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<h2 class="hd hd-2 unit-title">Subjective Probability</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> In this section I’ll tell you more about the notion of <strong>subjective probability,</strong> and about what it takes for a subject’s subjective probabilities to count as rational. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The central idea behind the notion of subjective probability, recall, is that people’s beliefs come in <em>degrees</em>. In other words: beliefs aren’t just “on” or “off”. Instead of just having two possibilities—believing something or not believing it—you have a whole range of options: for any real number [mathjaxinline]r[/mathjaxinline] between 0 and 1, you could in principle believe something to degree [mathjaxinline]r[/mathjaxinline]. So, for instance, if you are completely confident that it will rain, you believe that it will rain to degree 1; if you are somewhat confident, you might believe it to degree 0.7; if you fairly doubtful, you might believe it to degree 0.1; and so forth. (The expression " [mathjaxinline]x\%[/mathjaxinline]" is shorthand for "[mathjaxinline] {x}/{100}; [/mathjaxinline]" so believing something to degree 0.5 is the same as believing it to degree 50%.) </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">A natural way of modeling the credences of a subject [mathjaxinline]S[/mathjaxinline] is by using a <strong>credence function:</strong> a function that assigns to each proposition a real number between 0 and 1, representing the degree to which [mathjaxinline]S[/mathjaxinline] believes that proposition. In the philosophical literature, a degree of belief is usually called a <strong>credence</strong>. So instead of saying that Smith believes that it will rain to degree 0.6, one says that Smith’s <em>credence</em> that it will rain is 0.6.</span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> Having the wrong credences can make a subject irrational. Consider, for example, a subject who assigns credence 0.9 <em>both</em> to the proposition that it will rain today <em>and</em> to the proposition that it will not rain today. Such a subject would be confident that it will rain and confident that it will not rain, which is irrational. What sorts of constraints must a subject’s credences satisfy in order for the subject to count as <strong>perfectly rational?</strong> This is an important question, since the Objective-Subjective Connection relies on the notion of a perfectly rational subject. We will try to answer it in the remainder of this section.</span></p>
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<h2 class="hd hd-2 unit-title">Internal Coherence</h2>
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<p><span style="font-family: 'book antiqua', palatino;">A credence function, recall, is a function that assigns to each proposition a real number between <span class="math inline">\(0\)</span> and <span class="math inline">\(1\)</span>, representing the subject’s credence in that proposition. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">As I noted earlier, not just any assignment of credences to propositions is internally coherent. What does it take for a credence function to count as internally coherent? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">A standard answer is that it is internally coherent if and only if it is a <strong>probability function</strong>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">A probability function, <span class="math inline">\(p(\ldots)\)</span>, is an assignment of real numbers between <span class="math inline">\(0\)</span> and <span class="math inline">\(1\)</span> to propositions that satisfies the following two coherence conditions:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Necessity</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(A\)</span> is a necessary truth, then <span class="math inline">\(p(A) = 1\)</span>.</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Additivity</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are incompatible propositions, then <span class="math inline">\(p(A \mbox{ or } B)~=~p(A) + p(B)\)</span>.</span></p>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(\text{not-}A)=1-p(A)\)</span>, for any proposition <span class="math inline">\(A\)</span>.</span>
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<h2 class="hd hd-2 unit-title">Updating by Conditionalization</h2>
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<p><span style="font-family: 'book antiqua', palatino;">We have considered a constraint on rational belief: that one’s credence function internally coherent; and, more specifically, that it be a probability function. Notice, however, that this doesn’t tell us anything about what it takes for one to update one’s beliefs in a rational way, as one acquires additional information. That will be the topic of this subsection.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Just like one might use an <em>unconditional</em> probability function <span class="math inline">\(p(A)\)</span> to talk about the probability of <span class="math inline">\(A\)</span>, so one could also use a <em>conditional</em> probability function <span class="math inline">\(p(A|H)\)</span> to talk about the probability of <span class="math inline">\(A\)</span> <em>on the assumption that <span class="math inline">\(H\)</span> obtains</em>. Suppose, for instance, that <span class="math inline">\(S\)</span> thinks it’s unlikely to rain: <span class="math inline">\(p(\text{Rain}) = 0.2\)</span>. She thinks its even less likely that there’ll be a sudden drop in atmospheric pressure: <span class="math inline">\(p(\text{Drop}) = 0.1\)</span>. But <span class="math inline">\(S\)</span> also thinks there’s a strong correlation between rain and sudden pressure drops. In particular, she is confident to degree <span class="math inline">\(0.95\)</span> in the following conditional statement: it’ll rain, <em>assuming there’s a sudden drop in atmospheric pressure</em>. We can then say that <span class="math inline">\(S\)</span>’s <em>conditional</em> credence in rain given a pressure drop is <span class="math inline">\(0.95\)</span>; in symbols: <span class="math inline">\(p(\text{Rain}|\text{Drop}) = 0.95\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The notion of conditional credence puts us in a position to give an attractive answer to the question of what it takes for a subject to <em>update</em> her credences in a rational way, as she acquires additional information:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Update by Conditionalization</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(S\)</span> is rational, she will update her credences as follows upon learning that <span class="math inline">\(B\)</span>: <span class="math display">\[p^{new}(A) = p^{old}(A|B)\]</span> where <span class="math inline">\(p^{old}\)</span> is the function describing <span class="math inline">\(S\)</span>’s credences before she learned that <span class="math inline">\(B\)</span>, and <span class="math inline">\(p^{new}\)</span> is the function describing her credences after she learned that <span class="math inline">\(B\)</span>.