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Applied mathematics develops and uses mathematical models, analysis and computation to investigate problems in science, engineering and other fields. Choosing assumptions, deriving consequences and comparing predictions with observations are connected parts of the work. The Society for Industrial and Applied Mathematics describes the field through this cooperation between mathematical methods and practical scientific problems.^[ref]

For a worked example, a uniform-temperature cooling model assumeswhere $T_a$ is a constant ambient temperature. This is Newton's law of cooling. Setting $u=T-T_a$ gives $u^{\prime }=-ku$, so exponential decay yieldsWith illustrative values $T_0=100{}^{\circ }\mathrm{C}$, $T_a=20{}^{\circ }\mathrm{C}$ and $k=0.1{\min }^{-1}$, the predicted temperature after ten minutes is $20+80/e\approx 49.4{}^{\circ }\mathrm{C}$.

Solving the ordinary differential equation establishes a consequence of the assumptions. Testing the cooling prediction asks whether those assumptions fit the object and conditions. Parameter estimation uses measurements to infer $k$; disagreement can also reveal missing spatial variation or changing surroundings. This separates a correct mathematical derivation from the empirical adequacy of its mathematical model.

A computational model of cognition is an explicit set of mathematical or algorithmic rules connecting inputs and previous experience to predictions about behaviour. Its variables can represent internal quantities, while its predictions concern observable choices, reaction times or other measurements. This makes a proposed cognitive mechanism precise enough to simulate and test.^[ref]

A simple reinforcement learning example updates a predicted value $Q_t$ after a reward $r_t$:The learning rate $\alpha$ controls the contribution of the reward prediction error $r_t-Q_t$. With $Q_0=0$, constant reward $r_t=1$ and $\alpha =1/2$, successive values are $1/2$, $3/4$ and $7/8$. Equivalently, $Q_t=1-2^{-t}$. These are model predictions for constructed inputs, not measured human responses.^[ref]

Predicting choices additionally requires a rule linking values to actions. Parameter estimation can fit such a model, but a good fit alone does not establish that its internal variables are uniquely recoverable. Parameter recovery tests this by simulating data from known parameters and checking whether the fitting procedure retrieves them.^[ref] This distinguishes a useful formal account from a merely suggestive verbal description.

Model recovery tests whether a model selection procedure can identify the known model that generated simulated data. Generate observations from each candidate statistical model, fit the candidates using the intended pipeline, and record which candidate is selected. It is a check on the comparison procedure under stated simulation conditions.^[ref]

This differs from parameter recovery, which compares estimated parameters with the settings used to generate data within a model. Correctly identifying a model does not automatically imply accurate parameter estimation, and accurate fitting within one model does not establish that a procedure can distinguish it from its competitors.

For an invented example, suppose one hundred simulated datasets come from each of models A and B. The resulting counts areThis confusion matrix estimates selection behavior conditional on the generating model. Its empirical rate of choosing B when A generated is $20/100=0.20$.

The reverse question has a different denominator. Among the $85$ datasets for which B was selected, $65$ were generated by B, giving $65/85\simeq 0.765$. That calculation uses the example's balanced simulation design. It is not interchangeable with the $0.65$ recovery rate for B-generated datasets. Different generating-model frequencies can change the reverse proportion even if the conditional selection behavior stays fixed.^[ref]

Recovery depends on parameter ranges, observation count, noise and the selection criterion. Models with nearly indistinguishable predictions can be hard to recover despite successful numerical optimization. Repeat simulations in the regimes relevant to the planned inference. Finally, recovery among a chosen candidate set does not prove that the real system belongs to that set; checking how well the selected model reproduces relevant observations remains a separate task.^[ref]

Parameter estimation uses observations to infer the unknown parameters of a mathematical model. The result depends on both the observations and the model: one must specify which quantities are unknown, how observations relate to them, and how discrepancies are assessed. Least squares and maximum likelihood estimation provide two common criteria.^[ref]

For an explicit example, suppose $y_i=\theta +{\epsilon }_i$, where the same unknown level $\theta$ applies to all observations. MinimizinggivesSince $L^{\prime \prime }(\theta )=2n>0$, this is the unique minimum for $n>0$. Observations $2,4,6$ give the sample mean $\hat{\theta }=4$, with residuals $-2,0,2$. Independent normal distributions with a common variance also make this estimate the maximum likelihood estimation solution for the level.

An estimate is not the same as a known exact value. Under the independent equal-variance model, its standard error is estimated by $s/\sqrt{n}$, where $s^2$ is the sample variance. For the three illustrative observations, $s^2=4$ and the estimated standard error is $2/\sqrt{3}$. Correlated errors or a common measurement offset require different accounting; collecting more identically biased readings does not remove the offset.

Identifiability is a separate prerequisite. If a model depends only on $a+b$, data of that form can estimate their sum but cannot uniquely separate $a$ from $b$. Pairs $(1,3)$ and $(2,2)$ give the same sum. More repeated observations of that same relationship reduce uncertainty about the sum without identifying both parameters. State what the observations can determine before reporting numerical estimates.