Bayesian Estimation Fundamentals

4.2.5 Bayesian Estimation Fundamentals

Bayesian estimation provides the probabilistic foundation of modern spacecraft state estimation and sensor fusion. The observer-based methods introduced in the previous section estimate the spacecraft state by combining a dynamic model with measurement residuals. Bayesian estimation extends this idea by explicitly representing uncertainty.

Instead of asking only:

\[\boxed{\text{What is the spacecraft state?}}\]

Bayesian estimation asks:

\[\boxed{\text{What states are possible, and how probable is each one given all available information?}}\]

This distinction is fundamental.

A deterministic estimator may produce

\[\hat{\mathbf x}\]

as its best estimate. A Bayesian estimator represents knowledge about the unknown state through a probability distribution:

\[\boxed{p(\mathbf x \mid \mathbf z)}\]

where \mathbf z represents the available measurements.

The complete idea can be summarized as

\[\boxed{ \text{Prior Knowledge} + \text{New Sensor Measurement} \rightarrow \text{Updated Knowledge} }\]

or, mathematically,

\[\boxed{ \text{Prior} \times \text{Likelihood} \rightarrow \text{Posterior} }\]

after appropriate normalization.

This framework is the foundation for the Kalman Filter, Extended Kalman Filter (EKF), Error-State Kalman Filter (ESKF), Multiplicative EKF (MEKF), Unscented Kalman Filter (UKF), Particle Filter, and many other spacecraft navigation algorithms.

4.2.5.1 Why Probability Is Needed in Spacecraft State Estimation

A spacecraft navigation system never knows the exact physical state.

Suppose the state is

\[\mathbf x= \begin{bmatrix} \mathbf r\\ \mathbf v\\ \mathbf q\\ \boldsymbol\omega\\ \mathbf b_g\\ \mathbf b_a\\ \cdots \end{bmatrix}.\]

The flight computer may need estimates of:

• position \mathbf r,
• velocity \mathbf v,
• attitude \mathbf q,
• angular velocity \boldsymbol\omega,
• gyroscope bias \mathbf b_g,
• accelerometer bias \mathbf b_a,
• relative position and velocity during RPO,
• sensor alignment errors,
• clock states,
• disturbance parameters.

But every source of information is imperfect.

The dynamic model contains uncertainty:

\[\boxed{ \mathbf x_{k+1} = f(\mathbf x_k,\mathbf u_k) + \mathbf w_k }\]

and sensors contain uncertainty:

\[\boxed{ \mathbf z_k = h(\mathbf x_k) + \mathbf v_k }\]

where

\[\mathbf w_k\]

represents process uncertainty and

\[\mathbf v_k\]

represents measurement uncertainty.

Therefore, the navigation problem is inherently uncertain.

The estimator should not simply say

\[\mathbf x= \hat{\mathbf x}.\]

A more meaningful statement is:

\[\boxed{ \text{The state is believed to lie around } \hat{\mathbf x} \text{ with a certain uncertainty.} }\]

That is the motivation for probabilistic estimation.

4.2.5.2 Deterministic State vs Probabilistic State

Suppose a spacecraft is believed to be at

\[x=100\text{ m}.\]

A deterministic description would simply state

\[\boxed{x=100\text{ m}}.\]

But this does not indicate whether the estimate is accurate to:

\[\pm0.01\text{ m},\]

\[\pm1\text{ m},\]

or

\[\pm100\text{ m}.\]

These estimates clearly have very different navigation significance.

A probabilistic description might instead be

\[\boxed{ x\sim\mathcal N(100,\;2^2) }\]

meaning that position is represented by a Gaussian distribution with mean

\[\mu=100\text{ m}\]

and standard deviation

\[\sigma=2\text{ m}.\]

Now the estimate contains two kinds of information:

\[\boxed{\text{Estimated State}}\]

and

\[\boxed{\text{Uncertainty of the Estimated State}}.\]

This distinction becomes essential during operations such as rendezvous, proximity operations, docking, landing, and autonomous navigation.

4.2.5.3 Random Variables

Bayesian estimation treats unknown quantities as random variables.

A random variable does not mean that the physical spacecraft state itself behaves randomly in every sense.

Rather, it represents our uncertainty or incomplete knowledge about that state.

For example, let

\[X\]

represent an uncertain spacecraft position.

Before receiving a measurement, several values of X may be plausible.

The probability distribution of X describes how strongly we believe in those possibilities.

For a state vector,

\[\mathbf X= \begin{bmatrix} X_1\\ X_2\\ \vdots\\ X_n \end{bmatrix},\]

the joint probability distribution describes uncertainty in all state components and their statistical relationships.

This is particularly important because spacecraft navigation errors are often correlated.

For example, uncertainty in position may be correlated with uncertainty in velocity.

4.2.5.4 Probability Distribution

A probability distribution describes the relative plausibility of possible values of an uncertain variable.

