On The Evidential Import of Unification

Wayne C. Myrvold
Department of Philosophy

The University of Western Ontario
wmyrvold@uwo.ca

Forthcoming in Philosophy of Science.

June 28, 2016

Abstract

This paper discusses the evidential import of two senses in which
a hypothesis may be said to unify evidence. One is the ability of the
hypothesis to increase the mutual information of a set of evidence
statements; the other is the ability of the hypothesis to explain com-
monalities in observed phenomena by positing a common origin for
them. On Bayesian updating, it is only Mutual Information Unifica-
tion that contributes to the incremental support of a hypothesis by the
evidence unified. This poses a challenge for defenders of a view that
explanation ought to be taken as a confirmatory virtue that makes a
contribution in its own right to incremental support; in order for such
a view to be defensible, its advocates must ground it in some relevant
difference between humans and a Bayesian agent. Options for such a
defense are considered, and it is concluded that common origin uni-
fication has at best a limited heuristic role to play in confirmation.
Finally, it is shown how Reichenbachian common cause hypotheses fit
into the schema of mutual information unification. Keywords : Unifi-
cation, explanation, confirmation, Bayesianism, common cause.

1



1 Introduction

Myrvold (2003) identified what was described therein as “one inter-
esting sense” in which a theory can unify phenomena. This consists
of the ability of the theory to render distinct phenomena informative
(or more informative) about each other. Call this Mutual Information
Unification (MIU). This sense lends itself nicely to a probabilistic ex-
plication, and it can be shown that unification in this sense contributes
to incremental evidential support of the theory by the phenomena uni-
fied.

There is another sense of unification, having to do with hypothe-
ses that posit a common origin for the phenomena in question, be it
a common cause or some other type of explanation. Call this Com-
mon Origin Unification (COU). As emphasized by Lange (2004) and
Schupbach (2005), the two senses are logically independent; neither is
a necessary or a sufficient condition for the other, even though, in a
number of interesting cases, they are concomitants of each other.

In this paper, the respective roles of these two notions of unifi-
cation in connection with the bearing of evidence on a theory are
discussed. There are, of course, other questions one might ask, and
other roles for a notion of unification to play besides contributing to
confirmation. Having a common explanation for disparate phenomena
can contribute to deeper understanding, which is one goal of scientific
research. Insofar as it contributes to such understanding, Common
Origin Unification may play the role of a cognitive value.1 As such, it
can play a legitimate role in questions such as that of which research
programme to pursue; a theory might be regarded as more worthy of
development on account of its potential for affording understanding.2

This is a different matter from the question at issue in this paper,
which is whether unification ought to be regarded as contributing to
the evidential support of a theory by the phenomena unified.

On the question of the respective roles of these two notions of
unification in theory confirmation, on a Bayesian analysis, the answer
is clear: Mutual Information Unification contributes to incremental

1I am grateful to Michel Janssen for making this suggestion. See Myrvold (2011), and
references therein, on the subject of how to incorporate cognitive values into a Bayesian
framework.

2Cf. Salmon (2001, 130): “the scientist might say that Halley’s hypothesis is worth
pursuing, not because it is more likely to be true, but because, if it should turn out to be
true, it would be extremely valuable in terms of informational content.”

2



evidential support, and there is no scope, within Bayesian updating,
for Common Origin Unification to add to the evidential support of the
theory (see §4, below).

There may be some who do not take this to settle the normative
issue, and will maintain that, despite the Bayesian verdict, we ought
to take explanatory power of a hypothesis as a confirmatory virtue.
Advocates of such a view would have to reject the idea that we should
take consideration of a Bayesian agent updating via conditionalization
as normative for those of us who are not such agents. This presents
a challenge for such advocates. If it is rational, or reasonable, or oth-
erwise well and good for us to do what is impossible for a Bayesian
agent updating its credences via conditionalization, that is, to take
Common Origin Unification to be something that makes a contribu-
tion to evidential support, above and beyond what it contributes to
Mutual Information Unification, then this must be grounded in some
relevant difference between us and Bayesian agents. It is incumbent
on an explanationist to give an account of what that difference is.

In the following, these points are first illustrated by means of a sim-
ple example that, despite its artificiality, shares some salient features
with cases of actual scientific interest. Next, in §3, are presented the
probabilistic measures of MIU introduced in Myrvold (2003), and in
§4 their impact on evidential support is exhibited. In §5 are outlined
possible reactions to the Bayesian verdict regarding the respective con-
firmatory roles of the two types of unification. In §6 the question is
addressed whether there is still a role for Common Origin Unification
to play in hypothesis assessment, in assessing priors rather than in as-
sessing incremental evidential support (the answer is no). Finally, in
§7 it is shown how Reichenbachian common causes fit into the schema
of Mutual Information Unification.

2 Two Kinds of Unification

Consider the following toy example, of no use except for introducing
the issues at hand, though it does share some salient features with
a multitude of real-world cases of genuine scientific interest. You are
about to be presented with two data streams, S1 and S2, each of which
will be sequences of ten Heads or Tails. You know that they have been
produced by coin flipping, but you aren’t sure of exactly the procedure
used, or whether the coin or coins involved are fair.

3



Suppose that you have nonzero credences in both of the following
hypotheses:

H1: A fair coin was flipped ten times, and the results of this series of
coin flips are reported in both data streams.

H2: Two fair coins were flipped ten times each, and each data stream
reports the results of one of these series of coin flips.

I invite you to consider the effect of the evidence on these two hypothe-
ses. That evidence consists of specification of the two data streams:

S1: HHHTTHTHHT

S2: HHHTTHTHHT

Let E1 be the proposition that S1 is the string given above, and E2
the corresponding proposition about S2.

Now, if you have nonnegligible prior credence that the strings
might have been produced by radically unfair coins, E1 and E2 might
boost your confidence in the fairness of the coins, and hence condition-
alizing on each of E1 and E2, separately, might boost your credence
in both H1 and H2. But, when taken together, E1 and E2 strongly
favor H1 over H2.

