Prior probability

What Is Prior Probability?

Prior probability is the initial assessment of the likelihood of an event or hypothesis occurring before any new data or evidence is considered. It represents a belief about the state of the world, often derived from historical data, expert opinion, or logical reasoning. This concept is fundamental to Bayesian inference, a statistical method within the broader field of probability theory that updates probabilities as new information becomes available. Prior probability sets the groundwork for how initial assumptions are integrated into statistical analysis to inform subsequent conclusions.

History and Origin

The concept of prior probability is deeply rooted in the work of Thomas Bayes, an 18th-century English mathematician and Presbyterian minister. His seminal work, "An Essay Towards Solving a Problem in the Doctrine of Chances," published posthumously in 1763 by the Royal Society, laid the mathematical foundation for what is now known as Bayes' Theorem. In this essay, Bayes tackled the "inverse problem" of probabilities, aiming to determine the original probability of an event based on observed outcomes.¹⁶,¹⁵ While Bayes introduced the core idea, it was Pierre-Simon Laplace who, unaware of Bayes' earlier work, independently formulated a more general version of the theorem in 1774 and applied it to various scientific problems, significantly popularizing the approach., The use of prior probability, initially termed "inverse probability," has evolved considerably, especially with the advent of modern computing, leading to its widespread acceptance in diverse fields.

Key Takeaways

• Prior probability quantifies an initial belief or a known likelihood of an event before new data is observed.
• It is a critical component in Bayesian inference, where it is updated by new information.
• Prior probabilities can be based on historical data, expert judgment, or the principle of indifference.
• The choice of a prior probability can significantly influence the resulting posterior probability.
• It plays a role in decision making under uncertainty across various disciplines.

Formula and Calculation

Prior probability (P(H)) is a component of Bayes' Theorem, which describes how to update the probability of a hypothesis (H) given new evidence (E). The formula for Bayes' Theorem is:

$P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$

Where:

• (P(H|E)) is the posterior probability: the probability of hypothesis H given evidence E.
• (P(E|H)) is the likelihood: the probability of observing evidence E if hypothesis H is true.
• (P(H)) is the prior probability: the initial probability of hypothesis H before any evidence E is considered.
• (P(E)) is the marginal probability of evidence E: the probability of observing the evidence E under all possible hypotheses. It acts as a normalizing constant to ensure the posterior probability is a valid probability.

Interpreting Prior Probability

Interpreting prior probability involves understanding that it represents the starting point of a probabilistic analysis. It encapsulates all available knowledge or assumptions about an event or hypothesis before any new, specific observations are made. For instance, if a financial analyst believes there's a 60% chance of a company's earnings exceeding expectations based on past performance and industry trends, this 60% is their prior probability. This initial assessment guides the subsequent integration of new data. The strength and source of the prior can vary: it might be derived from extensive historical data, an expert's subjective judgment, or a "non-informative" prior used when there is little to no initial knowledge, such as assigning equal probabilities to all possible outcomes. A well-chosen prior helps to inform the model and can lead to more robust statistical inference.

Hypothetical Example

Consider a hypothetical scenario where an investor is evaluating the probability that a specific tech startup, "InnovateCo," will succeed (i.e., achieve an initial public offering within five years).

Before analyzing any specific financial statements or market data for InnovateCo, the investor uses their general knowledge of the startup industry. They know that historically, only about 15% of tech startups in this sector achieve an IPO within five years. This 15% becomes the investor's prior probability for InnovateCo's success.

• P(Success) = 0.15 (This is the prior probability).

Now, suppose the investor obtains new evidence: InnovateCo has just secured a significant Series C funding round from a reputable venture capital firm. The investor also knows that, historically, 80% of tech startups that receive Series C funding from such firms go on to achieve an IPO. However, they also know that the probability of any tech startup receiving Series C funding is relatively low, say 10%.

Using Bayes' Theorem, the investor can update their belief:

• (P(\text{IPO | Series C})) = Probability of IPO given Series C funding (what we want to find – posterior probability)
• (P(\text{Series C | IPO})) = Probability of Series C given IPO (this is not directly given, but we know 80% of those who IPO received Series C from reputable firms, so this can be interpreted as a likelihood of 0.8 if we assume the IPO happens)
• (P(\text{IPO})) = Prior probability of IPO = 0.15
• (P(\text{Series C})) = Marginal probability of Series C funding = 0.10

Let's assume (P(\text{Series C | IPO})) is 0.8 (meaning among companies that eventually IPO, 80% had Series C funding from reputable firms).

$P(\text{IPO | Series C}) = \frac{P(\text{Series C | IPO}) \cdot P(\text{IPO})}{P(\text{Series C})}$
$P(\text{IPO | Series C}) = \frac{0.80 \cdot 0.15}{0.10}$
$P(\text{IPO | Series C}) = \frac{0.12}{0.10}$
$P(\text{IPO | Series C}) = 1.2$

Self-correction: The calculation result of 1.2 is incorrect. This indicates an issue with how P(Series C | IPO) or P(Series C) is conceptualized in a simple example. Let's reframe.
The likelihood (P(E|H)) should be "probability of seeing this evidence if the hypothesis is true".
So, (P(\text{Series C funding | IPO success})) is the likelihood. Let's say, based on analysis, 80% of successful tech startups (those that IPO) had received Series C funding from a reputable firm.
(P(\text{IPO success})) is the prior (0.15).
(P(\text{Series C funding})) is the overall probability of any tech startup getting Series C funding (0.10).

The calculation implies a problem with the numbers if the result exceeds 1. This often happens if the likelihood or marginal probability is poorly estimated in a simplified example.

Let's re-align the numbers for a more plausible outcome.
Suppose:

• Prior Probability of InnovateCo IPOing (P(\text{IPO})) = 0.15
• Probability of getting Series C funding if they IPO (P(\text{Series C | IPO})) = 0.80 (This means, among companies that do IPO, 80% had Series C)
• Probability of getting Series C funding if they don't IPO (P(\text{Series C | No IPO})) = Let's say this is much lower, maybe 0.05 (5% of those that don't IPO still get Series C).
• (P(\text{No IPO})) = (1 - P(\text{IPO})) = (1 - 0.15 = 0.85)

Now, we need (P(\text{Series C})) which is the total probability of Series C funding:
(P(\text{Series C}) = P(\text{Series C | IPO}) \cdot P(\text{IPO}) + P(\text{Series C | No IPO}) \cdot P(\text{No IPO}))
(P(\text{Series C}) = (0.80 \cdot 0.15) + (0.05 \cdot 0.85))
(P(\text{Series C}) = 0.12 + 0.0425 = 0.1625)

Now, calculate the posterior probability:
$P(\text{IPO | Series C}) = \frac{P(\text{Series C | IPO}) \cdot P(\text{IPO})}{P(\text{Series C})}$
$P(\text{IPO | Series C}) = \frac{0.80 \cdot 0.15}{0.1625}$
$P(\text{IPO | Series C}) = \frac{0.12}{0.1625} \approx 0.738$

After observing the Series C funding, the investor's belief in InnovateCo's success increases significantly, from a 15% prior probability to approximately a 73.8% posterior probability. This demonstrates how new evidence updates the initial belief.

Practical Applications

Prior probability finds broad application in finance, risk assessment, and various analytical fields. In investing, prior probabilities are used to assess the initial risk of investments before considering new market trends or data., ¹⁴F¹³or instance, an investor might use the historical average annual return of a particular asset class as a prior to inform their expected value calculations for future returns.

¹²Financial institutions leverage prior probabilities in credit scoring models, using an applicant's credit history and other demographics to establish an initial likelihood of loan default. As new information, like current income or employment status, becomes available, this prior is updated to yield a more refined probability. Similarly, insurance companies rely on prior probabilities, derived from vast historical datasets on accidents, health conditions, and life expectancies, to calculate premiums for various policies.

¹¹Furthermore, in portfolio management, investors can use "base rates" to inform their prior probabilities, which are then updated with additional company or market-specific information. T¹⁰his approach helps in making informed investment decisions and managing diversified portfolios by assessing how different assets perform together.

⁹## Limitations and Criticisms

While prior probability is integral to Bayesian methods, it faces certain criticisms, primarily concerning its subjective nature and potential for undue influence on results. The most common critique is that the reliance on subjective prior assumptions can lead to conclusions that are heavily influenced by the initial beliefs of the analyst, rather than purely by the data., ⁸T⁷his subjectivity can make it challenging for different analysts to arrive at the same conclusions, even when presented with the same data, if their initial priors differ significantly.

⁶Another limitation arises when there is little to no prior information available, leading to the use of "uninformative" or "flat" priors. While these are intended to represent a lack of strong initial belief, their choice can still implicitly influence the posterior probability, and there's no universally agreed-upon method for choosing a truly objective non-informative prior.,

⁵Critics also point out computational complexity as a drawback, especially in complex models where calculating the posterior probability requires intensive computation. D⁴espite these challenges, proponents argue that Bayesian analysis, by explicitly requiring the specification of a prior, forces analysts to be transparent about their assumptions, which can be a distinct advantage over methods that do not explicitly account for initial beliefs.,
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2## Prior Probability vs. Posterior Probability

Prior probability and posterior probability are two fundamental concepts in Bayesian inference, representing the evolution of belief about a hypothesis as new information is acquired.

The distinction is crucial: prior probability is the starting point, reflecting what is known or believed before the current observation, whereas posterior probability is the end result of the Bayesian updating process, reflecting the refined belief after incorporating new evidence.

FAQs

What is the main purpose of prior probability?

The main purpose of prior probability is to incorporate existing knowledge, beliefs, or historical information into a statistical analysis before any new data is observed. It provides a baseline for updating probabilities using Bayes' Theorem.

How is prior probability determined if there's no historical data?

If no specific historical data is available, prior probability can be determined through expert judgment or by using "uninformative" priors. Expert judgment involves soliciting educated guesses from knowledgeable individuals. Uninformative priors, such as a uniform probability distribution, are used to express a state of minimal initial knowledge or to allow the data to dominate the posterior inference.

Can prior probability be subjective?

Yes, prior probability can be subjective. While sometimes based on objective historical data, it often reflects an individual's or expert's personal beliefs or educated guesses about the likelihood of an event. This subjective element is a core characteristic of Bayesian statistics, distinguishing it from frequentist approaches.

Does prior probability affect the final outcome of an analysis?

Yes, the choice of prior probability can significantly affect the final outcome, particularly when the amount of new evidence is limited. A strong or "informative" prior, based on robust existing knowledge, can lead to a posterior probability that closely resembles the prior. Conversely, a "weak" or "non-informative" prior allows the new data to have a greater influence on the posterior.

Is prior probability used in financial forecasting?

Yes, prior probability is used in financial forecasting and decision making. Investors and analysts might use historical market trends, company performance, or economic indicators to establish a prior probability for future asset returns, market movements, or the likelihood of a company achieving certain financial targets. This initial assessment is then updated as new financial reports, economic news, or market data become available.¹