Probability vs Likelihood: Key Differences in Statistics

A parameter is a constant characteristic of a population. For the normal distribution, the mean (mu) and standard deviation (sigma) are parameters that determine the shape of the distribution. These parameters come from data - with enough observations, you can estimate mu and sigma. The shape of the distribution is determined by the mathematical function and its parameters.

In the normal distribution, the mean parameter determines the center or location of the distribution along the x-axis. Increasing the mean shifts the entire probability density function to the right, while decreasing it shifts it to the left. The standard deviation parameter controls the spread or width of the distribution; a larger standard deviation results in a wider, flatter curve, while a smaller standard deviation produces a narrower, taller curve. These two parameters completely define the shape of the normal distribution.

Parameters define the mathematical model used as the probability distribution. In the case of a normal distribution, the shape and position are determined by two parameters: the population mean (which determines the center) and the population standard deviation (which determines the spread). Once these parameters are set, the normal distribution curve is fully defined with that specific mean and standard deviation.

The normal distribution is completely determined by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the location parameter - it determines where the center of the distribution is located on the number line. The standard deviation represents the shape parameter - it determines how spread out or peaked the distribution is. A smaller standard deviation results in a taller, narrower curve, while a larger standard deviation produces a shorter, wider curve.

Probability distributions require specific parameters (external values) to define them: (1) Normal distribution requires two parameters: the mean (μ) and the variance (σ², which is the square of the standard deviation), (2) Binomial distribution requires two parameters: the number of trials (n) and the probability of success (p), (3) Poisson distribution requires one parameter: the mean (λ or μ), which represents the average number of events occurring in a fixed interval. These parameters define the shape and characteristics of each distribution.

Statistical data describes a sample (part of the population), while parameter data describes the entire population. Statistical data uses words like 'sample,' 'some,' or 'part of,' while parameter data uses words like 'all,' 'entire,' or 'total.' Examples: 'The average weight of 200 salmon in a river is 60 cm' is statistical (sample of salmon). 'The average age of all marathon participants is 35 years' is a parameter (all participants). '53% of voters participated in the last municipal election' is a parameter (all voters). 'The average spending of all customers at a restaurant last month' is a parameter (all customers). The distinction depends on whether the data describes the entire population or just a portion of it.

Data (данные) refers to the actual observed values obtained when random variables are measured in a specific experiment. Each random variable Xᵢ (capital X) represents a potential value, while xᵢ (lowercase x) represents the actual observed value. A parameter (параметр) is a mathematical property of the population that is well-defined but unknown, typically denoted by Greek letters like θ. Examples include average height of all people on Earth or proportion of defective items. Since populations are often too large to analyze completely, we cannot determine parameters exactly but can estimate them using random samples.

Statistics encompasses methods for planning experiments, obtaining data, and organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions. Data are observations from various sources, distinguished by whether they come from a population (complete collection of all elements to be studied) or a sample (subset of the population). A census collects data from every population member, while sampling collects from a subset. A parameter is a numerical measurement describing a population characteristic, while a statistic describes a sample characteristic. The key distinction is that parameters relate to populations while statistics relate to samples.

Statistical analysis depends on whether data represents a population or a sample. A parameter is any value describing a population characteristic, represented by Greek letters like μ for population mean. A statistic describes a sample characteristic, represented by Latin letters like x̄ for sample mean. The same data (10 pages of typing errors) can be either a population or sample depending on context. When it's a population, values like mean=1.5 are parameters; when it's a sample from a larger document, they become statistics. Parameters are constants for a given population, while statistics vary across different samples.

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions. Data comes from observations, measurements, responses, and counts. A population is the entire group of interest (e.g., all adults in the United States), while a sample is a subset representing the population. Parameters describe populations (e.g., population mean), while statistics describe samples (e.g., sample mean). To distinguish between them, identify whether data comes from the entire population or a sample. For example, surveying all 37 teachers in a school yields a parameter, while surveying 1,780 adults yields a statistic.

The hypothesis output represents P(y = 1 | x, θ), which is read as 'the probability that Y equals 1 given X, parameterized by theta.' This notation indicates that the probability depends on the input features X and the learned parameters θ, and it estimates the likelihood of the positive class for a given input.

In practice, true probabilities are unknown and must be estimated through modeling. Notation conventions distinguish between true probabilities (P(X)) and model estimates (P(X|θ) or P(X;θ)), where θ represents model parameters or assumptions. The model P(X|θ) represents the estimated probability based on the model's assumptions, while P(X) would represent the true underlying probability. This distinction is crucial because real-world language modeling always involves making assumptions and building models to approximate true probabilities.

The conditional probability notation P(A|B) represents the probability of event A given that event B has already occurred. This can be remembered through deduction since if the notation is not for union (∪) or intersection (∩), it must represent 'given'. Alternatively, using a visual memory aid: imagine being a butterfly on one side of a wall - you can observe B but cannot see over the wall to observe A, representing the concept of observing A when B is already known.

Conditional probability notation P(X=x|condition) means the probability of X taking value x given a specific condition. For example, P(X=2|at least 2) means the probability that X equals 2 given that X is at least 2. This is calculated as P(X=2 and X≥2) / P(X≥2) = P(X=2) / P(X≥2).

When an event is known to have occurred, the probability model must be revised to reflect this new information. If event B has occurred, the new model should only include elements within B, with elements outside B assigned zero probability. Conditional probability is denoted as P(A|B), read as 'the probability of A given B'. This notation represents the probability of event A occurring under the condition that event B has already occurred. The vertical bar '|' indicates the condition or evidence provided.