What Is Bayes' Theorem and Why Does It Matter?

Let's find out if Joe uses drugs. We administer a drug test for heroin use, and the result comes back positive.

Given this result, how sure are we that Joe uses heroin?

• Good enough for me. Case closed!

• How good is that test?

• Whoa! Let's apply Bayes' Theorem!

Say what? What the heck is Bayes' Theorem, and why should we (or Joe) care?

Let's say we already knew that:

• 3% of the general population are heroin users. So Joe would have a 3% chance of getting a positive result to start with. This is called the baseline or "prior" probability.

• This particular test correctly identifies heroin users 95% of the time (5% of the time it would come back negative even though the sample was from a user). This is the test's "sensitivity".

• Using this test, non-users are correctly identified 90% of the time (10% of the time a non-user tests positive). This is the test's "selectivity".

Does this information, which we knew before we tested Joe, affect our determination about Joe's use or non-use of heroin? You bet it does!

Thomas Bayes

Fortunately, London Nonconformist minister Thomas Bayes devised a way to sort this out, back in the 1700s. His insight was rediscovered

Pierre-Simon Laplace

later in that century by French mathe­matician Laplace, and it has developed over the years as "Bayes' Theorem". It is used to estimate probabilities, given knowledge of certain related probabilities.

Joe's Case

In Joe's case, the relevant formula, applied to the facts above, tells us that there is just a 22.7% chance that Joe uses heroin, even though he tested positive. Based on this result, we can conclude that it is more likely that Joe is a user than that the average person is a user, but it is still much more likely that the test is wrong than that Joe uses heroin. The test provides incremental evidence that Joe uses heroin, but the total evidence we have so far still is far from supporting that conclusion. (This example was adapted from a site at Stanford.)

Bayes' Theorem

Bayes' Theorem is an important tool in understanding what we really know, given the evidence and other information we have. It helps incorporate "conditional probabilities" into our conclusions.

There are several mathematical formulas related to Bayes' Theorem, but they generally boil down to this:

conditional probability = prior probability × predictive power of evidence

Or,

What we know, given the evidence = what we knew even without the evidence, adjusted by how good that evidence is.

Bayes' Theorem tells us quantitatively how to update our prior information, given new evidence.

Another Example: Breast Cancer

Say we know, from lots of studies, that 1% of women at age forty have breast cancer. (This means that 99% of women of that age do not have breast cancer.) This is the "prior probability". Mammograms can give us additional information (the "evidence"). Say we know (about the quality of the evidence) that 80% of the women who do have breast cancer will get a positive mammogram (the test's "sensitivity"), and 9.6% of the women who do not have breast cancer will also get a positive mammogram (the test's "selectivity").

If a woman in this age group gets a positive mammogram, how likely is it that she actually has breast cancer? Most doctors, given this statement of the problem, get it wrong. Only 15% of doctors can answer this question correctly! (Doctors, like lawyers, are shockingly ignorant of Bayes' Theorem and its implications.)

The answer is that if a woman gets a positive result on her mammogram, there is a 7.8% chance that she actually has breast cancer. Of all the women who get positive results, only 7.8% actually have breast cancer.

This is calculated as follows:

Out of 10,000 women, 100 have breast cancer (the 1% "prior probability"). Eighty of those 100 will have positive mammograms (calculated from the 80% sensitivity of the test). From the same 10,000 women, 9,900 will not have breast cancer (the other 99%) and of those 9,900 women, 950 will also get positive mammograms (the test's 9.6% selectivity). This makes the total number of women with positive mammograms 950+80 or 1,030. Of those 1,030 women with positive mammograms, 80 will have cancer. Expressed as a proportion, this is 80/1,030 or 0.07767 or 7.8%.

So what good was it to have a mammogram? If a woman does have cancer, the prior probability of 1% is shifted substantially to 7.8% if her mammogram is positive. Only a small fraction of women who don't have cancer will have positive mammograms (9.6%). Obviously further tests will be called for, but the population of all women (or 10,000 in this example) has been focused down to 1,030 women with positive tests, reduced by 90%. And the proportion of that smaller population who do have cancer has shifted from 1% (the prior probability) to 7.8% (the conditional probability). The probability of cancer, if you are in this smaller sample, has increased by almost 8 times. That's useful information for you and your doctor, though not conclusive.