For a continuous random variable X, we use a probability density function:

\[\boxed{p(x)}.\]

The probability that X lies between a and b is

\[\boxed{ P(a\le X\le b) = \int_a^b p(x)\,dx }\]

and the total probability satisfies

\[\boxed{ \int_{-\infty}^{\infty}p(x)\,dx=1. }\]

The density must satisfy

\[\boxed{p(x)\ge0}.\]

For a multidimensional spacecraft state,

\[\boxed{p(\mathbf x)}\]

describes uncertainty throughout the state space.

4.2.5.5 Mean — The Expected State

One important property of a probability distribution is its mean or expected value.

For a scalar random variable,

\[\boxed{ \mu = E[X] = \int x\,p(x)\,dx }.\]

For a vector state,

\[\boxed{ \boldsymbol\mu = E[\mathbf X]. }\]

In many state estimators, this mean becomes the reported state estimate:

\[\boxed{ \hat{\mathbf x} = E[\mathbf X]. }\]

However, the mean alone is not sufficient.

We must also describe uncertainty around it.

4.2.5.6 Variance

For a scalar variable,

\[\boxed{ \sigma^2 = E[(X-\mu)^2] }\]

is the variance.

The standard deviation is

\[\boxed{ \sigma=\sqrt{\sigma^2}. }\]

Small variance means the probability distribution is concentrated near the mean.

Large variance means there is greater uncertainty.

Conceptually:

\[\boxed{ \text{Small Variance} \Rightarrow \text{High Confidence} }\]

and

\[\boxed{ \text{Large Variance} \Rightarrow \text{Low Confidence}. }\]

4.2.5.7 Covariance Matrix

A spacecraft state normally contains many variables, so uncertainty is represented using a covariance matrix.

For state

\[\mathbf x\]

with estimate

\[\hat{\mathbf x},\]

define estimation error

\[\boxed{ \mathbf e = \mathbf x-\hat{\mathbf x}. }\]

The covariance matrix is

\[\boxed{ \mathbf P = E[ \mathbf e\mathbf e^T ] }.\]

For a simple two-state system,

\[\mathbf x= \begin{bmatrix} x\\ v \end{bmatrix},\]

the covariance is

\[\boxed{ \mathbf P = \begin{bmatrix} \sigma_x^2 & \operatorname{cov}(x,v)\\ \operatorname{cov}(v,x) & \sigma_v^2 \end{bmatrix} }.\]

The diagonal terms describe individual state uncertainties.

The off-diagonal terms describe correlations between estimation errors.

Therefore,

\[\boxed{ \mathbf P \neq \text{just a collection of variances}. }\]

It also contains information about how state uncertainties are coupled.

This becomes extremely important in Kalman filtering.

4.2.5.8 Probability Ellipse and Ellipsoid

For multiple states, covariance has a geometric interpretation.

For two states, uncertainty may be represented by an ellipse:

\[\boxed{ (\mathbf x-\boldsymbol\mu)^T \mathbf P^{-1} (\mathbf x-\boldsymbol\mu) = c }\]

for some selected confidence contour c.

In three dimensions this becomes an uncertainty ellipsoid.

The orientation of the ellipse or ellipsoid is determined by covariance between state components.

This has direct physical significance in orbital and relative navigation.

For example, RPO relative-position uncertainty may be larger along one direction than another:

\[\sigma_x\neq\sigma_y\neq\sigma_z.\]

Thus a single scalar "navigation accuracy" cannot always describe the true uncertainty geometry.

4.2.5.9 Joint Probability

Suppose we have two uncertain quantities:

\[X\]

and

\[Z.\]

Their joint probability distribution is

\[\boxed{ p(x,z) }.\]

It describes the probability structure of the two variables together.

Joint probability is fundamental because Bayesian estimation concerns relationships between:

\[\boxed{\text{State}}\]

and

\[\boxed{\text{Measurement}}.\]

The estimator needs to reason about how likely a particular measurement is for a particular spacecraft state.

4.2.5.10 Marginal Probability

Suppose we know the joint distribution

\[p(x,z).\]

The probability distribution of x alone is obtained by integrating over all possible measurements:

\[\boxed{ p(x) = \int p(x,z)\,dz }.\]

Similarly,

\[\boxed{ p(z) = \int p(x,z)\,dx }.\]

These are called marginal distributions.

Marginalization becomes particularly important in advanced Bayesian estimators when nuisance variables or hidden states are removed from a probability description.

4.2.5.11 Conditional Probability

Conditional probability asks:

What is the probability of one quantity when another quantity is known?

For continuous variables,

\[\boxed{ p(x|z) = \frac{p(x,z)}{p(z)} }.\]

In spacecraft navigation,

\[\boxed{ p(\mathbf x|\mathbf z) }\]

means:

What is our probability distribution for the spacecraft state given the sensor measurement?

This is exactly the quantity a Bayesian state estimator seeks.

