(1)

for each model where is just the parameter dependent likelihood function we used to find the posterior probabilities for the parameters of the model.

The integration over the parameters can be accomplished using the Markov Chain Monte Carlo, which I summarized previously here. We will start by defining the partition function

   (2)

where is an inverse temperature. The derivative of the log of the partition function gives

   (3)

which is equal to the ensemble average of . However, if we assume that the MCMC has reached stationarity then we can replace the ensemble average with a time average .  Integrating (3) over from 0 to 1 gives

From (1) and (2), we see that  , which is what we want to compute  and .

Hence, to perform Bayesian model comparison, we simply run the MCMC for each model at different temperatures (i.e. use as the likelihood in the standard MCMC) and then integrate the log likelihoods over at the end. For a Gaussian likelihood function, changing temperature is equivalent to changing the data “error”. The higher the temperature the larger the presumed error. In practice, I usually run at seven to ten different values of and use a simple trapezoidal rule to integrate over .  I can even do parameter inference and model comparison in the same MCMC run.

Erratum, 2013-5-2013,  I just fixed an error in the final formula

A salient example is the current idea that fiscal austerity will lead to greater economic growth. GDP is defined as the sum of  consumption, investment, government spending and exports minus imports. If consumption or investment were to decline in an economic contraction, as in the Great Recession, then the simple linear idea would be that GDP and growth can be bolstered by increased government spending. This was the standard government response immediately after the financial crisis of 2008. However, starting in about 2010 when the recovery wasn’t deemed fast enough instead of considering the simple idea that the stimulus wasn’t big enough, the idea that policy makers, especially in Europe, adopted was that government spending was crowding out private spending so that a decrease in government spending would lead to a net increase in GDP and growth. This is very nonlinear thinking because it requires a decrease in GDP to induce an increase in GDP. Thus far this idea is not working and austerity has led to lower GDP growth in all countries that have tried it.  This idea was reinforced by a famous, now infamous, paper by Reinhart and Rogoff, which claimed that when government debt reaches 90% of GDP, growth is severely curtailed. This result has been taken as undisputed truth by governments and the press even though there were many economists who questioned it.  However, it turns out that the paper has major errors (including an Excel coding error). See here for a summary.  This is case where the nonlinear idea (as well as conflating correlation with causation) is probably wrong and has inflicted immense hardship on a large number of people.

Diane C Adler-Wailes, Vipul Periwal, Asem H Ali, Sheila M Brady, Jennifer R McDuffie, Gabriel I Uwaifo, Marian Tanofsky-Kraff, Christine G Salaita, Van S Hubbard, James C Reynolds, Carson C Chow, Anne E Sumner, Jack A Yanovski

Section on Growth and Obesity (D.C.A.-W., A.H.A., S.J.R.M., G.I.U., M.T.-K., J.A.Y.), Program in Developmental Endocrinology and Genetics, Eunice Kennedy Shriver National Institute of Child Health and Human Development; Mathematical Cell Modeling Section (V.P., C.C.C.), Division of Extramural Activities (C.G.S.), Division of Nutrition Research Coordination (V.S.H.), and Laboratory of Endocrinology and Receptor Biology (A.E.S.), National Institute of Diabetes and Digestive and Kidney Diseases; and Nuclear Medicine Department (J.C.R.), Hatfield Clinical Research Center, National Institutes of Health, U.S. Department of Health and Human Services, Bethesda, Maryland 20892.

The Journal of clinical endocrinology and metabolism (impact factor: 6.5). 02/2013; DOI:10.1210/jc.2012-3817