So use of Bayes' Theorem helps us know what we know. The results are often counterintuitive. That is why Bayes' Theorem is so important.

For a revealing illustration of how different Bayesian thinking can be from "common sense", and the impact that difference might have in the real world, look at the Case of the Careless Cab. There's a link there to bring you back here when you are done.

Further Resources

A lot of this discussion was lifted from this informative and fun site.

You might find this post on visualizing Bayes' theorem helpful.

Wikipedia also has a good article.

The portraits are from Wikimedia Commons at https://secure.wikimedia.org/wikipedia/commons/wiki/File:Thomas_Bayes.gif and https://secure.wikimedia.org/wikipedia/commons/wiki/File:Pierre-Simon_Laplace.jpg

Technorati tags: science, science education, statistics,
math, Science In Action

The Nose Knows . . . Stem Cells!

"And the nose [pause] has been cloned!" Miles Monroe (Woody Allen), Sleeper, 1973

For several years neuroscientists have been exploring the use of the stem cells found in the nose to help address injury and disease. Currently several teams of scientists are competing to develop clinical methods to help paralyzed patients.

[See update on success in pet dogs, below.]

The olfactory mucosa is one of the few sites where adults continue to grow new nerve cells from stem cells. Smell receptors can quickly regenerate after a cold or other damage. Professor Alan Mackay-Sim, deputy director of the Institute for Cell and Molecular Therapy at Griffith University, Queensland, has been studying smell and the unique properties of the cells he found in the nose for years. His team has been studying potential use of olfactory stem cells in treating Parkinson’s disease and is using stem cells from Parkinson’s sufferers to investigate the causes of Parkinson's.

Since nasal stem cells are abundantly available in a patient's own nose, concerns about transplant rejection are minimal, and no cultured stem cells from anyone's embryo are needed.

According to Mackay-Sim,

Our goal is to repair the brain and spinal cord by taking small pieces of tissue from the nose and transplanting these cells back into the same person in a manner similar to a skin graft. The cells could be grown in a dish to expand their numbers or they could be genetically engineered to cause them to express therapeutic molecules. The nose is the only place where neurons, and their associated cells, are easily accessible. Nasal transplants would overcome many ethical issues associated with cell therapy, such as the use of embryonic cells. [source]

Dr. Geoffrey Raisman and his team at University College London have experimented with rats, using nasal stem cells to repair spinal injuries. Dr Carlos Lima at the Egaz Moniz Hospital in Lisbon has performed similar operations on dozens of human patients.

Neural crest stem cells are another type of embryonic stem cell that persist into adulthood in hair follicles. Maya Sieber-Blum of the Medical College of Wisconsin and Milos Grim of Charles University Prague have previously shown that follicles might provide stem cells for some types of cell replacement therapy. (Abstract of a report of their work.)

The team at Griffiths University in Queensland, Australia, recently announced results of research where they transplanted cells from the olfactory mucosa of humans, rats, and mice in to chicken embryos. They demonstrated the cells can give rise not only to nerve cells but also to heart, liver, kidney, and muscle cells. A paper on their work is to be published online this week in Developmental Dynamics. “Multipotent stem cell in adult olfactory mucosa”, Wayne Murrell et al. (Abstract here. Subscription required for full text.)

This announcement got a lot of ink, partly because it comes in the middle of a debate in Australia about the legality of embryonic stem cell research, and partly because the research was partly funded by the Catholic Archdiocese of Sydney. Naturally anything about stem cells becomes political these days.

Dr. Murrell has said he hopes the findings will help advance adult stem cell research. "It's not that they can do more than the bone marrow or brain stem cells; it's just that, we hope, they will be easier to work with."

Comment

The citizens of California have recently committed to tax themselves $6 billion to establish a stem cell research center and sponsor stem cell research in their state, since the U.S. federal government has stopped funding research using embryonic stem cell lines.