There are two features of this example that I would like to draw
your attention to. The first feature is that H1, if true, renders E1
informative about what data stream S2 will be. Conditional on H1,
knowing E1 permits one to anticipate the truth of E2. That is, H1
exhibits Mutual Information Unification (MIU ) with respect to the
evidence set {E1,E2}. A hypothesis has this property, with respect to
a set of evidential propositions, if conditionalizing on that hypothesis
increases the mutual informativeness of the set. Obviously, this is the
sort of thing that comes in degrees. In our toy example, conditional
on H1, knowledge of E1 permits one to anticipate, with certainty, all
details of E2. In more interesting cases the increase of informativeness
will be less than maximal. Probabilistic measures of degree of this sort
of unification will be introduced below.

The second feature is that H1 posits a common origin of the two
data streams, and thus is ripe to be the subject of what Janssen (2002)
has called a COI story, for Common Origin Inference. In addition to
MIU, H1 also exhibits Common Origin Unification (COU ).

The two concepts are of a manifestly very different sort. One be-
longs to a cluster of concepts involving information, states of knowl-
edge, and the like; the other is related to concepts of cause and ex-

4



planation.3 As already mentioned, they are logically independent. A
hypothesis can posit a common origin for two (or more) evidential
propositions without making them mutually informative about each
other, as the propositions could be about independent aspects of their
posited common origin; thus, we can have COU without MIU. Fur-
thermore, once two or more evidential propositions are known, that
is, have been absorbed into one’s background knowledge, they are no
longer informative about each other, though any common origin they
might have remains, and again we have COU without MIU. One can
also trivially construct hypotheses that exhibit MIU without COU.
With respect to our toy example, consider the hypothesis,

H3: Two fair coins were flipped ten times each, each data stream
reports the results of one of these series of coin flips, and the
results of each series of flips just happened to be the same.

This hypothesis, if true, also renders one data stream informative
about the other. Of course, prior to the evidence, one would expect
credence in H3 to be low, lower than credence in H2 by a factor of
1, 024.

Though artificial, our toy example has a multitude of parallels in
actual science. One is the case of heliocentric v geocentric world sys-
tems, discussed by Janssen (2002) and Myrvold (2003). The analog
of H1 is what was called hC in Myrvold (2003), that is, the heliocen-
tric hypothesis that all planets have circular or nearly circular orbits
centred at or near the sun, and the analog of H2 is the bare-bones
geocentric hypothesis hP , which posits that, for each planet, there is
a deferent circle centered near the earth, and that the planet travels on
an epicycle whose center travels on the deferent, with no assumption
made about any connections between the motions of different planets
or between planetary motions and the motion of the sun. The ana-
log of H3 is the geocentric hypothesis conjoined with the sun-planet
parallelism condition; this is the hypothesis hSP , or the strengthened
Ptolemaic hypothesis.

3Similar remarks apply to probabilistic measures of explanatory power such as those
proposed by Popper (1954, 1959), Good (1960), Schupbach and Sprenger (2011), and
Crupi and Tentori (2012). Glymour (2015) has argued that it would be a grave mistake to
take any of these probabilistic notions as an explication of explanatory power. This seems
to be generally accepted by recent authors; Schupbach and Sprenger, for example, are
clear that what is proposed is a measure of strength of explanation between propositions
bearing an antecedently identified explanatory relation to each other.

5



One can find analogs in cases in which a hypothesis turns dis-
parate, prima facie unrelated phenomena into agreeing measurements
of some theoretical parameter. The classic case is Perrin’s argument
for the existence of atoms. Perrin (1913, §119; 1916, §120) adduces 13
distinct phenomena that, on the atomic hypothesis, count as measure-
ments of Avogadro’s number. The analog of H1 is that atoms exist,
and hence there is a common origin explanation of the agreement of
these measurements; the analog of H2 would be the hypothesis that
matter is continuously divisible, and the analog of H3 would be the
hypothesis that adds to H2 the stipulation that Perrin’s 13 phenomena
yield values that just happen to agree within experimental error, even
though they are not agreeing measurements of any physically mean-
ingful quantity. Another example is the quantum hypothesis, which
turns disparate phenomena into agreeing measurements of Planck’s
constant; see Kao (2015).

3 Probabilistic Measures of Unification

Consider a Bayesian agent whose credences are represented by a prob-
ability function Cr. We define the mutual information of a pair of
propositions, {p1,p2}, relative to background b, by4

I(p1,p2|b) = log2
(
Cr(p2|p1b)
Cr(p2|b)

)
= log2

(
Cr(p1 p2|b)

Cr(p1|b)Cr(p2|b)

)
. (1)

If p1 and p2 are probabilistically independent on b, then I(p1,p2|b)
is zero; it is positive if conditionalizing on one boosts credence in the
other, negative, if conditionalizing on one lowers credence in the other.

For a larger set, p = {p1,p2, . . . ,pn}, we add up the information
yielded by p1 about p2, the information yielded by p1p2 about p3, and
so on, up to the information about pn yielded by the conjunction of

4A note on notation. We will use concatenation for conjunction, and the overbar p̄ for
the negation of p. We use boldface letters to denote sets of propositions. Note that these
are sets and are not replaceable by a single proposition that is their conjunction. Thus,
{p1,p2} is not the same set as {p1p2,T}, where T is the logically true proposition, though
the conjunction of their members is the same. This matters because we will be concerned
with the mutual informativeness of members of a set of propositions; p1 and p2 may be
mutually informative though the logically true proposition is not informative about their
conjunction or anything else.

6



all the others.5

I(p1, . . . ,pn|b) = I(p1,p2|b) + I(p1p2,p3|b) + . . . + I(p1 . . .pn−1,pn|b)

=

n−1∑
k=1

I

(
k∧

i=1

pi,pk+1|b

)
. (2)

Although the form of (2) does not make this obvious, this quantity is
independent of the order in which the elements of the set p are taken,
and we have,

I(p1, . . . ,pn|b) = log2
(

Cr(p1 p2 . . . pn|b)
Cr(p1|b)Cr(p2|b) . . .Cr(pn|b)

)

= log2

(
Cr(

∧n
i=1 pi|b)

Πni=1Cr(pi|b)

)
. (3)

With a slight abuse of notation, we will write I(p|b) for I(p1, . . . ,pn|b).
We will also drop, as irrelevant, the base of the logarithm, since chang-
ing base is only a matter of a constant multiplicative factor.

I(p|b) is the logarithm of the quantity that appears in Keynes’
Treatise on Probability (1921, §XIV.8) as the coefficient of dependence,
with an attribution to unpublished work by W.E. Johnson.6 It was
called a measure of similarity by Wayne (1995) and Myrvold (1996),
and taken by Shogenji (1999) as a measure of coherence of a set of
propositions.