ABSTRACT Context: In obesity, increases in free fatty acid (FFA) flux can predict development of insulin resistance. Adult women release more FFA relative to resting energy expenditure (REE) and have greater FFA clearance rates than men. In adolescents, it is unknown whether sex differences in FFA flux occur. Objective: Our objective was to determine the associations of sex, REE, and body composition with FFA kinetics in obese adolescents. Participants: Participants were from a convenience sample of 112 non-Hispanic white and black adolescents (31% male; age range, 12-18 years; body mass index SD score range, 1.6-3.1) studied before initiating obesity treatment. Main Outcome Measures: Glucose, insulin, and FFA were measured during insulin-modified frequently sampled iv glucose tolerance tests. Minimal models for glucose and FFA calculated insulin sensitivity index (SI) and FFA kinetics, including maximum (l0 + l2) and insulin-suppressed (l2) lipolysis rates, clearance rate constant (cf), and insulin concentration for 50% lipolysis suppression (ED50). Relationships of FFA measures to sex, REE, fat mass (FM), lean body mass (LBM) and visceral adipose tissue (VAT) were examined. Results: In models accounting for age, race, pubertal status, height, FM, and LBM, we found sex, pubertal status, age, and REE independently contributed to the prediction of l2 and l0 + l2 (P < .05). Sex and REE independently predicted ED50 (P < .05). Sex, FM/VAT, and LBM were independent predictors of cf. Girls had greater l2, l0 + l2 and ED50 (P < .05, adjusted for REE) and greater cf (P < .05, adjusted for FM or VAT) than boys. Conclusion: Independent of the effects of REE and FM, FFA kinetics differ significantly in obese adolescent girls and boys, suggesting greater FFA flux among girls.

Hepatitis, which literally means inflammation of the liver, was characterized by Hippocrates and known to be infectious since the 8th century. The disease had been postulated to be viral at the beginning of the 20th century and by the 1960′s two viruses termed Hepatitis A and Hepatitis B had been established. However, there still seemed to be another unidentified infectious agent which was termed Non-A Non-B Hepatitis NANBH.

Michael Hougton, George Kuo and Qui-Lim Choo were all working at the Chiron corporation in the early 1980′s.   Houghton started a project to discover the cause of NANBH in 1982 with Choo joining a short time later. They made significant process in generating mouse monoclonal antibodies with some specificity to NANBH infected materials from chimpanzee samples received from Daniel Bradley at the CDC. They used the antibodies to screen cDNA libraries from infected materials but they had not isolated an agent. George Kuo had his own lab at Chiron working on other projects but would interact with Houghton and Choo. Kuo suggested that they try blind cDNA immunoscreening on serum derived from actual NANBH patients. This approach was felt to be too risky but Kuo made a quantitative assessment that showed it was viable. After two years of intensive and heroic screening by the three of them, they identified one clone that was clearly derived from the NANBH genome and not from human or chimp DNA. This was definitive proof that NANBH was a virus, which is now called Hepatitis C. Kuo then developed a prototype of a clinical Hepatitis C antibody detection kit and used it to screen a panel of NANBH blood provided by Harvey Alter of the NIH. Kuo’s test was a resounding success and the blood test that came out of that work has probably saved 300 million or more people from Hepititis C infection.

The question then is who deserves the prizes. Is it Bradley and Alter, who did careful and diligent work obtaining samples or is it Houghton, Choo, and Kuo, who did the heroic experiments that isolated the virus? For completely unknown reasons, the Lasker was awarded to just Houghton and Alter, which primed the pump for more prizes to these two. Now that the Lasker and Gairdner prizes have been cleared, that leaves just the Nobel Prize. The scientific community could get it right this time and award it to Kuo, Choo, and Houghton.

Addendum added 2013-5-2:  I should add that many labs from around the world were also trying to isolate the infective agent of NANBH and all failed to identify the correct samples from Alter’s panel.  It is not clear how long it would have been and how many more people would have been infected if Kuo, Choo, and Houghton had not succeeded when they did.

The second line of thought was with regards to the question of how to generalize the Wilson-Cowan equation, which is a phenomenological population activity equation for a set of neurons, which I summarized here. That paper built upon the work that Michael had started in his PhD thesis with Jack Cowan. The Wilson-Cowan equation is a mean field theory of some system but it does not specify what that system is. Michael considered the variable in the Wilson-Cowan equation to be the rate (stochastic intensity) of a Poisson process and prescribed a microscopic stochastic system, dubbed the spike model, that was consistent with the Wilson-Cowan equation. He then considered deviations away from pure Poisson statistics. The expansion parameter in this case was more obscure. Away from a bifurcation (i.e. critical point) the statistics of firing would be pure Poisson but they would deviate near the critical point, so the small parameter was the inverse distance to criticality. Michael, Jack and I then derived a set of self-consistent set of equations for the mean rate and rate correlations that generalized the Wilson-Cowan equation.