</span></p>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose, for example, that <span class="math inline">\(S\)</span> starts out fairly confident that it won’t rain: <span class="math inline">\(p^{old}(\text{Rain}) = 0.1\)</span>. But she is highly confident that it’ll rain, given that there’s a sudden pressure drop: <span class="math inline">\(p^{old}(\text{{Rain}}|\text{{Drop}}) = 0.95\)</span>. Now suppose that <span class="math inline">\(S\)</span> learns that there’s been a sudden pressure drop. How confident should she be that it’ll rain, in light of the new information. According to Update by Conditionalization, she should be highly confident: <span class="math inline">\(p^{new}(\text{Rain}) = 0.95\)</span>.</span></p>
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<h2 class="hd hd-2 unit-title">Bayes' Law</h2>
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<p><span style="font-family: 'book antiqua', palatino;">We have considered two different constraints on rational belief: (1) one’s <em>unconditional</em> credences can be represented by a probability function, and (2) one must respond to new information in accordance with one’s <em>conditional</em> credences. Are there any constraints on how a subject’s unconditional credences ought to be related to her conditional credences? Yes! It is natural to think that a rational subject should satisfy the following principle:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Bayes’ Law</strong></span><br /><span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(AB) = p(A)\cdot p(B|A)\)</span></span></p>
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<p><span style="font-family: 'book antiqua', palatino;">Notice that whenever the subject assigns non-zero credence to <span class="math inline">\(A\)</span>, Bayes’ Law entails: <span class="math display">\[p(B|A) = \frac{p(AB)}{p(A)}\]</span> which allows one to determine the subject’s conditional credences on the basis of her unconditional credences. More specifically, one can determine the subject’s (conditional) credence in <span class="math inline">\(B\)</span> given <span class="math inline">\(A\)</span> by looking at her (unconditional) credence in <span class="math inline">\(A\)</span> and her (unconditional) credence in <span class="math inline">\(A\)</span>-and-<span class="math inline">\(B\)</span>.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Consider a subject <span class="math inline">\(S\)</span> whose unconditional credences regarding rain (<span class="math inline">\(R\)</span>), and regarding a sudden drop in atmospheric pressure (<span class="math inline">\(D\)</span>), are as follows: <span class="math display">\[\begin{aligned}
p(R) &amp;=&amp; 0.2 \\
p(D) &amp;=&amp; 0.1 \\
p(RD) &amp;=&amp; 0.09\end{aligned}\]</span> Use Bayes&#8217; Law to calculate <span class="math inline">\(p(R|D)\)</span> and <span class="math inline">\(p(D|R)\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(R|D)\)=</span>?</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(D|R)\)=</span>?</span>
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<span style="font-family: 'book antiqua', palatino;">Now suppose that <span class="math inline">\(p(RD) = 0.02\)</span>. How do the conditional probabilities <span class="math inline">\(p(R|D)\)</span> and <span class="math inline">\(p(D|R)\)</span> compare to the unconditional probabilities <span class="math inline">\(p(R)\)</span> and <span class="math inline">\(p(D)\)</span>?</span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(R)&gt; p(R|D)\)</span> and <span class="math inline">\(p(D)\geq p(D|R)\)</span></span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(R) \leq p(R|D)\)</span> and <span class="math inline">\(p(D) \leq p(D|R)\)</span></span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(R)= p(R|D)\)</span> and <span class="math inline">\(p(D)= p(D|R)\)</span></span>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(p(R)&gt; p(R|D)\)</span> and <span class="math inline">\(p(D)\leq p(D|R)\)</span></span>
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<span style="font-family: 'book antiqua', palatino;">Suppose that <span class="math inline">\(S\)</span> is perfectly rational. (In particular, her credences are described by a probability function, she updates by conditionalizing on her evidence, and she respects Bayes&#8217; Law.)</span>
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<span style="font-family: 'book antiqua', palatino;">Suppose, moreover, that she is certain of <span class="math inline">\(A\)</span> (i.e. <span class="math inline">\(p(A) = 1\)</span>), and that she assigns a non-trivial probability to <span class="math inline">\(B\)</span> (i.e. <span class="math inline">\(p(B) \neq0\)</span>).</span>
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<span style="font-family: 'book antiqua', palatino;">Could learning <span class="math inline">\(B\)</span> cause her to be less than certain about <span class="math inline">\(A\)</span>?</span>
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<h2 class="hd hd-2 unit-title">The Principle of Indifference</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Although the constraints on rational belief that we have considered so far are non-trivial, they leave a lot of leeway. Notice, in particular, they don’t fully constrain a rational subject’s <em>initial credences</em>: her credences before she has received any information about the way the world is. All our constraints tell us is that her unconditional credences will constitute a probability function, and that her conditional and unconditional credences will be related by Bayes’ Law. Some people think this is too much leeway, and hope to identify further constraints on rational belief—constraints that might help reduce the range of admissible initial credences, and it is natural to suppose that the following might be added as a further constraint:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Principle of Indifference</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Consider a set of propositions, and suppose one knows that exactly one of them is true. Suppose, moreover, that one has no more reason to believe any one of them than any other. Then, insofar as one is rational, one should assign equal credence to each proposition in the set.</span></p>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose, for example, that we have a coin, and that we have no more reason to believe that it will land Heads than that it will land Tails, and vice-versa. The Principle of Indifference tells us that we should divide our credence equally, and assign credence <span class="math inline">\(0.5\)</span> to both the proposition that the coin will land Heads and the proposition that it will land Tails.</span></p>
<h4><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> The Bad News </span></h4>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The bad news is that it is not clear that there is a coherent way of formulating the Principle of Indifference. It is certainly incoherent in the formulation I used above. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Philosopher Bas van Fraassen came up with a nice way of making the point. Imagine a cube factory. We know that the factory produces cubes with a side-length of less than 1 meter, but we haven’t the slightest idea how the cube sizes are chosen. What is the probability that the next cube produced will have a side-length of less than half a meter? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Here is an argument based on the Principle of Indifference. Since the distance between 0 and 1/2 is the same as the distance between 1/2 and 1, our reasons for thinking that the factory will produce a cube with a side-length of less than half a meter are exactly analogous to our reasons for thinking that the factory will produce a cube with a side-length of more than half a meter. So the Principle of Indifference tells us that we should assign credence 0.5 to the proposition that the next cube produced will have a side-length of less than half a meter, and credence 0.5 to the proposition that the next cube will have a side-length of more than half a meter. (To keep things simple, I ignore the possibility that the factory will next produce a cube whose side-length is exactly half a meter.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;"> Everything sounds plausible so far. But notice that the Principle of Indifference delivers a different conclusion when we focus on <em>volume</em> rather than side-length. To see this, we again start with the observation that the distance between 0 and 1/2 is the same as the distance between 1/2 and 1. But this time we use this observation as grounds for thinking that our reasons for thinking that the factory will next produce a cube with a <em>volume</em> of less than half a cubic meter are exactly analogous to our reasons for thinking that the factory will next produce a cube with a volume of more than half a cubic meter. </span>So the Principle of Indifference tells us that we should assign credence 0.5 to the proposition that the next cube produced will have a volume of less than half a cubic meter, and credence 0.5 to the proposition that the next cube will have a volume greater than half a cubic meter.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The problem, of course, is that cubes with a side-length of half a meter do not have a volume of half a cubic meter, but a volume of 1/8 cubic meters. Accordingly, cubes with a side-length of less than half a meter make up a small fraction of the cubes whose volume is less than half a cubic meter. So if we are 50% confident that the next cube will have a volume of less than half a cubic meter, we should be <em>less</em> than 50% confident that the next cube will have a side-length of less than half a meter, which contradicts our earlier result.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Perhaps there is a version of the Principle of Indifference that allows us to avoid this kind of problem. I myself do not know of any that seems very convincing.</span></p>
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<span style="font-family: 'book antiqua', palatino;">God has picked a real number, and that you have no more reason for thinking that She picked one number than you have for thinking that She picked any other.</span>
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<span style="font-family: 'book antiqua', palatino;">Use the Principle of Indifference to show that you ought to assign credence 0 to the proposition that God picked the number <span class="math inline">\(\pi\)</span>.</span>
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<h2 class="hd hd-2 unit-title">Summing Up</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> This concludes our discussion of subjective probability. Our discussion focused on the question of what a subject’s credences must look like in order for the subject to count as fully rational. We identified three important constraints:</span></p>
<ul>
<li><span style="font-family: 'book antiqua', palatino;">the subject’s credence function must be internally coherent (i.e. it must be a probability function);</span></li>
<li><span style="font-family: 'book antiqua', palatino;">as she acquires additional information, the subject must update her credences by conditionalization;</span></li>
<li><span style="font-family: 'book antiqua', palatino;">the subject’s conditional and unconditional credences must be related in accordance with Bayes’ Law.</span></li>
</ul>
<p><span style="font-family: 'book antiqua', palatino;">We also considered the hypothesis that the subject’s initial credences satisfy a Principle of Indifference, but noted that formulating a consistent version of the principle is not as easy as one might have hoped.</span></p>
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