We will say that a hypothesis h MIUnifies a set e = {e1, . . . ,en},
relative to background b, if and only if

I(e|hb) > I(e|b). (4)

This suggests a way to measure the degree to which a hypothesis
MIUnifies a set of evidential propositions.7

MIU1(e; h|b) = I(e|hb) − I(e|b). (5)
5Obviously, a single number cannot capture all the informational relations there could

be between elements of a set of more than two members. This would require a specification
of all I(q,q′|b), where q and q′ range over all conjunctions of elements of p. But it is this
quantity that will be useful for the purposes at hand.

6I am indebted to Brössel (2015) for pointing this out.
7This quantity is the logarithm of a quantity that was referred to as an “interaction

term” in Myrvold (1996), and is called focussed correlation in Wheeler (2009), Schlosshauer
and Wheeler (2011), and Wheeler and Scheines (2013). What we are calling MIU1 was
called U (for unification) in Myrvold (2003). MIU2 was discussed therein, though not
given its own name.

7



We might also be interested in whether a hypothesis does a better job
of unifying a set of propositions than its negation. Define

MIU2(e; h|b) = MIU1(e; h|b) −MIU1(e; h̄|b)
= I(e|hb) − I(e|h̄b). (6)

The two are not ordinally equivalent, and, indeed, need not agree as
to sign. Suppose a hypothesis h unifies a body of evidence, relative
to background b. That is, suppose the evidence is more mutually
informative conditional on hb than on b alone. Then MIU1(e; h|b)
is positive. But whether MIU2(e|b) is negative or positive depends
on whether or not h̄ unifies the evidence more. If I(e; h̄|b) is greater
than I(e; h|b), then, even if MIU1(e; h|b) is positive, MIU2(e; h|b) is
negative. In fact, all four combinations of signs of MIU1 and MIU2
are possible, though it is easy to show that, unless e1 and e2 are,
when taken individually, oppositely relevant to h (that is, unless one
of them is positively relevant and the other negatively relevant), if
MIU1(e1,e2; h|b) is positive, MIU2(e1,e2; h|b) is also positive. See
Appendix for details.

Both of these quantities are special cases of a comparative measure
of unification,

MIUc(e; h1,h2|b) = I(e|h1 b) − I(e|h2 b). (7)

On McGrew’s account of consilience, h1 is said to be more consilient
than h2 with respect to e to the extent that I(e|h1 b) > I(e|h2 b), or,
equivalently, to the extent that MIUc(e; h1,h2|b) > 0 (McGrew, 2003,
562).

Readers are asked to kindly refrain from engaging in a battle of
the intuitions over whether MIU1 or MIU2 is the One True Measure
of degree of unification. They are simply measuring different things,
and if you have intuitions that are incompatible with properties that
one or another of these quantities possesses, then your intuitions are
about some other concept.8

4 The Evidential Value of Unification

To some readers, it might seem obvious that what counts when it
comes to confirmation is Common Origin Unification, with Mutual

8And if your intuitions find it repugnant to use the word “unification” in connection
with either of these, then feel free to use a different word.

8



Information Unification being a poor cousin that hardly merits the
illustrious family surname. This view is expressed by Marc Lange,
who writes,

the examples I have given suggest that insofar as theories
that unify in the stronger,9 ontological-explanatory sense
derive greater support in virtue of the unification they
achieve, they do so not solely in virtue of their achiev-
ing unification in the weaker, creating-mutual-positive rel-
evance sense. The stronger sense of unification is epistemi-
cally significant. In the case of the light-quantum hypoth-
esis, hC and hL both supply unity in the weaker sense,
but Einstein took hL to receive greater support from the
phenomena than hC by virtue of hL’s unifying those phe-
nomena in an ontological-explanatory sense (Lange 2004,
212).

Here hL is Einstein’s light quantum hypothesis, and hC is the hypoth-
esis that hL is false but nevertheless, by sheer coincidence, light be-
haves as if it were quantized. According to Lange, hL receives greater
support from the phenomena unified than does hC .

It is not entirely clear whether incremental or absolute support
is meant, where incremental support has to do with an increase in
credibility lent to a hypothesis by the evidence, and absolute support
with the credibility of the hypothesis, taking all known considerations
into account. If absolute, this suggests that the case of hC is analogous
to that of our toy example’s H3, which is accorded a low prior because
it posits an improbable coincidence. One the other hand, if the claim
is to be a counterexample to anything in Myrvold (2003), incremental
support must be what is meant. Let us therefore consider the position
that, when it comes to incremental support, it is COUnification, not
MIUnification, that counts.

A Bayesian analysis renders the opposite verdict: when it comes to
incremental support of a hypothesis, it is MIUnification, rather than
COUnification, that matters.

One popular measure of the degree to which an evidential propo-
sition e lends incremental confirmation to a hypothesis h, relative to
background b, is the ratio of posterior probability of h to its prior
probability. This is, of course, ordinally equivalent to its logarithm.

9This is a slip; the two senses are, as Lange emphasizes, logically independent.

9



Let us define

R(h; e|b) = log
(
Cr(h|eb)
Cr(h|b)

)
. (8)

Another is the ratio of the posterior odds of h to its prior odds, or,
equivalently, the logarithm of this, called weight of evidence by Good
(1950). Define

W(h; e|b) = log
(
Cr(h|eb)/Cr(h̄|eb)
Cr(h|b)/Cr(h̄|b)

)
= log

(
Cr(e|h)
Cr(e|h̄)

)
. (9)

As Myrvold (2003) pointed out, on either way of measuring incre-
mental confirmation, we have a contribution of unification to confir-
mation.10 The incremental support, as measured by R, of h by e can
be decomposed into a sum of increments due to the individual mem-
bers of e, plus an additional term that is the degree of MIUnifcation
(positive or negative) of e by h, as measured by MIU1.

R(h;

n∧
i=1

ei|b) =
n∑

i=1

R(h; ei|b) + MIU1(e; h|b). (10)

The result for W takes the same form, with MIU2 in place of MIU1.

