Detection theory

Conceptually, sensitivity refers to how hard or easy it is to detect that a target stimulus is present from background events. For example, in a recognition memory paradigm, having longer to study to-be-remembered words makes it easier to recognize previously seen or heard words. In contrast, having to remember 30 words rather than 5 makes the discrimination harder. One of the most commonly used statistics for computing sensitivity is the so-called sensitivity index or d'. There are also non-parametric measures, such as the area under the ROC-curve.[6]

Bias is the extent to which one response is more probable than another, averaging across stimulus-present and stimulus-absent cases. That is, a receiver may be more likely overall to respond that a stimulus is present or more likely overall to respond that a stimulus is not present. Bias is independent of sensitivity. Bias can be desirable if false alarms and misses lead to different costs. For example, if the stimulus is a bomber, then a miss (failing to detect the bomber) may be more costly than a false alarm (reporting a bomber when there is not one), making a liberal response bias desirable. In contrast, giving false alarms too often (crying wolf) may make people less likely to respond, a problem that can be reduced by a conservative response bias.

Another field which is closely related to signal detection theory is called compressed sensing (or compressive sensing). The objective of compressed sensing is to recover high dimensional but with low complexity entities from only a few measurements. Thus, one of the most important applications of compressed sensing is in the recovery of high dimensional signals which are known to be sparse (or nearly sparse) with only a few linear measurements. The number of measurements needed in the recovery of signals is by far smaller than what Nyquist sampling theorem requires provided that the signal is sparse, meaning that it only contains a few non-zero elements. There are different methods of signal recovery in compressed sensing including basis pursuit , expander recovery algorithm[11], CoSaMP[12] and also fast non-iterative algorithm.[13] In all of the recovery methods mentioned above, choosing an appropriate measurement matrix using probabilistic constructions or deterministic constructions, is of great importance. In other words, measurement matrices must satisfy certain specific conditions such as RIP (Restricted Isometry Property) or Null-Space property in order to achieve robust sparse recovery.

In the case of making a decision between two hypotheses, H1, absent, and H2, present, in the event of a particular observation, y, a classical approach is to choose H1 when p(H1|y) > p(H2|y) and H2 in the reverse case.[14] In the event that the two a posteriori probabilities are equal, one might choose to default to a single choice (either always choose H1 or always choose H2), or might randomly select either H1 or H2. The a priori probabilities of H1 and H2 can guide this choice, e.g. by always choosing the hypothesis with the higher a priori probability.

When taking this approach, usually what one knows are the conditional probabilities, p(y|H1) and p(y|H2), and the a priori probabilities ${\displaystyle p(H1)=\pi _{1}}$ and ${\displaystyle p(H2)=\pi _{2}}$. In this case,

${\displaystyle p(H1|y)={\frac {p(y|H1)\cdot \pi _{1}}{p(y)}}}$,

${\displaystyle p(H2|y)={\frac {p(y|H2)\cdot \pi _{2}}{p(y)}}}$

where p(y) is the total probability of event y,

${\displaystyle p(y|H1)\cdot \pi _{1}+p(y|H2)\cdot \pi _{2}}$.

H2 is chosen in case

${\displaystyle {\frac {p(y|H2)\cdot \pi _{2}}{p(y|H1)\cdot \pi _{1}+p(y|H2)\cdot \pi _{2}}}\geq {\frac {p(y|H1)\cdot \pi _{1}}{p(y|H1)\cdot \pi _{1}+p(y|H2)\cdot \pi _{2}}}}$

${\displaystyle \Rightarrow {\frac {p(y|H2)}{p(y|H1)}}\geq {\frac {\pi _{1}}{\pi _{2}}}}$

and H1 otherwise.

Often, the ratio ${\displaystyle {\frac {\pi _{1}}{\pi _{2}}}}$ is called ${\displaystyle \tau _{MAP}}$ and ${\displaystyle {\frac {p(y|H2)}{p(y|H1)}}}$ is called ${\displaystyle L(y)}$, the likelihood ratio.

Using this terminology, H2 is chosen in case ${\displaystyle L(y)\geq \tau _{MAP}}$. This is called MAP testing, where MAP stands for "maximum a posteriori").

Taking this approach minimizes the expected number of errors one will make.

In some cases, it is far more important to respond appropriately to H1 than it is to respond appropriately to H2. For example, if an alarm goes off, indicating H1 (an incoming bomber is carrying a nuclear weapon), it is much more important to shoot down the bomber if H1 = TRUE, than it is to avoid sending a fighter squadron to inspect a false alarm (i.e., H1 = FALSE, H2 = TRUE) (assuming a large supply of fighter squadrons). The Bayes criterion is an approach suitable for such cases.[14]

Here a utility is associated with each of four situations:

• ${\displaystyle U_{11}}$: One responds with behavior appropriate to H1 and H1 is true: fighters destroy bomber, incurring fuel, maintenance, and weapons costs, take risk of some being shot down;
• ${\displaystyle U_{12}}$: One responds with behavior appropriate to H1 and H2 is true: fighters sent out, incurring fuel and maintenance costs, bomber location remains unknown;
• ${\displaystyle U_{21}}$: One responds with behavior appropriate to H2 and H1 is true: city destroyed;
• ${\displaystyle U_{22}}$: One responds with behavior appropriate to H2 and H2 is true: fighters stay home, bomber location remains unknown;

As is shown below, what is important are the differences, ${\displaystyle U_{11}-U_{21}}$ and ${\displaystyle U_{22}-U_{12}}$.

Similarly, there are four probabilities, ${\displaystyle P_{11}}$, ${\displaystyle P_{12}}$, etc., for each of the cases (which are dependent on one's decision strategy).

The Bayes criterion approach is to maximize the expected utility:

${\displaystyle E\{U\}=P_{11}\cdot U_{11}+P_{21}\cdot U_{21}+P_{12}\cdot U_{12}+P_{22}\cdot U_{22}}$

${\displaystyle E\{U\}=P_{11}\cdot U_{11}+(1-P_{11})\cdot U_{21}+P_{12}\cdot U_{12}+(1-P_{12})\cdot U_{22}}$

${\displaystyle E\{U\}=U_{21}+U_{22}+P_{11}\cdot (U_{11}-U_{21})-P_{12}\cdot (U_{22}-U_{12})}$

Effectively, one may maximize the sum,

${\displaystyle U'=P_{11}\cdot (U_{11}-U_{21})-P_{12}\cdot (U_{22}-U_{12})}$,

and make the following substitutions:

${\displaystyle P_{11}=\pi _{1}\cdot \int _{R_{1}}p(y|H1)\,dy}$

${\displaystyle P_{12}=\pi _{2}\cdot \int _{R_{1}}p(y|H2)\,dy}$

where ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$ are the a priori probabilities, ${\displaystyle P(H1)}$ and ${\displaystyle P(H2)}$, and ${\displaystyle R_{1}}$ is the region of observation events, y, that are responded to as though H1 is true.

${\displaystyle \Rightarrow U'=\int _{R_{1}}\left\{\pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)-\pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)\right\}\,dy}$

${\displaystyle U'}$ and thus ${\displaystyle U}$ are maximized by extending ${\displaystyle R_{1}}$ over the region where

${\displaystyle \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)-\pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)>0}$

This is accomplished by deciding H2 in case

${\displaystyle \pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)\geq \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)}$

${\displaystyle \Rightarrow L(y)\equiv {\frac {p(y|H2)}{p(y|H1)}}\geq {\frac {\pi _{1}\cdot (U_{11}-U_{21})}{\pi _{2}\cdot (U_{22}-U_{12})}}\equiv \tau _{B}}$

and H1 otherwise, where L(y) is the so-defined likelihood ratio.

Das and Geisler [15] extended the results of signal detection theory for normally distributed stimuli, and derived methods of computing the error rate and confusion matrix for ideal observers and non-ideal observers for detecting and categorizing univariate and multivariate normal signals from two or more categories.