Subjective Probability

An approach concerned with the consequences of modifying our previous beliefs as a result of receiving new data. By contrast with the 'classical' approach which begins with a hypothesis test that proposes a specific value for an unknown parameter, θ, Bayesian inference proposes a prior distribution, p(θ), for this parameter. Data x₁, x₂,..., x_n are collected and the likelihood f(x₁, x₂,..., x_n|θ) is calculated. Bayes's theorem is now used to calculate the posterior distribution, g(θ| x₁, x₂,..., x_n). The change from the prior to the posterior distribution reflects the information provided by the data about the parameter value. For any particular event the initial probability is described as the prior probability and the subsequent probability as the posterior probability.

If nothing is known about the value of a parameter, then a non-informative prior is used — typically, this is a uniform distribution over the feasible set of values of the parameter. Another approach, the empirical Bayes method, utilizes the data to inform the prior distribution.

In a similar way, if nothing is known about the underlying distribution, then the principle of indifference effectively states that all possible values should be assigned the same probability of occurrence. This is also called the principle of insufficient reason.

Subjective probability measures the degree of belief an individual has in an uncertain proposition. This could form the basis for a prior distribution. Another term is personal probability, though this may be used to suggest that the person's selected probability is misguided.

Often, however, a more useful choice for the form of a prior distribution is a member of a family of distributions which is such that the posterior distribution is another member of that family, so that the effect of the data can be interpreted in terms of changes in parameter values. Such a prior is called a conjugate prior.

Sometimes useful information is available. For example, an appropriate prior for the amount taken by a supermarket on a Saturday might be a normal distribution centred on the amount taken the previous Saturday. This would be an informative prior.

Jeffreys argued that an appropriate prior should be unaffected by the way a model is expressed: this leads to the Jeffreys prior which is proportional to √I(θ), where I(θ) is the Fisher information.

A probability derived from an individual's personal judgment about whether a specific outcome is likely to occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions and past experience.

Investopedia Says:
Subjective probabilities differ from person to person. Because the probability is subjective, it contains a high degree of personal bias. An example of subjective probability could be asking New York Yankees fans, before the baseball season starts, the chances of New York winning the world series. While there is no absolute mathematical proof behind the answer to the example, fans might still reply in actual percentage terms, such as the Yankees having a 25% chance of winning the world series.

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Estimate of the relative frequency of each of various future outcomes, based on the intuition or experience of the accountant making the estimate. Subjective probability is used in many business situations (i.e., estimating rates and/or dollar returns on investment decisions). Based on his or her experience, assume a treasurer is considering possible interest rates the company can expect to pay on bonds it intends to issue next week. The treasurer feels that interest rates can have only four possible values. The subjective assessment is as follows: