Statistical inference

Statistical inference is a branch of statistics that involves drawing conclusions about a population based on a sample of data taken from that population. It provides the framework for estimating population parameters, testing hypotheses, and making predictions based on sample data. The primary goal of statistical inference is to infer properties about a population when it is impractical or impossible to collect data from every member of that population.

Bayesian inference is a statistical method that applies Bayes' theorem to update the probability of a hypothesis based on new evidence or data. It is grounded in the principles of Bayesian statistics, which interpret probability as a measure of belief or certainty rather than a frequency of occurrence. ### Key Components: 1. **Prior Probability (Prior):** This is the initial belief about a hypothesis before observing any data. It reflects the information or assumptions we have prior to the analysis.

Statistical forecasting is a method that uses historical data and statistical theories to predict future values or trends. It employs various statistical techniques and models to analyze past data patterns, relationships, and trends to make informed predictions. The core idea is to identify and quantify the relationships between different variables, typically focusing on time series data, which involves observations collected at regular intervals over time.

Data transformation in statistics refers to the process of converting data from one format or structure into another to facilitate analysis, improve interpretability, or meet the assumptions of statistical models. This can involve a variety of techniques and methods, depending on the objectives of the analysis and the nature of the data involved.

The empirical characteristic function (ECF) is a statistical tool used in the analysis of random variables and processes. It is a nonparametric estimator of the characteristic function of a distribution based on a sample of observations. The characteristic function itself is a complex-valued function that provides useful information about a probability distribution, such as the moments and the behavior of sums of random variables.

Exact statistics typically refers to methods in statistical analysis that provide precise probabilities or exact solutions to statistical problems, often under specific conditions or constraints. This can involve the use of parametric or non-parametric methods that offer exact results rather than approximate or asymptotic solutions. Here are a few examples where the term "exact statistics" might be applicable: 1. **Exact Tests**: These are statistical tests that yield an exact p-value based on the distribution of the test statistic under the null hypothesis.

Fiducial inference is a statistical framework developed by the mathematician Ronald A. Fisher in the early 20th century. It is intended for making inferences about parameters of a statistical model based on observed data without relying on the subjective probabilities associated with prior distributions, which are common in Bayesian statistics.

Frequentist inference is a framework for statistical analysis that relies on the concept of long-run frequencies of events to draw conclusions about populations based on sample data. In this approach, probability is interpreted as the limit of the relative frequency of an event occurring in a large number of trials. Here are some key characteristics and concepts associated with frequentist inference: 1. **Parameter Estimation**: Frequentist methods often involve estimating parameters (such as means or proportions) of a population from sample data.

Group size measures refer to the quantification and analysis of the size of a group in various contexts, such as social sciences, psychology, biology, and organizational studies. The concept can encompass different metrics and statistics to evaluate the number of individuals within a group and how that affects interactions, behavior, dynamics, and outcomes.

Informal inferential reasoning refers to the process of drawing conclusions or making inferences based on observations and experiences without employing formal statistical methods or rigorous logical arguments. This type of reasoning relies on informal logic, personal judgments, and anecdotal evidence rather than structured data analysis or established scientific principles. Key characteristics of informal inferential reasoning include: 1. **Contextual Understanding**: It takes into account the context in which observations are made.

Inverse probability, often referred to in the context of Bayesian probability, is the process of determining the probability of a hypothesis given observed evidence. In other words, it involves updating the probability of a certain event or hypothesis in light of new data or observations. This concept contrasts with "forward probability," where one would calculate the likelihood of observing evidence given a certain hypothesis.

Nonparametric statistics refers to a branch of statistics that does not assume a specific distribution for the population from which the samples are drawn. Unlike parametric methods, which rely on assumptions about the parameters (such as mean and variance) of a population's distribution (often assuming a normal distribution), nonparametric methods are more flexible as they can be used with data that do not meet these assumptions.

Parametric statistics refers to a category of statistical techniques that make specific assumptions about the parameters of the population distribution from which samples are drawn. These techniques typically assume that the data follows a certain distribution, most commonly the normal distribution. Key features of parametric statistics include: 1. **Assumptions**: Parametric tests often assume that the data is normally distributed, that variances are equal across groups (homogeneity of variance), and that the observations are independent.

Pseudolikelihood is a statistical technique used in the context of estimating parameters for models where traditional likelihood methods may be computationally intractable or where the full likelihood is difficult to specify. It is particularly useful in cases involving complex dependencies among multiple variables, such as in spatial statistics, graphical models, and certain machine learning applications. The idea behind pseudolikelihood is to approximate the full likelihood of a joint distribution by breaking it down into a product of conditional likelihoods.

A randomised decision rule (also known as a randomized algorithm) is a decision-making framework or mathematical approach that incorporates randomness into its process. It involves making decisions based on probabilistic methods rather than deterministic ones. This can add flexibility, enhance performance, or help manage uncertainty in various contexts. **Key Characteristics of Randomised Decision Rules:** 1. **Randomness:** The decision rule involves an element of randomness where the outcome is not solely determined by the input data.

Resampling in statistics refers to a collection of methods for repeatedly drawing samples from observed data or a statistical model. The main purpose of resampling techniques is to estimate the distribution of a statistic and to validate models or hypotheses when traditional parametric assumptions may not hold. Resampling is particularly useful in situations where the sample size is small or the underlying distribution is unknown.

Rodger's method, often referred to in the context of statistics and research methodology, is not a widely recognized or standard term. However, it could refer to various methods or techniques depending on context.

A sampling distribution is a probability distribution of a statistic (such as the sample mean, sample proportion, or sample variance) obtained from a large number of samples drawn from a specific population. In essence, it shows how a statistic would vary from sample to sample if you were to take repeated samples from the same population.

The "Sunrise problem" typically refers to a problem in the field of optimization, particularly in the context of scheduling and resource management, although the term might also appear in various contexts. One interpretation of the "Sunrise problem" is related to determining the optimal way to schedule tasks or activities based on the availability of daylight. This involves maximizing the use of daylight hours (i.e., the time from sunrise to sunset) to perform certain tasks.

The Transferable Belief Model (TBM) is a theory in the field of evidence theory, particularly dealing with the representation and management of uncertain information. It was introduced by Philippe Smets in the context of artificial intelligence and decision-making. ### Overview of the Transferable Belief Model: 1. **Foundation on Belief Functions**: The TBM is based on belief functions, which provide a framework for managing uncertainty.

In statistics, a "well-behaved" statistic generally refers to a statistic that has desirable properties such as consistency, unbiasedness, efficiency, and robustness. These properties make the statistic reliable for inference and analysis. Here are some aspects that typically characterize a well-behaved statistic: 1. **Unbiasedness**: A statistic is considered unbiased if its expected value is equal to the parameter it is estimating, meaning that on average, it hits the true value.