The unifying theme of both approaches is that these systems can be described by either a hierarchy of moment equations or equivalently as a functional or path integral. This all boils down to the fact that any stochastic system is equivalently described by a distribution function or the moments of the distribution. Generally, it is impossible to explicitly calculate or compute these quantities but one can apply perturbation theory to extract meaningful quantities. For a path integral, this involves using Laplace’s method or the method of steepest descents to approximate an integral and in the moment hierarchy method it involves finding ways to truncate or close the system. These methods are also directly related to WKB expansion, but I’ll leave that connection to another post.

Markets are not perfect and often fail but they are clearly superior to central planning for the distribution of most resources (particularly consumer goods). However, they need to be monitored and regulated. When markets fail, government should intervene. Even the staunchest libertarian would support laws that prevent the elimination of your competitors by violence. Organized crime and drug cartels are an example of how businesses would run in the absence of laws. However, regulations and laws should have built-in sunset clauses that force them to be reviewed after a finite length of time. In some cases, a freer market makes sense. I believe that the government is bad in picking winners so if we want to promote alternative energy, we shouldn’t be helping nascent green industries but rather tax fossil fuel use and let the market decide what is best. Making cars more fuel-efficient may not lead to less energy use but just encourage people to drive more. If we want to save energy, we should make energy more expensive. We should also make regulations as universal and simple as possible to minimize  regulatory capture. I think means testing for social services like medicare is a bad idea because it will just encourage people to find clever ways to circumvent it. The same probably goes for need-based welfare. We should just give everyone a minimum income and let everyone keep any income above it. This would then provide a safety net but not a disincentive to work. Some people will choose to live on this minimum income but as I argued here, I think they should be allowed to. If we want to address wealth inequality then we should probably tax wealth directly rather than income. We want to encourage people to make as much money as possible but then spend it to keep the wealth circulating. By the same reasoning, I don’t like a consumption tax. Our economy is based on consumer spending so we don’t want to discourage that (unless it is for other reasons than economic).

People do not suddenly become selfless and rational when the political system changes but systems can mitigate the effects of their irrational and selfish tendencies. As the work of Kahneman, Tversky, Ariely, and others have shown, rational and scientific thinking does not come naturally to people. Having the market decide what is the most effective medical treatment is not a good idea. A perfect example is in a recent Econtalk podcast with libertarian leaning economist John Cochrane on healthcare. Cochrane suggested that instead of seeing a doctor first, he should just be allowed to buy antibiotics for his children whenever they had an earache. The most laughable part was his idea that we have rules against self-administering of drugs to protect uneducated people. Actually, the rules are to protect highly educated people like him who think that expertise in one area transfers to another. The last thing we want is for even more antibiotic use and more antibiotic resistant bacterial strains. I definitely do not want to live in a society where I have to wait for the market to penalize companies that provide unsafe food or build unsafe buildings. It doesn’t help me if my house collapses in an earthquake because the builder used inferior materials. Sure they may go out of business but I’m already dead.

There is no single perfect system or set of rules that one should always follow. We should design laws, regulations, and governments that are adaptable and adjust according to need. The US Constitution has been amended 27 times. The last time was in 1992, which just changed the rules on salaries for elected officials. The 26th amendment in 1971 made 18 the universal threshold age for voting. We are thus due for another amendment and I think the 2nd amendment, which guarantees the right to bear arms, is a place to start. We could make it more explicit what types of arms are protected and what types can be regulated by local laws. If we want to reduce gun violence then gun regulation makes sense. People will do things they later regret. If one is in the heat of an argument and there is a gun available then it could be used inadvertently. It takes a lot of training and skill to use a gun effectively. Accidents will happen. In the case of guns, failure often leads to death. I would prefer to live in a society where guns are scarce rather than one where everyone carries a weapon like the old wild west.