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<h2 class="hd hd-2 unit-title">Probability, Subjective, and Objective</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The concept of probability is ubiquitous. We feel safe traveling by plane because we believe that the probability of an accident is small; we buy fire insurance because we believe that, although the probability of a fire is small, it is not small enough to be ignored. But what <em>is</em> probability? What does it mean to say, for example:</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">The probability that the coin will land Heads is 50%.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">The answer is that it might mean two different things, depending on whether one takes “probability” to mean <em>subjective probability</em> or <em>objective probability.</em></span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Subjective Probability</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Claims about subjective probabilities are claims about someone’s <em>beliefs</em>, and, in particular, about the <em>degree</em> to which someone believes something. For example, the claim: “Smith’s subjective probability that the coin will land Heads is 50%.” means something like: “Smith believes to degree 0.5 that the coin will land Heads”. (I’ll have more to say later about what it means to believe something to a degree.)</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Objective Probability</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Claims about objective probability, in contrast, are <em>not</em> claims about what someone believes. When one talks about the objective probability of an event, one is meant to be describing a feature of the world that does not depend on the beliefs of any particular subject. Consider, for example, the chemical element Seaborgium. One of its isotopes,[mathjaxinline]^{265}\mbox{Sg}[/mathjaxinline], has a half-life of 8.9 seconds. This means that if you take a particle of [mathjaxinline]^{265}\mbox{Sg}[/mathjaxinline] the probability that it will decay in the next 8.9 seconds is 50%. When one speaks of probability in this context one isn’t talking about the subjective credences of any particular subject. One is describing an objective feature of the world: the half-life of [mathjaxinline]^{265}\mbox{Sg}[/mathjaxinline].</span></p>
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<h2 class="hd hd-2 unit-title">How are Subjective and Objective Probability Related?</h2>
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<p><span style="font-family: 'book antiqua', palatino;">There is an important connection between subjective and objective probabilities:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>The Objective-Subjective Connection</strong></span><br /><span style="font-family: 'book antiqua', palatino;">The objective probability of <span class="math inline">\(A\)</span> at time <em>t</em> is the subjective probability that a perfectly rational agent would assign to <span class="math inline">\(A\)</span>, if she had perfect information about the way the world is before or at <em>t</em> and no information about the way the world is after <span class="math inline">\(t\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">This principle can be used in two different ways:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Subjective <span class="math inline">\(\rightarrow\)</span> Objective</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Start with information about what an agent’s subjective probabilities ought to be, and use the Objective-Subjective Connection to get information about what the objective probabilities are.</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Objective <span class="math inline">\(\rightarrow\)</span> Subjective</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Start with information about what the objective probabilities are, and use the Objective-Subjective Connection to get information about what an agent’s subjective probabilities ought to be.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">To illustrate the first kind of usage, suppose that you assign a subjective probability of 30% to a particular event. Suppose, moreover, that you have all relevant information about the case at hand, and that you’ve made no mistakes in your reasoning. Then your credences are the credences that a perfectly rational agent would have, if she had perfect information about the past. So the Objective-Subjective Connection entails that the objective probability of the relevant event is 30%.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">To illustrate the second kind of usage, suppose you know that <span class="math inline">\(^{265}\mbox{Sg}\)</span> has a half life of 8.9 seconds, and therefore that the objective probability that a particle of <span class="math inline">\(^{265}\mbox{Sg}\)</span> will decay within the next 8.9 seconds is 50%. Then the Objective-Subjective Connection entails that a perfectly rational agent with access to all available evidence would set her subjective probability that the particle will decay within the next 8.9 seconds at 50%. But such an agent is at least as good at assessing evidence as you are, because she is perfectly rational. And (unless you have access to information about the future) she has at least as much evidence as you do, because she has access to all evidence about the past. So you should defer to her judgment, and set your subjective probability that the particle will decay within the next 8.9 seconds at 50%.</span></p>
<p><span style="font-family: georgia, palatino;">(Nerdy observation: Here I am tacitly presupposing that a perfectly rational agent is always certain about the objective probabilities at \(t\), given full information about how the world is before \(t\). So, in particular, for each complete history of the world up to \(t\), \(H_t\), there is a specification \(P_t\) of the objective probabilities at \(t\) such that the agent treats \(H_t\) and \(H_t P_t\) as equivalent. This assumption is potentially controversial but adds simplicity to our discussion.)</span></p>
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<span style="font-family: 'book antiqua', palatino;">We have seen that it follows from the Objective-Subjective Connection, and from the fact that <span class="math inline">\(^{265}\mbox{Sg}\)</span> has a half life of 8.9 seconds, that if you have no information about the future, you should set your subjective probability that a particle of <span class="math inline">\(^{265}\mbox{Sg}\)</span> will decay within the next 8.9 seconds at 50%. Suppose, however, that a trusted time traveler comes back from the future and informs you that she has witnessed the fact that the particle does, in fact, decay within the the next 8.9 seconds. What should your subjective probability that the particle will decay within the next 8.9 seconds be?</span>
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