If, as seems increasingly likely, use of embryonic stem cells turns out to be a minor detour on the road to effective stem cell therapies, and the National Institutes of Health and other federal funding agencies get back in the act, will the California program reap the benefits its backers have promised?

Here is the web site of The California Institute for Regenerative Medicine.

[Update 2012-11-19: Recent work with pet dogs suggests that some degree of paralysis can be partly overcome by transplantation of cloned olfactory sheathing cells from the lining of the nose to sites of spinal lesion. BBC article on the research here. We always knew dogs had good noses, but we didn't know everything they were good for! Enjoy the video in the BBC item.]

[ITN video of successful spinal cord repair in Guardian item here.]

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Further Reading:

This site describes research at Griffith U. Queensland Australia

Here are some images of the olfactory epithelium and its stem cells from Thomas Schoenfeld's site at University of Massachusetts.


Technorati tags: science, science education, stem cells, neurobiology, Science In Action

How Do Batteries Work?

Batteries (electrochemical cells) produce electricity by spontaneous chemical reactions. If you understand how batteries work, you understand a key part of chemistry and biochemistry. That is because batteries work by oxidation/reduction (redox) reactions, involving the transfer of electrons. All biochemical energy reactions, such as photosynthesis, chemosynthesis, respiration, phosphorilation, etc., depend on redox chemistry.

The terms used in redox chemistry are very confusing (I've never quite understood them). I'll try to line them all up so they make sense.

Electrons moving along a conductor is what we call electricity, or electric current. It can be used to generate magnetic fields (as in a motor) or heat things up (as in a light bulb), to drive other chemical reactions (e.g. electrolysis or electroplating), or for many other purposes. (What are electrons, anyway?)

History

Until the late 1700s nobody knew there was such a thing as electric current, or how to generate it. Electricity was all the rage, but principally static electricity (surface charges). The discharge of a static charge results in a brief current (lightning, static sparks, "shock" troops and jumping monks), but it cannot be sustained to do useful work.

In the 1780s Dr. Luigi Galvani, at the University of Bologna, noticed that muscles could be stimulated to twitch by electric discharges or by contact with dissimilar metals. These studies suggested a connection between electricity and chemistry. His colleague physicist Alessandro Giuseppe Antonio Volta, professor at the University of Pavia, invented the galvanic (electrochemical) cell in about 1800. A stack of discs of metal separated by blotters wet with salt water (a "Voltaic pile") generated significant current. Each element is a "cell", and a collection of cells is a "battery". Today we use the word "battery", in English, to refer to a single cell. (In French it is a "pile".) Here is what they looked like in Volta's day. The University of Pavia has a beautiful site showing some of Volta's apparatus.

The lead-acid battery was invented in 1859 by Gaston Plante, and the dry cell between 1867 and 1877 by George Leclanché, both of France. The alkaline cell was invented in 1914 by Thomas Edison. Dozens of other types of batteries have been developed since then. They all work by oxidation/reduction chemistry. Here are some pictures of cells of yesteryear.

An electrochemical cell has three main parts: two electrodes and an electrolyte. The electrodes are the dissimilar metals mentioned above. There is an electrolyte that allows ions to move between them. Outside the cell they can be connected by a circuit through which electrons will flow.

A lemon can be used to make a simple cell. The cell has a zinc strip, a copper strip, and the acidic juice of the lemon as the electrolyte. It generates about one volt, but only a very small amount of current. (The voltage of a battery is determined by the materials used as electrodes and electrolyte.)

Lemon Cell

Here is the site this picture came from, where you can learn how to make a lemon battery.

Redox Reactions

Redox reactions happen all the time. Hydrocarbon fuels (such as methane, which was also discovered and isolated by Alessandro Volta) can react with oxygen to be oxidized to carbon dioxide and water. The carbon is oxidized, and gives up electrons, and the oxygen accepts them. Oxidation and reduction are simultaneous. The trick in a battery is to separate the oxidation from the reduction, so the electrons have to go on a trip through the external circuit, where we can make use of them.

"Oxidation" and "reduction" reactions make a battery work. Oxidation/reduction reactions are electron-transfer reactions. For a battery to work, both an oxidation and a reduction must happen. One generates electrons at one electrode, and the other uses them up at the other electrode. Each of these is called a "half reaction". If the electrodes are connected outside the cell by a circuit, electrons flow and the full reaction is completed.

Oxidation is when electrons are transferred from a substance to oxygen or some other compound. Oxidation doesn't have to involve oxygen, and can be thought of as "de-electronation."

Since electrons are negatively charged particles you can see how this might be related to electricity. Remember that electrons moving along a conductor is electric current. The electrode where oxidation (loss of electrons) takes place is called the anode. On a commercial battery it is marked as the "-" side.

Reduction is when a chemical reactant accepts electrons. It ends up with more electrons than it started with. Reduction could be called "electronation."

Summary of Battery Terms

Anode	Cathode
Oxidation	Reduction
De-electronation	Electonation
Gives up electrons	Accepts additional electrons
- (minus) side	+ (plus) side

The substance that loses electrons is the "reducing agent" or "reductant", while the substance that gains electrons is the "oxidizing agent" or "oxidant". Anions are negative ions moving toward the anode. Cations are positive ions that move toward the cathode.

Summary Illustration

(The picture is from this site.)

OK, So What Makes Electrochemical Cells Work?

Different chemical reactions occur at the anode and the cathode in a cell. The reaction at the anode releases electrons, and leaves behind positively-charged ions. The reaction at the cathode soaks up electrons. Different materials have different tendencies to give up or accept electrons. By choosing the materials for the anode and the cathode carefully, cells with different properties can be designed.

The two dissimilar metals that form the electrodes of a cell have different "reduction potentials". Reduction potential expresses the tendency for a materiel to accept electrons (to be reduced). This reduction potential is expressed in volts. Explanation of "reduction potential". Here is a table of the reduction potentials of various half-reactions.

Here is how the lead-acid battery in a car works:

The anode of each cell is metallic lead, Pb. The cathode is solid lead oxide, PbO2. The electrolyte is sulfuric acid, H2SO4 (which is dissociated into hydrogen (H+) ions and hydrogen sulfate (HSO4-) ions in solution.)

When electrons are allowed to flow (when the battery is under load) this reaction takes place at the anode (- side):

Lead metal is oxidized to bivalent Pb(II), giving up 2 electrons, and reacts with sulfate ion (SO42-) to form lead sulfate and a hydrogen ion:

Pb + SO4²¯ → PbSO4 + 2e- + H+

The reduction potential of this reaction is +0.356 Volts.

Simultaneously at the cathode (+ side):

The lead in lead oxide is reduced from tetravalent Pb(IV) to bivalent Pb(II) when the lead oxide reacts with sulfate ions in the electrolyte and hydrogen ions, accepting electrons to form lead sulfate and water:

PbO2 + SO4²¯ + 4 H+ + 2 e- → PbSO4 + H2O

The potential of this reaction is 1.685 volts.

Combined, these two half reactions make up the redox reaction

Pb + PbO2 + 2H2SO4 → 2 PbSO4 + 2 H2O

Two electrons were liberated at the anode and flowed through the external circuit to the cathode. The total potential of the reaction is about +2 volts. A 12-volt car battery therefore has six cells in series, each contributing two volts for a total electric potential of 12 volts.

This site explains the reactions in a lead-acid cell in more detail.

This site has half-reactions and other information on all kinds of commercial batteries. Highly recommended. Check there to see how a nickel-cadmium, nickel-metal hydride, or alkaline battery works.

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major sources:
Powerstream
Electrochem Encyclopedia at Case Western Reserve and
Electrochemical Dictionary at Case Western Reserve

I don't have a link to Volta's original article. Maybe you can find one:

On the Electricity Excited by the Mere Contact of Conducting Substances of Different Kinds, A. Volta, "Philosophical Transactions" Vol. 2, pp 403-431, 1800.

good article on Volta and his invention

cool information about early electrical devices

beautiful site summarizing electrochemical cell concepts

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David Wheat's Science In Action site has articles about science and math in the real world, weird science, science news, unexpected connections, and other cool science stuff. There is an index of the articles by topic here.

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