W(h;
n∧

i=1

ei|b) =
n∑

i=1

W(h; ei|b) + MIU2(e; h|b). (11)

These relations can be readily verified by the reader.
Since the MIU-term is not the only contribution to the increment

of confirmation, it would be incorrect to gloss these results as say-
ing that hypotheses that are more unifying receive more confirmation.
Although it would not be incorrect to say that ceteris paribus, a hy-
pothesis that achieves a higher degree of MIUnification of the evidence
is accorded greater incremental support, this is strictly weaker than
what is conveyed in equations (10) and (11), and there is no advantage
in making the ceteris paribus claim when it is a trivial matter to say
how things stand when all else is not equal.

Imagine, now, a Bayesian agent that had numerical credences,
which it11 updated by conditionalizing on new items of evidence.

10Equation (10) corresponds to (6) of Myrvold (1996) and to (12) of Myrvold (2003); (11)
corresponds to (13) of Myrvold (2003). Closely related results appear already in Keynes
(1921, 151–154); in particular, our equation (11) is essentially the same as Keynes’ (48).

11I say “it,” because a being with precise numerical credences would be far from human.

10



Then, depending on how we measured degree of incremental confir-
mation, the confirmational boost accorded to h by a set e of evidential
propositions would be given by either (10) or (11). In each case the
additional confirmational boost, beyond that attributable to the items
of evidence taken singly, is given by the MIUnification term.

Applied to our toy example: The fact that H1 and H3 make E1
and E2 informative about each other is reflected in the likelihoods,
Cr(E1E2|H1) and Cr(E1E2|H3), which are higher than Cr(E1E2|H2)
by a factor of 1, 024. Thus, relative to H2, credence in H1 and H3 is
boosted:

Cr(H1|E1E2)
Cr(H1)

=
Cr(H3|E1E2)

Cr(H3)
= 1, 024 ×

Cr(H2|E1E2)
Cr(H2)

. (12)

It doesn’t follow, of course, that H3 gets final credence comparable to
that of H1. Since H3 posits an improbable coincidence, it is accorded
a lower prior probability, lower than that of H2 by a factor of 1, 024;
the additional confirmational boost it receives is just enough to bring
it up to posterior credence equal to that of H2 (which, of course, must
be the case, since, given the evidence, H3 is true if and only H2 is).

There is a close parallel between this case and the case of geocentric
v heliocentric world systems, and also the case of the light quantum,
considered by Lange.

In the case of planetary motion, on both the heliocentric hypothe-
sis and the strengthened Ptolemaic hypothesis, features of one planet’s
apparent motion are informative about features of others’ (see Janssen
2002 and Myrvold 2003 for discussion). In the case of the heliocentric
hypothesis, HC , these have a common origin in the motion of our van-
tage point as observers on earth; for HSP , they are the consequence
of the posited sun-planet parallelism. Against a background that in-
cludes little or no information about observed planetary motions, both
of these get a confirmational boost from the celestial phenomena, due
to the MIU-component of incremental confirmation. It doesn’t follow
that they end up with equal posterior credence. Arguably, HSP , on
that background, should be accorded markedly lower prior credence
than HP , as it posits a relation that HP by itself would not lead one
to anticipate. HC and HSP get the same incremental confirmation on
the evidence. Therefore, posterior credence in HC will be markedly
higher than posterior credence in HSP unless prior credence in HC is
markedly lower than prior credence in HP .

Something similar can be said in regards to Lange’s case of the

11



light quantum hypothesis. Let us grant that the light quantum hy-
pothesis plays a unificatory role. Lange asserts that Einstein took the
observed phenomena to lend greater support to the light quantum hy-
pothesis than the hypothesis that, by sheer coincidence, all observable
phenomena are as if the light quantum hypothesis is true. The sugges-
tion is that that such a judgment is the right one, given the evidence
available to Einstein in 1905. In order for this claim to be relevant to
the issue at hand, this must mean that the phenomena lend greater
incremental support to the light-quantum hypothesis than to the co-
incidence hypothesis. One might also regard hC as so implausible as
to be dismissed out of hand. But this would mean according it a low
prior, which is consistent with the Bayesian account of the virtue of
unification.

5 Possible Reactions to the Bayesian

Verdict

Bayesian updating leaves no room for an additional confirmatory boost
to be attached to hypotheses with greater explanatory power; the
contribution to incremental support comes via the MIUnification term.
There is a tension between this Bayesian verdict and the thought
that COUnification should play a role in incremental confirmation
above and beyond its contribution to MIUnification. We have here an
exact parallel with van Fraassen’s argument against those who would
take explanatory power of a theory to yield an extra confirmatory
boost, beyond that yielded by conditionalization on the evidence (van
Fraassen, 1989, 166–169).

One reaction might be to downplay the distinction, focussing on
cases in which explanationist and Bayesian judgments agree. One
might be tempted to declare that hypotheses that provide ‘lovelier’
explanations are precisely those that bestow higher likelihood on a
hypothesis. This is not tenable as a general thesis. Although, in
many interesting cases, explanation and likelihood go together, the
connection is not so tight that they never come apart. The interesting
question is what the explanationist will say about the cases in which
they do come apart.

One possible reaction, in my opinion the correct one, is to use the
Bayesian verdict to correct any intuitions one might have that are in
tension with it. The ability of a theory to unify disparate phenomena

12



by positing a common origin plays a confirmatory role only insofar
as the posited common origin renders distinct phenomena informative
(or more informative) about each other. A temptation to assign it
a stronger role in confirmation might be ascribed, in part, to a con-
flation of distinct questions (a conflation encouraged by philosophers’
overuse of the phrase “theory choice,” a phrase that conflates distinct
sorts of choices). Certainly, a hypothesis’ power to explain, if true,
can contribute to making it worthwhile to pursue a project of devel-
oping a theory that includes that hypothesis, and it can contribute to
the value of accepting the hypothesis, if true; we should only be wary
of thinking that everything that contributes to making a hypothesis
pursuit-worthy also lends it greater credibility. The temptation might
also be ascribed, in part, to not distinguishing between incremental
confirmation and overall credibility in the light of all evidence. The
most obvious examples that exhibit MIUnification without COUni-
fication are those such as our H3, that achieve it by brute fiat, by
tacking on an improbable conjunct, and we rightly regard these as
implausible.

This suggests one way in which an explanationist might retrench;
the import of COUnification might be relegated to informing priors.
While, certainly, common-origin considerations sometimes play a role
in assessing prior credibility, I am skeptical that anything beyond a
very limited role can be defended; more on this in the next section.

