Noise

Plenty of research these days has shown that our standard quick firing mental heuristics fail to grasp the subtlety of probability in our everyday world. This can pose some serious issues when attempting to meld the perspectives of science and journalism for example. Statistical statements are prolific in modern writing, and yet we often try to portray or interpret these probabilistic statements with more certainty than they deserve. Furthermore, understanding probabilistic concepts has typically been approached via our visual and symbolic processing systems. This post, I hope, gives a taste of the role sound can play in our standard perceptions of randomness, as well as re-conceptualizing randomness’ role in society.

The Monty Hall Problem

A prolific example of the counter-intuitive nature of probability is the Monty Hall problem. The Monty Hall problem sets outs to describe the ideal strategy in the game show Let’s Make a Deal. The game begins with the participant being shown three doors. Behind one of these doors is a valuable prize such as a sports car, but behind the other two doors lies, say, a goat. The goal here is to pick the door which hides the sports car.

You begin by choosing a door, say Door 1. Next, Monty will reveal what is behind one of these doors, namely what is behind Door 2 or Door 3. Suppose he reveals that a goat lies behind Door 3. Now, Monty asks you if you would like to switch your bet to Door 2 or keep your initial guess of Door 1. Which door should you choose?

If you’re like me, when first presented with this problem you’ll be inclined to say “well, it doesn’t matter which door you choose, because there is a 50% chance the car is behind either door!” That is, the car is behind one of the remaining two doors, so obviously your odds in either case are no better than guessing heads or tails in a (fair) coin flip.

You should always choose to switch doors, however. Why? Monty is not playing the same game as you in a sense — he knows which door hides the car after all! He certainly is not going to ruin the show’s appeal by revealing the car before you are asked whether or not you would like to switch doors. This imbalance of information changes the odds of winning from that of a coin flip.

A Typical Solution

Let’s look at a couple of tree diagram to try and make sense of this. The left hand diagram shows the potential outcomes with probabilities when you choose to stay with your initial choice (so Door 1 for us), and the right hand diagram shows the outcomes when you choose to switch (to Door 3).

This diagram is courtesy of the online course on game theory and probability at https://www3.nd.edu/~apilking/Math10120/Links/Simulations/SolMontyHall.htm

If you’re unfamiliar with this sort of representation, the branches each state the probability of the respective outcome. So, in the first step there is a 1/3 chance of each possible outcome. That is, you are equally likely to have picked the door with the car, goat A, and goat B. If you choose to stay with Door 1, then only if Door 1 originally hid the car would you win. However, in two of the three possible outcomes Door 1 really hid a goat. Here you have a 1/3 chance of winning the car in total. In the case that you switch, you instead have a 2/3 chance of winning by a similar argument. So you should always switch!

There is quite a bit more that can be said here. If you’re interested the link above has some good information with simulations, and https://en.wikipedia.org/wiki/Monty_Hall_problem provides a nice breakdown of the problem.

Why Do We Care?

The point here though is not to demonstrate how to solve somewhat involved probability problems, but rather to demonstrate the necessity for multiple representations of information in order to provide some level of explanation for the problem and solution itself. Here we have a contextual explanation of the game show, two tree diagrams with probabilities (fractions) on each branch associated with potential decisions of the player, and required arithmetic to answer this question. This is tricky even when we take our time, draw pictures and perform the calculations ourselves.

So, how well do we really do in interpreting a probabilistic statement made ‘out in the wild’? In our day-to-day lives it is often impractical to take the time and care that something like the Monty Hall problem implicitly demands. While useful in this context, these visual and symbolic approaches can be bloated and unnatural.

The Role of Sound

We are quite used to the cluttered overlay of sounds in our everyday lives. Cars rushing by, birds chirping, people chatting as they walk down the street — a soundscape consists of familiar clutter. Music also consists of layers of sounds and instruments, often in a relatively chaotic fashion. We can work to attune to the subtlety of these sounds in more ways than one.

Embodied Listening

We don’t just listen with our ears. Sound experts and rhetoricians including Steph Ceraso have written extensively about the many ways in which our bodies can become attuned to sound. For instance, the deaf percussionist Evelyn Glennie has become a famous example of someone who communicates her art through the vibratory nature of sound. Listening with our whole body provides a necessary depth in sensation that extends in many ways beyond the functions of our visual systems. We often try to juggle many visuals at once (as we did above), and yet with sound we have a capacity to hold many layers of information at one time.

Allowing our bodies to become more in tune with sounds around us makes us more susceptible to the rhetorical nature of sound. That is, the sounds around us force a ‘turning’ in which we implicitly or explicitly are affected by their presence. This idea can be broken down into the rhetorical categories of ambiance and salience. An ambient sound is one that we do not perceive, like the kitchen fan you’ve left on all day and don’t explicitly notice. This sound can become salient, however, once you realize its presence.

Probability: Ambient or Salient?

When presented with a numerical statement relating to social justice, the economy, scientific discovery, etc. there is necessarily some margin of error that comes along. Much work in statistical reporting has been done to try and make this error a salient entity. The American Statistical Association (ASA) has even pushed for multiple reporting: including confidence intervals along with familiar statistics such as the mean. In doing so, we explicitly report a margin of error rather than just a singular numeric statement.

