Bayesian statistics

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  1. Bayesian Statistics: A Beginner's Guide

Introduction

Bayesian statistics is a powerful framework for statistical inference that differs fundamentally from the more commonly taught frequentist statistics. While frequentist statistics focuses on the frequency of events in repeated trials, Bayesian statistics focuses on updating beliefs in light of new evidence. It's a way of reasoning under uncertainty, and it has applications in a vast range of fields, from medicine and machine learning to finance and even everyday decision-making. This article aims to provide a comprehensive, yet accessible, introduction to Bayesian statistics for beginners. We will cover the core concepts, the underlying mathematics (simplified where possible), and illustrate its practical applications, particularly within the context of Technical Analysis.

The Core Concepts

At the heart of Bayesian statistics lie three core concepts:

  • **Prior Probability:** This represents your initial belief about a hypothesis *before* observing any evidence. It's your starting point. This is often based on existing knowledge, past experience, or even a reasoned guess. For example, before looking at any stock price charts, you might have a prior belief about the probability that a particular stock will rise tomorrow.
  • **Likelihood:** This measures how well the observed evidence supports the hypothesis. It's the probability of observing the data *given* that the hypothesis is true. In our stock example, the likelihood would be the probability of seeing the actual price movements given that the stock *did* rise. This is closely related to Candlestick Patterns.
  • **Posterior Probability:** This is your updated belief about the hypothesis *after* observing the evidence. It's calculated by combining the prior probability and the likelihood. This is the key output of Bayesian analysis – a refined belief based on both your initial assumptions and the new data.

These concepts are mathematically expressed through **Bayes' Theorem**:

P(A|B) = [P(B|A) * P(A)] / P(B)

Where:

  • P(A|B) is the **posterior probability** – the probability of event A happening given that event B has happened.
  • P(B|A) is the **likelihood** – the probability of event B happening given that event A has happened.
  • P(A) is the **prior probability** – the probability of event A happening before observing event B.
  • P(B) is the **marginal likelihood** or **evidence** – the probability of event B happening. This often acts as a normalizing constant. Calculating P(B) can be the most challenging part of Bayesian analysis.

Understanding Bayes' Theorem with an Example

Let's consider a simple example: A medical test for a rare disease.

  • **Event A:** Having the disease.
  • **Event B:** Testing positive for the disease.

Suppose:

  • P(A) = 0.01 (The disease affects 1% of the population – the prior probability)
  • P(B|A) = 0.95 (The test correctly identifies 95% of people *with* the disease – the sensitivity of the test)
  • P(B|¬A) = 0.05 (The test incorrectly identifies 5% of people *without* the disease as having it – the false positive rate)

We want to know P(A|B) – the probability of *actually having* the disease given a positive test result.

First, we need to calculate P(B):

P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A) P(B) = (0.95 * 0.01) + (0.05 * 0.99) P(B) = 0.0095 + 0.0495 P(B) = 0.059

Now we can apply Bayes' Theorem:

P(A|B) = (0.95 * 0.01) / 0.059 P(A|B) = 0.0095 / 0.059 P(A|B) ≈ 0.161

This means that even with a positive test result, there's only about a 16.1% chance of actually having the disease! This is because the disease is rare, and the test has a non-negligible false positive rate. This highlights the importance of considering prior probabilities – a positive test result is much more meaningful if the disease is common.

Applying Bayesian Statistics to Trading & Technical Analysis

Bayesian statistics offers a powerful approach to improving your trading strategies. Here's how:

  • **Prior Beliefs About Strategies:** You can start with a prior belief about the effectiveness of a particular Trading Strategy. For example, you might believe that a moving average crossover strategy has a 60% chance of being profitable.
  • **Likelihood Based on Backtesting:** Backtesting provides the likelihood. The more consistently a strategy performs well during backtesting, the higher the likelihood that it will perform well in the future. A robust backtest will consider Market Volatility and various Time Frames.
  • **Posterior Beliefs and Strategy Optimization:** As you gather more data from live trading, you update your posterior belief. If the strategy continues to perform well, your belief in its profitability increases. If it performs poorly, your belief decreases, and you might consider modifying or abandoning the strategy. This is a form of dynamic Risk Management.
  • **Incorporating Expert Opinion:** Bayesian statistics allows you to incorporate the opinions of experienced traders as prior probabilities. This is particularly valuable when dealing with subjective assessments of market conditions.
  • **Assessing the Probability of Trends:** You can use Bayesian methods to assess the probability of a particular trend continuing, based on historical data and current market signals. For instance, what's the probability that an Uptrend will continue given the current Relative Strength Index (RSI) reading?
  • **Evaluating the Effectiveness of Indicators:** Bayesian analysis can help determine whether an Indicator (e.g., MACD, Bollinger Bands, Fibonacci Retracements) actually provides valuable information for predicting future price movements. You can quantify the likelihood of a profitable trade given a specific indicator signal.
  • **Portfolio Optimization:** Bayesian methods can be used to optimize portfolio allocation by considering the uncertainty in expected returns and correlations between assets.

Practical Considerations & Common Distributions

While the concept is straightforward, applying Bayesian statistics can be computationally challenging. Here are some practical considerations:

  • **Choosing a Prior:** Selecting an appropriate prior is crucial. A "non-informative" prior (e.g., a uniform distribution) expresses minimal prior belief and lets the data speak for itself. However, even non-informative priors can influence the results. Carefully consider the implications of your prior choice.
  • **Computational Complexity:** Calculating the posterior distribution often requires complex integration. In many cases, closed-form solutions are not available, and we must rely on numerical methods like Markov Chain Monte Carlo (MCMC).
  • **Conjugate Priors:** Using a conjugate prior simplifies calculations. A conjugate prior is a prior distribution that, when combined with the likelihood, results in a posterior distribution of the same family. For example, the Beta distribution is a conjugate prior for the Bernoulli distribution.

Several common probability distributions are frequently used in Bayesian analysis:

  • **Beta Distribution:** Used to model probabilities (e.g., the probability of a coin landing heads).
  • **Normal Distribution:** Used to model continuous variables (e.g., stock prices). Understanding the Normal Distribution is fundamental.
  • **Gamma Distribution:** Used to model waiting times or positive continuous variables.
  • **Poisson Distribution:** Used to model the number of events occurring in a fixed interval of time or space.

Bayesian Networks

Bayesian Networks (also known as belief networks or directed acyclic graphical models) are a powerful tool for representing and reasoning about probabilistic relationships between variables. They are particularly useful in complex trading scenarios where multiple factors influence price movements. For example, a Bayesian network could model the relationships between economic indicators, news sentiment, trading volume, and price changes.

Frequentist vs. Bayesian Statistics – Key Differences

| Feature | Frequentist Statistics | Bayesian Statistics | |---|---|---| | **Focus** | Frequency of events | Updating beliefs | | **Probability** | Long-run frequency | Degree of belief | | **Parameters** | Fixed, unknown constants | Random variables | | **Inference** | Hypothesis testing, p-values | Posterior distributions, credible intervals | | **Prior Knowledge** | Generally ignored | Explicitly incorporated | | **Interpretation** | "If we repeat this experiment many times,..." | "Given the evidence, how likely is this hypothesis?" |

Resources for Further Learning



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