Is the functional relationship that polygraph artists claim to exist even mathematically possible? Is it possible to have a single binary independent variable (truth/deceit) affect three variables (respiration, blood pressure, galvanic skin response) in any way that isn’t completely simplistic?

… meaning what, exactly? Polygraphs don’t detect fine truthfulness/deceit, AFAIK–they’re inherently simplistic. They’re detecting anxiety, really, and if lying about a particular subject doesn’t make you measurably (to the instruments) anxious, it doesn’t work.

A polygraph just detects metabolic processes. The polygrapher has to make interpretations about what that means, based in part on comparison to a baseline. There’s no formula. You could get differing interpretations of the same data from different polygraphers.

f(x) = (x[sup]2[/sup] + 1, 2x - 1, 2[sup]x[/sup]). That’s one variable that determines three others.

This question isn’t really mathematical, though–it’s more biological and physical. Read what **toadspittle** said.

Yes, they’re bullshit. But that’s not what I asked. Is it mathematically possible for the sort of relationship posited by polygraphers to exist?

Maybe I’m missing the question, but as I understand it, the answer is:

sure, why not?

ultrafilter gave a mathematical example. Though maybe it would be clearer as:

p independent variable, range 0,1

a = 1 +2 * p

b = 47.3 - 51*p
c = p*1,147

To put in a biological context, Alive/Dead is a simple binary variable, and I don’t think you’ll disagree that it has a pretty strong effect on blood pressure, respiration, and just about every other physiological variable you can think of.

I’m not sure I’m getting what you’re asking, but isn’t this part backwards? I mean, what a polygraph machine (and reader) does is measure your respiration,…, then determine whether you’re telling a truth or a lie. So respiration,… are the independent variables, and whether you tell a truth or a lie is the dependent variable.

I don’t think any one could seriously argue that respiration ®, blood pressure (bp), and galvanic skin response (gsr) are functions of whether you lie or tell the truth, there are way too many other variables involved (does telling the truth completely determine your r, bp, and gsr? I don’t think so). However, the other way around (which is what a polygraph machine and reader do), where you determine whether a statment is true or not as a function of r, bp, and gsr, certainly seems reasonable to me. (I should clarify that–I don’t really think you *can* determine lie/truth telling in such a simplistic way, but it’s certainly a *mathematical* possibility).

As I understand it the claim is that deceit elicits certain bodily functions (either directly or throught an intermediary, like stress or whatever). From what I’ve been able to learn about it, the polygraph is portrayed by proponents as being all scientific and reliable and whatnot. So if we assume for a moment that it is reliable and scientific, then there should be some knowable relationship in the direction claimed by the proponents; i.e. deceit causes knowable changes in the three measured variables. Presumably, the relationship will need to be invertible since the detection of deceit goes the other way–the way you describe.

If physicist Jean Bricmont is correct, then to really be scientific the proponents should be able to tell us the form and specifics of this function–at least to an approximation. But it seemed to me as if the only sort of function you could get would be pretty simplistic. You either have truth or deceit, so even if you use a function like

f(x) = (x^2 + 1, 2x - 1, 2^x)

the results will be, for truth=0 and deceit=1, (1, -1, 1) or (2, 3, 2), respectively. But if you’ve ever seen a polygraph, the meters are all over the place. I get the impression that the signal output (as opposed to the noise) is still supposed to be more complicated than the hypothetical output I’ve described. (IIRC, there are two schools of polygraph artist: one looks only at the chart, the other looks at behavior during questioning and all that sort of stuff.)

So it doesn’t seem to me like anything more nuanced than the function given above is possible with an input of just 0 or 1. That’s what I’m wondering about.

That’s not quite what the proponents of polygraphs are claiming. The responses being measured are certainly functions of variables other than truth/lie: for example, the temperature of the room, the person’s metabolic processes, other things he might be thinking about… And the truth/falsehood is probably not a simple binary variable, either; people probably have different responses to telling different sorts of falsehoods or partial truths.

Imagine the three measured variables as defining a point in a cube. (The sides of the cube are where one of the variables is as large or as small as it can get.) The claim made by the pro-polygraph people is that the responses fall into two (mostly) separable clusters, one for people telling the truth and one for people who are lying. These clusters might be complicated regions within the cube, and there may also be “uncertain” regions, where the result is indeterminate. The various other unknown and uncontrolled variables (like the circadian rhythms or the other, unrelated, thoughts of the testee) mean that the clusters aren’t single points, even if the truth variable is binary true/false. They may even make the clusters overlap substantially, in which case the reliability of the polygraph would obviously be reduced.

I personally don’t know anything about polygraphs specifically, so I don’t know what these purported clusters look like. I doubt that there’s much to the polygraph as currently implemented. But the general idea is basically statistical parameter estimation and is mathematically sound.