Confidence intervals can be hard to interpret for those less familiar with the subject of statistics, so we can utilize sound to convey a similar amount of information by ‘playing’ error (noise) alongside reported averages/measures instead. Including variability and/or error through sound functions to give a voice to randomness.

Let’s now explore this idea in a specific setting. That is, let us look at two quotes from New York Times articles related to racial injustice in America; one from 1881 and one from 2016.

Certainty or Accuracy?

First, give a listen to the sound bite below (it is pretty quiet, but just familiarize yourself with the ‘noise’ that we will be using moving forward):

Here is a sound of static noise. This represents 5% relative error in the system.

We will use this standard volume level of noise to allow us to quantify the amount of error explicitly included in the statements below. That is, if the volume of noise doubles from the above, then there is a present error of 10% in the statement/system.

New York Times, 1881

Below are two quotes taken from Growth of the Colored Population, 1881 (New York Times). The first is purely for context:

“The Census Bureau furnishes some interesting figures regarding the increase of the colored population of the country in the last ten years. Their value for purposes of comparison is somewhat vitiated by the fact that the census of 1870 for the Southern States was very imperfect, its deficiencies being probably greater in regard to the colored than the white population. But they retain sufficient significance to dissipate the notion formerly somewhat prevalent that the negro race in a state of freedom was destined gradually to die out.”

Note they mention the imperfections of their own data collection, and yet have no means of truly quantifying the present randomness. Furthermore, in the passage below they report numbers to the one’s place! They both claim to have an imperfect data collection method, and that reporting a population count of 50,152,866 is reasonable:

“The total population of the country in 1880 was 50,152,866, against 38,558,371 in 1870, a gain of about 30 per cent. During the same period the colored population increased from 4,880,000 to 6,577,151, a gain of nearly 34.8 per cent. This would appear to indicate that natural increase went on among negroes at a considerably higher rate than among the whites.”

Here is the above quote read aloud with 5% error.

Above we have simply included the original 5% error rate which is fairly common in modern statistical reporting. However, based on their own statements, 5% seems like an insubstantial amount of error. Listen below for an included 20% error rate:

Here is the same quote but with a 20% error.

It is certainly more difficult to make out what is being said. The issue with this type of statement is we are forced to accept or reject how much we agree with the study by a single assumption. That is, whether or not we think the effect they describe is significant. Our own biases play a much larger role in the amount of error we perceive than any kind of quantified or meaningful information.

New York Times, 2016

So what has changed in these 135 years? How have we as a society made probability a salient object that is not purely dictated by our own prior beliefs?

A quote from Black America and the Class Divide, 2016 (Henry Louis Gates Jr., NYT):

“The Harvard sociologist William Julius Wilson calls the remarkable gains in black income ‘the most significant change’ since Dr. King’s passing. When adjusted for inflation to 2014 dollars, the percentage of African-Americans making at least $75,000 more than doubled from 1970 to 2014, to 21 percent. Those making $100,000 or more nearly quadrupled, to 13 percent (in contrast, white Americans saw a less impressive increase, from 11 to 26 percent). Du Bois’s ‘talented 10th’ has become the ‘prosperous 13 percent.’ But, Dr. Wilson is quick to note, the percentage of Black America with income below $15,000 declined by only four percentage points, to 22 percent.”

While the present margin of error in the statement above is not immediately made clear, I tracked down a supporting paper from the National Bureau of Economic Research (NBER) titled “Divergent Paths: Structural Change, Economic Rank, and the Evolution of Black-White Earnings Differences, 1940-2014” (https://www.nber.org/papers/w22797) which supports the claims of Dr. Wilson. Here the pooled standard error for their estimates over the time period 1940-2014 comes out at 11.3%. Hear what this sounds like below:

Here we calculated a cumulative summed error (an overestimate) of 11.3% in this study.

One thing to note is that a pooled standard error is not necessarily the best way to quantify the overall error in the study, but instead plays the role of providing a rough (probably over) estimate of the noise accumulated in performing so many calculations based on probabilistic models.

Regardless, this still accomplishes our goal of giving a voice to randomness. As we listen to the statement, we are explicitly aware there is some margin of error in what we are hearing without having to perceive the factual and random elements as distinct entities. We can choose to listen to these arguments with the same care we might listen to our favorite (or least favorite) song.

Is This Helpful?

Importantly, the present noise in the 2016 example is not a direct reflection of bias or beliefs, but instead a product of agreed upon standards that can be compared, criticized, and reproduced by other interested parties. As our quantitative tools increase in complexity and accuracy, we can open ourselves to these arguments through the same processes we use to practice embodied listening.

We can utilize sound to re-conceptualize our consumption of complex information. Forcing the presence of error layered on top of the arguments themselves can give each statement its respective ‘grain of salt,’ so to speak. In this manner we make randomness explicit, giving some leeway to the opposing side, and yet solidifying the important aspects of the argument itself.

Being aware of the error that is associated with a statement as it is being said helps us understand what we are hearing. That is, pairing noise with the associated statement being made dissuades listeners from hearing a personally beneficial potential error. Moving beyond the ways of thinking present in 1881 requires the acceptance of an objective ever-present notion of randomness, which can be made digestible via sound. Giving randomness its own voice allows it to object to internal biases.