The only other avenue of defense for an advocate of an explana-
tionist thesis would be to deny that considerations of how a Bayesian
agent would update have normative force for the judgments of human
scientists.

A line of defense along these lines of thought would have to ground
it in some relevant difference between us and Bayesian agents. We are
certainly different from Bayesian agents in a number of ways. We
do not have precise numerical degrees of belief; our judgments about
how likely or unlikely a hypothesis is tend to be vague. Moreover, as
an abundance of empirical evidence shows, routinely our qualitative
judgments of the relative credibility of various propositions are not
even compatible with the existence of numerical credences satisfying
the axioms of probability, and our changes in credences are often not
in accord with Bayesian conditionalization.

The usual understanding of facts of this sort is that they are due
to cognitive limitations, and that some of them can be understood as
resulting from usually reliable heuristics, of the sort that any agent

13



with limited cognitive capacities would be well-advised to employ as
an alternative to spending excessive time on cogitation. In taking
such limitations into account, one does not ipso facto abandon the
domain of normativity for descriptive psychology. From a decision-
theoretic point of view, deployment of such heuristics can be regarded
as rational behavior for a cognitively limited agent. This involves what
I. J. Good (1971, 1976) called “Type II Rationality”: decision-making
that takes into account the cost in time and cognitive effort of the act
of deliberation.

Peter Lipton has offered a limited defense of explanationism along
these lines. We are often not very good, he notes, at judging likeli-
hoods correctly.

My thought is this. In many real life situations, the calcu-
lation that the Bayesian formula would have us make does
not, in its bare form, meet the requirement of epistemic ef-
fectiveness: it is not a recipe we can readily follow. . . . My
suggestion is that explanatory considerations of the sort to
which Inference to the Best Explanation appeals are often
more accessible than those probabilistic principles to the
inquirer on the street or in the laboratory, and provide an
effective surrogate for certain components of the Bayesian
calculation. On this proposal, the resulting transition of
probabilities in the face of new evidence might well be just
as the Bayesian says, but the process that actually brings
about the change is explanationist (Lipton 2004, 113-114;
see also Lipton 2001, 110–111).

On such a view, when a judgment needs to be made on the fly, it is
better to invoke an explanationist heuristic than to spend time think-
ing through likelihoods; this will, one hopes, provide judgments that
are not too far off, either most of the time or in the most significant
cases. Though Lipton suggests that the division of labor between
Bayesian and explanationism maps onto the distinction between nor-
mative and descriptive accounts, he also uses language that suggests
that we cognitively limited agents are well-advised to employ explana-
tionist considerations as a surrogate for doing a Bayesian calculation:
“explanatory considerations help us to perform what is in effect a
Bayesian calculation” (Lipton 2004, 120). This suggests that consid-
erations of Type II rationality are in play.

Using a heuristic of this sort as a surrogate for a considered evalua-

14



tion of likelihoods carries with it a risk of error, in those cases in which
COU and MIU come apart. Presumably, Lipton would agree that,
in such cases, if an accurate appraisal of the import of the evidence
matters, one should correct the explanationist judgment by reference
to the Bayesian one. On Lipton’s view, the role of explanationist
considerations is severely constrained.

Can a stronger defense of explanationism be mounted? It is doubt-
ful. Since such a defense would have to be grounded in some difference
between cognitively limited humans and Bayesian agents, it’s hard to
see any role for explanationist consideration beyond the limited heuris-
tic role envisaged by Lipton.

6 A Prior Preference for Unifying Hy-

potheses?

We have considered cases (in the toy example, H1 and H3, in the
case of planetary motion, HC and HSP , and in the light quantum
case, hL and hC ), in which each of a pair of hypothesis possesses
the same ability to render items of evidence informationally relevant
to one another, but they do so in different ways. In each of these
cases one does it by virtue of positing a common origin for prima
facie unrelated phenomena, the other, by brute fiat, in positing an
unexplained correlation between the phenomena. In each of these
cases, the hypothesis that involves a common origin is, arguably, less
implausible than the one that posits brute coincidence.

One might be tempted to generalize, positing, that, whenever we
have MIU without COU, there will be a corresponding hypothesis that
achieves precisely the same MIUnification via COUnification, and we
should accord much less prior credence to the hypothesis that exhibits
MIU without COU than to the one that achieves it via COU. This
would mean that there is a role for COU, not in incremental confir-
mation, but in setting priors.

Anything so sweeping would be a mistake, I think. There are
patterns in the world of all sorts, some due to some sort of common
origin, some not. We should not demand that a common origin be
found for every similarity between two phenomena. Given any pattern
in the phenomena, however, it will be possible to cook up an artificial
MIUnifying hypothesis. We ought not seek a common origin lurking
behind every such hypothesis!

15



Perhaps, then, the generalization should be that, when we do have
a pair of hypotheses that both induce the same informational relevance
relations among a body of phenomena, one doing it via COUnification
and the other by brute fiat, we should attach higher prior credence to
the COUnifying hypothesis.

This is still too sweeping. When we have a case of two hypothe-
ses h1 and h2 of roughly equal prior credibility, and create a third h3
by tacking on to h2 some conjunct with low prior plausibility, then,
indeed, in such a case, we should place lower credence in h3 than in
h1. But not all cases will be like that, and a COUnifying hypothesis
might be deemed implausible on other grounds. Take, for example,
Ptolemy’s attitude towards heliocentric hypotheses. Since Ptolemy
recognized that in the observed phenomena there is a connection be-
tween the apparent motion of the sun and that of the other planets,
he was in a position to appreciate the COUnifying power of helio-
centrism. But, since he accepted Aristotelian physics for terrestrial
phenomena, he thought that terrestrial phenomena ruled out a diur-
nal rotation of the earth (see Ptolemy 1984, Bk. I, §7); for him, it was
reasonable to place low credence in heliocentric theories that posited
such a rotation.

One can exhibit plenty of hypothesis pairs in which the less unify-
ing, less explanatory hypothesis has less prior credibility, because the
less explanatory hypothesis posits an implausible coincidence. But the
emphasis should be on the credibility-diminishing role of coincidence,
rather than any prior conviction that nature is unified. What H3, the
strengthened Ptolemaic hypothesis, and Lange’s hC have in common
is that, in each case, we have a hypothesis to which is tacked on some
additional condition that one would not expect to hold in the absence
of evidence that it does, and hence we have a hypothesis that ought
to be accorded low prior credence. Rather than a sweeping prefer-
ence for COUnification, I suggest that the methodological adage that
underwrites low prior credence in such hypotheses is:

Place little prior credence in things you take to be improb-
able.

This is, I hope, unobjectionable! It is, of course, utterly empty, but I
am skeptical that anything stronger could be defended as a maxim of
more than very limited scope.

It would be a mistake to raise this bland but unobjectionable
maxim into a global rejection of hypotheses that posit coincidences.

16



Improbable things do happen, after all. Moreover, in some cases it is
reasonable to accept hypotheses that posit an improbable coincidence.
The evidence available to you in the toy example strongly suggests a
common cause. But, if you were to obtain strong evidence that the
two data streams were the results of independent tosses of two fair
coins, then it would be reasonable to accord high credence to H3. For
a real-world case: Ptolemy propounded a geocentric system with an
unexplained sun-planet parallelism, because he thought he had strong
evidence to rule out hypotheses that involved a moving earth.

7 Unification and Reichenbachian Com-

mon Causes

Among unifying hypotheses are those that posit a Reichenbachian
common cause to explain some observed statistical correlation (Re-
ichenbach, 1956, §19). This type of hypothesis fits well within the
schema of the Bayesian account of unification, but, since this might
not be obvious, it is worth showing how it fits.

Consider two sequences of propositions, {Ai, i = 1, . . . ,n}, and
{Bi, i = 1 . . . ,n}. Given such sequences, let n(A) be the number of
true instances of the Ais, and let f(A) = n(A)/n be the relative fre-
quency of true instances of the Ai. Define f(B) and f(AB) similarly.
Let E1 be a proposition expressing which of the Ais are true, and
which are false. For example, in our toy example, Ai could be the
proposition that the ith element of S1 is Heads, and E1 would be

A1A2A3Ā4Ā5A6Ā7A8A9Ā10.

Let E2 be the evidence statement specifying the B-sequence.
A statistically significant difference between f(AB) and the prod-

uct f(A)f(B) is thought to call for explanation. A Reichenbachian
Common Cause of an observed correlation between A and B is a
third sequence Ci that screens off their correlation. That is,

Pr(AiBi|Ci) = Pr(Ai|Ci)Pr(Bi|Ci);

Pr(AiBi|C̄i) = Pr(Ai|C̄i)Pr(Bi|C̄i).
(13)

A hypothesis that posits a common cause of this sort, if it leads one to
expect correlations close to those observed, clearly, can be supported

17



by evidence in which there is an observed statistical correlation be-
tween two sequences of events. Such a hypothesis can be a MIUnifying
hypothesis, in the sense of making the evidence statements E1 and E2
mutually informative.

This might seem paradoxical. A common cause screens off the
correlations between the Ais and Bis; how can it be that, at the same
time, there is a confirmational boost associated with rendering them
informative about each other?

The answer to this is: the hypothesized common causes Ci screen
off the correlations, but a hypothesis Hcc that posits that there are
common causes of the right sort can render the truth or falsity of
Ai informative about the truth or falsity of Bi, and hence render E1
and E2 mutually informative. That is, a hypothesis that there is a
common cause of the right sort will lead one to expect correlations
between the Ais and Bis, and so count as MIUnifying with respect to
the evidence set {E1,E2}, relative to a background against which the
observed correlations are unexpected.

Moreover, each event Ci can count as a common origin of Ai and
Bi. Let Hcc be some hypotheses according to which there exists a
sequence {Ci} satisfying (13). Suppose that, on the supposition of
Hcc, Ci is a probability raiser for both Ai and Bi, as a cause should
be, and suppose that, according to Hcc, for each i, Ci and C̄i both
have nonzero probability. Then, even though, for each Ci, the truth
or falsity of Ci screens off informational relations between Ai and Bi,
the supposition of Hcc leads one to expect correlations between the
Ais and the Bis.

Pr(AiBi|Hcc) > Pr(Ai|Hcc) Pr(Bi|Hcc). (14)

Let us now see in more detail how this works. We consider the
bearing of the statistical evidence stemming from observation of the
A-sequence and the B-sequence on members of a family of hypotheses,
each of which posits the existence of a Reichenbachian common cause.
For simplicity, we consider only hypotheses on which distinct Ais are
independent and identically distributed, as are {Bi} and {AiBi}. The
statistical data can be accounted for on a hypothesis positing Cis that
are also independently and identically distributed. Any hypothesis
positing a common cause of this sort can be characterized by five

18



parameters:

p = Pr(Ci),

a1 = Pr(Ai|Ci), a0 = Pr(Ai|C̄i),

b1 = Pr(Bi|Ci), b0 = Pr(Bi|C̄i).

(15)

Probabilities for the Ais, Bis, conditional on a hypothesis of this sort,
are

Pr(Ai|Hcc) = pa1 + (1 −p) a0,

Pr(Bi|Hcc) = pb1 + (1 −p) b0,
(16)

and their covariance is,

Cov(Ai,Bi|Hcc) = Pr(AiBi|Hcc) −Pr(Ai|Hcc)Pr(Bi|Hcc)
= p(1 −p)(a1 −a0)(b1 − b0). (17)

As pointed out by Reichenbach (1956, 159–161), and as can be readily
seen from (17), if p ∈ (0, 1) and a1 −a0 and b1 −b0 are both positive,
then, conditional on the hypothesis Hcc, the Ais are positively corre-
lated with the Bis. Obviously, the same conclusion follows if a1 −a0
and b1 − b0 are both negative; also, the Ais are negatively correlated
with the Bis if a1 − a0 and b1 − b0 have opposite sign, and they are
uncorrelated if the Cis are irrelevant to either the Ais or the Bis, that
is, if a1 = a0 or b1 = b0.

12 The family of all such hypotheses, thus,
includes as a special case those that posit no common cause for Ai
and Bi.

We inquire into the degree of support lent to common-cause hy-
potheses, with various values of the parameters, by the pair {E1,E2}.
Let Hcc be some hypothesis of the form considered above. We have,
from (10),

R(Hcc; E1E2) = R(Hcc; E1) + R(Hcc; E2) + MIU1({E1,E2}; Hcc).
(18)

Since we’re interested in comparing degrees of support for different
hypotheses on a fixed body of evidence, it is useful to compare log-
likelihoods, as, for two different hypotheses, the differences between

12These probabilistic facts were familiar in the statistical literature well before Reichen-
bach’s use of them; see Yule (1911, §§IV.6–7).

