happens behind the scene.
In both cases, the improvement allows the customer to get more benefits for lower costs (less or cheaper resources and labor).

Yet, just as Luddites were afraid of the effects of technology (automated looms), protectionists are afraid of the effects of trade. They both wrongly see temporarily displaced workers and their pain of adjustment as a threat to employment levels, they ignore the benefits such developments and specialization bring (see Ricardo's difficult idea), and they seek special privileges for the party most visibly affected (despite long-term consequences and principles of voluntary exchange).

David Friedman and Steve Landsburg use the Iowa car crop to illustrate this point:

There are two technologies for producing automobiles in America. One is to manufacture them in Detroit, and the other is to grow them in Iowa. Everybody knows about the first technology; let me tell you about the second. First, you plant seeds, which are the raw material from which automobiles are constructed. You wait a few months until wheat appears. Then you harvest the wheat, load it onto ships, and send the ships eastward into the Pacific Ocean. After a few months, the ships reappear with Toyotas on them.

International trade is nothing but a form of technology. The fact that there is a place called Japan, with people and factories, is quite irrelevant to Americans’ well-being. To analyze trade policies, we might as well assume that Japan is a giant machine with mysterious inner workings that convert wheat into cars.

Any policy designed to favor the first American technology over the second is a policy designed to favor American auto producers in Detroit over American auto producers in Iowa. A tax or a ban on ‘imported’ automobiles is a tax or a ban on Iowa-grown automobiles. If you protect Detroit carmakers from competition, then you must damage Iowa farmers, because Iowa farmers are the competition.

Xeni points out that healthcare in the US is screwed and expensive, which she illustrates with the story of a sick father and her own. No contention with that. But she follows with the typical anti-capitalist case (greedy providers) and a link to Sicko, which offers similar arguments and promotes government-run healthcare.
My comment is admittedly analytical and argumentative, it focuses on economics and ethics. I point out how government intervention made such situations worse and question the morality of compulsory "charity" (welfare programs).

I was shaken by the moderator's feedback. But upon reflection, I stand by my comment. It is copied below with a link to the original post and the moderator's email.
BB is a forum for exchanging ideas honestly and openly (or at least reasonably so), and it is crucial to respond to incorrect economic and social arguments which result in deeply harmful effects.

Specifically, I cannot see how promoting a solution that helps people sustainably is somehow less compassionate than a bad solution promoted by someone ill (even tragically so). Call me callous in my delivery, but I care more about the well-being of millions of people than hurting the feelings of one.

1. The poor and sick are better helped by lowered costs and improved quality. Healthcare is not so different from food or other necessities that laws of economics would suddenly flip around.
2. There is no morality in a solution that bring 14 trillion dollars of debt burden (not counting multiples of that in unfunded of welfare liabilities) onto coming generations. How difficult will it be for them to afford medical treatment when they are still paying back this generation's debt?
3. Think of how well the poor, the old and the sick will cope in debt- and tax-ridden states. Seen the news from Greece recently? Keep an eye on Spain, France, the UK, and soon the US which are similarly on the verge of bankruptcy. How sustainable are the current benefits? Should people try to understand the source of their hardships, or simply feel satisfied with scapegoats?

I understand the illusion and temptation of government and that we are all only human, but if I fall seriously ill tomorrow, I will still curse the interventions that made the situation un-necessarily worse.

Xeni, I would like to understand your concepts of charity and respect for individuals. More generally your morals.

There are four ways for dealing with a catastrophic event: (1) you have savings/buffer to deal with it, (2) you have insurance (voluntary deal to mitigate the risk), (3) you rely on charity (voluntary donations) or (4) you rely on government (forced "charity"). The last one is really not charity, just as taking is not giving.
To the extent that government is the one solution out of four that involves using force on others and it is also government that screwed up our current healthcare system (high costs which you deplore), I don't understand why you reject the other three options and in particular charity.

Cancer treatments could be inherently expensive, but they are also clearly more costly than they should as a result of government interventions (Medicare, Medicaid, AMA licensing, FDA).
By shifting healthcare costs onto taxpayers (by force) you are not promoting charity and solidarity, but you are allowing providers to charge more. It's economics 101, a simple subsidy.
By restricting supply, through AMA licensing, FDA and medical patents, you are restricting competition to bring costs down.

While I understand your outrage, it is misdirected. The profits of providers are not the root of the problem, merely a symptom. The root cause lies in the 80 years of government intervention.

Economist Hans-Hermann Hoppe outlines a four-step healthcare solution (http://mises.org/daily/3643), which would bring back normal market dynamics to healthcare: competition, innovation and falling costs.

Here is the moderators reply:

All of your comments on BB over the years have been textbook libertarian talking points, which I have left to other commenters to deconstruct. When you respond to a post by someone who's going through cancer treatment by giving her an empathy-less, kneejerk lecture on libertarian economics, you have gone outside the pale of acceptable interaction. I have suspended your account for the time being.

Antinous

If you're interested in healthcare, here are a few presentations and articles on the topic.
How did healthcare work before the welfare state (starting roughly in the 1930's)?
What effects do various policies and institutions have on healthcare (AMA, FDA, Medicaid/Medicare, patents, medical "insurance" regulations, tax rules)?

Mutual-aid society and lodge practice by David Beito.

Why costs are rising?

The analogy of food insurance.

Swedish healthcare.

Walter Block on health economics:

Milton Friedman Q&A on healthcare in the freed-market: