How to calculate probabilities: The Bayesian calculator

the tl;dr key points

• This post describes and provides a Bayesian calculator to supplement a philosophy of science course taught at Stanford University
• The calculator is potentially useful for a variety of purposes, including calculating the probability of propositions in philosophical, scientific and mundane contexts
• The calculator also features some examples of Bayesian calculations, just to help others get an intuition for how to use the calculator

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THE BAYESIAN CALCULATOR: WHY YOU SHOULD CARE ABOUT IT

Tomorrow, I'll be giving my last lecture on Bayesianism for the course "Phil 60: Introduction to Philosophy of Science" at Stanford University. 

There, I'll be talking about a Bayesian solution to the problem of underdetermination, associated with Pierre Duhem and Willard van Orman Quine.

The problem essentially concerns the limited ability of evidence to support or rule out isolated hypotheses. For example, if you run an experiment to test whether a putative piece of iron melts at 1538 degrees Celsius, and the piece doesn't melt at that temperature, then you have at least two possible responses: you could rule out the hypothesis that iron melts at 1538 degrees Celsius, or you could instead rule out the hypothesis that the piece of metal was actually iron as opposed to another substance. As Duhem put it, the experiment itself does not tell you which specific hypothesis is false: 
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“In sum, the physicist can never subject an isolated hypothesis to experimental test, but only a whole group of hypotheses; when the experiment is in disagreement with his predictions, what he learns is that at least one of the hypotheses constituting this group is unacceptable and ought to be modified; but the experiment does not designate which one should be changed.” (Duhem, [1914] 1954, p. 187)
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Quine then generalized the point to concern not just physics, but all of our beliefs and theories in general.  He states that "our statements about the external world face the tribunal of sense experience not individually, but only as a corporate body” (Quine, 1953, p. 41). Elsewhere, he makes the radical claim that “any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system” (Quine, 1953, p. 43). How should one respond to these claims? That is the topic of what is now called the Duhem-Quine ​problem.

Not all philosophers have been entirely comfortable with Quine's claims, a notable example being Larry Laudan. And that's fair; after all, Quine's claims threaten to undermine the very notions of progress, reliability and rationality that we justifiably associate with science. 

In any case, the purpose of the lecture tomorrow will not be to discuss the problem at length and various possible responses to it. I already did that yesterday, and it was pretty fun. Rather, the purpose for tomorrow will be to explore a Bayesian approach.
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A BAYESIAN SOLUTION TO THE DUHEM-QUINE PROBLEM

That is where the Bayesian calculator comes in. You can access it here.

It can help us calculate probabilities for a variety of hypotheses, not just in scientific contexts, but also in other contexts that might be of interest. (For instance, to try stimulate students' interest in the subject, I provide an example of how to calculate the probability that someone has a crush on you given various pieces of evidence. From experience lecturing on this topic before, that tends to hold people's attention more than, say, whether iron melts at a particular temperature.)

Regardless, the calculator can also shed light on contexts involving underdetermination. There, the key insight is that the question of how to distribute blame given a false prediction depends crucially on the prior probabilities of the relevant hypotheses. Consider our earlier example about the iron.  According to the Bayesian calculator, if you are 95% confident that the metal is iron, but only 90% confident in the hypothesis that all iron melts at 1538 degrees Celsius, then the outcome will make you much less confident in the hypothesis about iron's melting point (which has a posterior probability of 19%) than in the substance being iron (which has a posterior probability of 79%).

However, a disclaimer: I am not the first to advocate this solution to the Duhem-Quine problem. From what I can tell, it was first articulated by the philosopher Jon Dorling in 1979. His paper, in my opinion, is a very underrated piece of Bayesian genius. You can find the references for this and the other works below.
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SOME REFERENCES

Dorling, J., 1979, "Bayesian personalism, the methodology of scientific research programmes, and Duhem's problem", Studies in History and Philosophy of Science Part A, 10(3): 77-187.

Duhem, P., [1914] 1954, The Aim and Structure of Physical Theory, trans. from 2nd ed. by P. W. Wiener; originally published as La Théorie Physique: Son Objet et sa Structure (Paris: Marcel Riviera & Cie.), Princeton, NJ: Princeton University Press.

Quine, W. V. O., 1951, “Two Dogmas of Empiricism”, Reprinted in From a Logical Point of View, 2nd Ed., Cambridge, MA: Harvard University Press, pp. 20–46.