19



their R-values will be the same as the differences between the re-
spective log-likelihoods. The log-likelihoods can be partitioned in a
manner parallel to our partitioning of R:

log Pr(E1E2|Hcc) = log Pr(E1|Hcc) + log Pr(E2|Hcc) +I(E1,E2|Hcc).
(19)

The first two terms of this are

log Pr(E1|Hcc) = n(A) log Pr(Ai|Hcc) + n(Ā) log Pr(Āi|Hcc);

log Pr(E2|Hcc) = n(B) log Pr(Bi|Hcc) + n(B̄) log Pr(B̄i|Hcc).
(20)

These are maximized by a hypothesis Hcc that has Pr(Ai|Hcc) =
f(A) and Pr(Bi|Hcc) = f(B). That is, these terms are largest for
hypotheses that posit probabilities for the Ais and Bis that are equal
to the observed relative frequencies.

The mutual information of E1 and E2, conditional on a hypothesis
Hcc, is

I(E1,E2|Hcc) = n(AB)I(Ai,Bi|Hcc) + n(AB̄)I(Ai, B̄i|Hcc)
+ n(ĀB)I(Āi,Bi|Hcc) + n(ĀB̄)I(Āi, B̄i|Hcc). (21)

Once Pr(Ai|Hcc) and Pr(Bi|Hcc) are fixed, this is maximized by tak-
ing

Pr(AiBi) = f(AB). (22)

Thus, in the expression (19) for the log-likelihood, we see that the first
two terms reward hypotheses whose probabilities for Ai and Bi are
close to the observed relative frequencies of these, and the last term,
which corresponds to unification in the Mutual Information sense,
rewards hypotheses with theoretical correlations close to the observed
statistical correlations. What goes for log-likelihoods goes also for the
evidential support R. Thus, when there is a difference between f(AB)
and f(A)f(B), a common-cause hypothesis on which this difference is
expected, by virtue of appropriate values of the parameters, counts as
a MIUnifyng hypothesis, and thereby achieves greater support.

For example, consider a case in which we have two sequences {Ai},
{Bi}, with a significant positive correlation between them: f(AB) is
much larger than f(A)f(B). Consider two hypotheses, Hcc and H

′
cc,

which posit the existence of sequences {Ci} and {C′i}, respectively,
such that

Pr(Ai|Hcc) = Pr(Ai|H′cc) ≈ f(A);
Pr(Bi|Hcc) = Pr(Bi|H′cc) ≈ f(B).

(23)

20



Suppose, now that, Hcc correctly predicts the correlations, but H
′
cc

doesn’t. That is, Pr(AiBi|Hcc) is close to f(AB), but Pr(AiBi|H′cc)
is not. In such a case we will have

MIU1({E1,E2}; Hcc) > MIU1({E1,E2}; H′cc). (24)

Thus, for appropriate values of the parameters, the hypothesis Hcc
affords MIUnification to the evidence set {E1,E2}, even though, in
individual cases, the supposition Ci does not render Ai informative
about Bi.

This does not prevent Ci from being regarded as a common origin
of Ai and Bi. To take an example used by Lange in §3 of his paper,
suppose that we take the clinical evidence to establish that some dis-
ease C can cause symptoms A and B. Then, if we observe A and B
in some patient, this will raise our credence that C also occurs in that
patient, even if the symptoms A and B are independent, conditional
on C. In such a case, the support provided by the symptoms A and
B to the hypothesis that the patient has disease C is just the sum
of the supports given to the hypothesis by the individual items by
themselves.

Lange raises the question of whether we should place more credence
in a hypothesis that posits a single disease than in one that posits two
independent origins of the symptoms A and B. Suppose there are
two other diseases D1 and D2, such that A but not B is a symptom
of D1, and B but not A is a symptom of D2, and suppose further
that the chance that a patient with D1 exhibits symptom A is the
same as that of a patient with C, and that the chance that a patient
with D2 exhibits symptom B is the same as that of a patient with C.
Then, upon observation of both symptoms, the confirmational boost
afforded to the hypothesis that the patient has C is the same as the
boost afforded to the hypothesis that the patient has both D1 and D2.
The issue then comes down to priors. Is the joint occurrence of D1
and D2 much rarer than the occurrence of C? If the answer is yes—as
would be the case if the three diseases are equally rare, and D1 and D2
uncorrelated—then we should place more credence in the hypothesis
that the patient has C. If not—if the disease C is so rare, and D1 and
D2 so common that more patients contract both D1 and D2 than C—
then our credences should favor the two-disease hypothesis. It would
clearly be a mistake for one’s credences to favor the C-hypothesis
merely on the basis of a preference for common origin explanations.

21



8 Conclusion

Mutual Information Unification is not the same as common origin
explanation, and is neither a necessary nor sufficient condition for a
hypothesis to play an explanatory role. Nevertheless, in a host of
interesting cases, MIUnification is a concomitant of common origin
explanation. Moreover, when a hypothesis that renders an otherwise
puzzling coincidence comprehensible by providing a common origin
explanation does receive an incremental confirmational boost from a
body of evidence, beyond that provided by the individual items of
evidence, that boost stems from MIUnification.

So, at least, is the verdict delivered by a Bayesian analysis; there
is no room in Bayesian conditionalization for an extra confirmatory
boost that is due to Common Origin Unification. A proponent of an
explanationist thesis, to the effect that we ought to take hypotheses
that involve common origin explanations to receive greater incremen-
tal support than hypotheses that achieve the same degree of Mutual
Information Unification without explanation, should be in a position
to explain why what is impossible for a Bayesian agent is rational for
us. As we have seen, there is a limited heuristic role for considerations
of Common Origin Unification, based on considerations of Type II ra-
tionality. It is doubtful whether any stronger explanationist thesis can
be defended.

9 Appendix

Given a probability function Pr, and propositions h, e1, e2, define,

U1 =
Pr(e1e2|h)

Pr(e1|h)Pr(e2|h)
Pr(e1)Pr(e2)

Pr(e1e2)
; (25)

U2 =
Pr(e1e2|h̄)

Pr(e1|h̄)Pr(e2|h̄)
Pr(e1)Pr(e2)

Pr(e1e2)
. (26)

Then we have
MIU1(e1,e2; h) = log U1; (27)

MIU2(e1,e2; h) = log (U1/U2) . (28)

Thus, MIU1(e1,e2; h) is positive iff U1 > 1, negative iff U1 < 1, and
zero iff U1 = 1, and MIU2(e1,e2; h) is positive iff U1 > U2, negative
iff U1 < U2, and zero iff U1 = U2.

22



We want to show that each of the following four alternatives can
be realized by some probability function.

1. MIU1 > 0 and MIU2 > 0; that is, U1 > 1 and U1 > U2.

2. MIU1 > 0 and MIU2 < 0; that is, 1 < U1 < U2.

3. MIU1 < 0 and MIU2 > 0; that is, U2 < U1 < 1.

4. MIU1 < 0 and MIU2 < 0; that is, U1 < 1 and U1 < U2.

It is easy to show (see Lemma 1, below), that, if either e1 or e2
is irrelevant to h, then, if U1 > 1, U2 < 1, and vice versa. Thus, it
is easy to construct examples that satisfy conditions 1 and 4. Take
Pr(e1|h) = Pr(e1). Then, on an any probability function with U1 > 1,
we will have U2 < 1 < U1, and condition 1 will be satisfied. Similarly,
if Pr(e1|h) = Pr(e1), on any probability function with U1 < 1, we will
have U1 < 1 < U2, and condition 4 will be satisfied.

For condition 2, we need to have both U1 and U2 greater than
1. As is shown in Lemma 1, below, this is possible only if e1 and
e2 are relevant to h in opposite directions; that is, only if R(h; e1)
and R(h; e2) have opposite sign. Here’s one way to do it. Take, for
simplicity, Pr(h) = Pr(e1) = Pr(e2) = 1/2, and take Pr(e1e2) =
1/4. Take Pr(e1|h) = 0.7, Pr(e2|h) = 0.3, and Pr(e1e2|h) = 0.24.
The reader can readily verify that these are consistent, and that they
determine the full probability function on boolean combinations of
{h,e1,e2}. In particular, they entail that Pr(e1|h̄) = 0.3, Pr(e2|h̄) =
0.7, and Pr(e1e2|h̄) = 0.26. We thus have U1 = 24/21 and U2 = 26/21,
satisfying the desired conditions.

For condition 3, we can take the probability assignment described
in the previous paragraph and create a new one by interchanging e2
and ē2. We have, once again, Pr(h) = Pr(e1) = Pr(e2) = 1/2,
Pr(e1e2) = 1/4, Pr(e1|h) = 0.7, and Pr(e1|h̄) = 0.3. We also have
Pr(e2|h) = 0.7, and Pr(e1e2|h) = 0.46. These further entail that
Pr(e2|h̄) = 0.3, and Pr(e1e2|h̄) = 0.04. We thus have U1 = 46/49,
and U2 = 4/9, and so U2 < U1 < 1, and condition 4 is satisfied.

Having shown that all four alternatives are possible, we now prove
the Lemma alluded to above.

Lemma 1. Let {h,e1,e2} be logically independent propositions, and
let Pr be a probability function on the boolean algebra generated by
this set. We assume that the denominators of the relevant fractions
are nonzero, and define U1 and U2 as above.

23



a) If Pr(h|e1) = Pr(h) or Pr(h|e2) = Pr(h), then, if U1 > 1,
U2 < 1, and vice versa.

b) If U1 and U2 are both less than one, then either e1 and e2 are
both positively relevant to h, or they are both negatively relevant
to h.

c) If U1 and U2 are both greater than one, then one of {e1,e2} is
positively relevant to h, and the other negatively relevant.

Proof. Let

p = Pr(h); q = Pr(h̄) = 1 −p;

α1 = Pr(h|e1)/Pr(h); α2 = Pr(h|e2)/Pr(h);

β1 = Pr(h̄|e1)/Pr(h̄); β2 = Pr(h̄|e1)/Pr(h̄).

(29)

This allows us to write

U1 =
1

α1α2

Pr(e1e2|h)
Pr(e1e2)

; U2 =
1

β1β2

Pr(e1e2|h̄)
Pr(e1e2)

. (30)

Once p, α1, α2, β1, and β2 are fixed, this yields a constraint on U1
and U2:

pα1α2 U1 + q β1β2 U2 = 1. (31)

It is convenient to write this in terms of a weighted average of U1 and
U2. Define

w1 =
pα1α2

pα1α2 + q β1β2
; w2 =

q β1β2
pα1α2 + q β1β2

. (32)

Then (31) becomes,

w1 U1 + w2 U2 =
1

pα1α2 + qβ1β2
, (33)

with w1 and w2 both nonnegative, and

w1 + w2 = 1. (34)

It is instructive to rewrite the right-hand side of (33), using the fact
that pα1 + q β1 = pα2 + q β2 = 1. A bit of algebraic manipulation
yields,

w1 U1 + w2 U2 = 1 −
pq(α1 −β1)(α2 −β2)

pα1α2 + qβ1β2
. (35)

24



From (35) it is readily apparent that, if either e1 or e2 is irrelevant to
h—that is, if α1 = β1 or α2 = β2, then

w1 U1 + w2 U2 = 1, (36)

and in such a case, if U1 > 1, then U2 < 1, and vice versa. If we
want to construct a case in which U1 and U2 are both greater than
one, this requires the right-hand side of (35) to be greater than one,
which means that α1 −β1 and α2 −β2 must have opposite sign: one
of {e1,e2} must be positively relevant to h, and the other negatively
relevant. If we want to construct a case in which U1 and U2 are both
less than one, then α1 −β1 and α2 −β2 must have the same sign: e1
and e2 are either both positively relevant, or both negatively relevant,
to h.

10 Acknowledgments

I thank Michel Janssen, Marc Lange, Bill Harper, and Molly Kao
for helpful discussions. I am grateful to Clark Glymour for raising
the question, addressed in §7, of how common-cause explanations fit
into the framework. This work was supported, in part, by a grant
from the Social Sciences and Humanities Research Council of Canada
(SSHRC).

25



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