4.2.5.12 Bayes' Theorem

Bayes' theorem follows from conditional probability:

\[\boxed{ p(\mathbf x|\mathbf z) = \frac{ p(\mathbf z|\mathbf x) p(\mathbf x) }{ p(\mathbf z) } }.\]

This is the central equation of Bayesian estimation.

Each term has a specific meaning.

4.2.5.12.1 Prior

\[\boxed{p(\mathbf x)}\]

represents what we believe about the spacecraft state before processing the new measurement.

4.2.5.12.2 Likelihood

\[\boxed{p(\mathbf z|\mathbf x)}\]

represents how likely the observed sensor measurement is if the spacecraft were in state \mathbf x.

4.2.5.12.3 Evidence

\[\boxed{p(\mathbf z)}\]

is the total probability of observing the measurement.

4.2.5.12.4 Posterior

\[\boxed{p(\mathbf x|\mathbf z)}\]

represents our updated belief about the state after incorporating the measurement.

Therefore:

\[\boxed{ \text{Posterior} = \frac{ \text{Likelihood}\times\text{Prior} }{ \text{Evidence} } }.\]

Or conceptually:

\[\boxed{ \text{What We Believed} + \text{What the Sensor Told Us} \rightarrow \text{What We Believe Now} }.\]

4.2.5.13 Why the Evidence Term Is Needed

The denominator is

\[p(\mathbf z).\]

It can be obtained by marginalization:

\[\boxed{ p(\mathbf z) = \int p(\mathbf z|\mathbf x) p(\mathbf x) \,d\mathbf x }.\]

Its role is to normalize the posterior so that

\[\boxed{ \int p(\mathbf x|\mathbf z)d\mathbf x=1. }\]

For many estimation derivations we write

\[\boxed{ p(\mathbf x|\mathbf z) \propto p(\mathbf z|\mathbf x)p(\mathbf x) }\]

where

\[\propto\]

means "proportional to."

The normalization can then be calculated afterward.

4.2.5.14 Understanding the Prior

Suppose a spacecraft estimator predicts

\[x=100\text{ m}\]

with uncertainty

\[\sigma=5\text{ m}.\]

The prior may be

\[\boxed{ p(x) = \mathcal N(100,5^2) }.\]

This prior is not arbitrary.

In recursive navigation it normally comes from the previous state estimate propagated through the spacecraft dynamics.

Therefore:

\[\boxed{ \text{Previous Posterior} \rightarrow \text{Dynamics Propagation} \rightarrow \text{New Prior}. }\]

This is the first half of recursive Bayesian estimation.

4.2.5.15 Understanding the Likelihood

Suppose a sensor reports

\[z=104\text{ m}\]

with measurement noise standard deviation

\[\sigma_z=2\text{ m}.\]

The measurement model may be

\[z=x+v\]

where

\[v\sim\mathcal N(0,2^2).\]

Therefore,

\[\boxed{ p(z|x) = \mathcal N(x,2^2) }\]

when viewed as a distribution over possible measurements for a given state.

Once the actual measurement z=104 is observed, the same model defines a likelihood function over candidate values of x.

The likelihood answers:

If the spacecraft were actually at state x, how plausible would the observed measurement be?

This is different from the posterior.

A likelihood is not itself the probability distribution of the state.

4.2.5.16 Understanding the Posterior

The posterior combines the prior with the measurement information:

\[\boxed{ p(x|z) \propto p(z|x)p(x). }\]

If the prior strongly favours values near

\[100\text{ m}\]

and the sensor measurement favours values near

\[104\text{ m},\]

the posterior typically lies somewhere between them, depending on their uncertainties.

If the sensor is very accurate, the posterior shifts more toward the measurement.

If the sensor is noisy, the posterior remains closer to the prior.

Thus Bayesian fusion is fundamentally uncertainty-weighted information fusion.

4.2.5.17 Bayesian Fusion Is Not Simple Averaging

Suppose:

\[x_{prior}=100\text{ m}\]

with

\[\sigma_{prior}=1\text{ m},\]

while a noisy sensor reports

\[z=110\text{ m}\]

with

\[\sigma_z=20\text{ m}.\]

A simple average would give

\[105\text{ m}.\]

That would ignore the fact that the prior is much more precise than the measurement.

Bayesian estimation gives much more influence to the high-confidence information.

Therefore,

\[\boxed{ \text{Fusion} \neq \text{Arithmetic Averaging} }\]

but rather

\[\boxed{ \text{Fusion} = \text{Combination According to Information and Uncertainty}. }\]

This concept will later appear explicitly in the Kalman gain.

4.2.5.18 Gaussian Distribution

The Gaussian or normal distribution is central to many spacecraft estimators.

For a scalar variable,

\[\boxed{ p(x) = \frac{1} {\sqrt{2\pi\sigma^2}} \exp \left[ -\frac{(x-\mu)^2}{2\sigma^2} \right] }.\]

It is completely described by

\[\boxed{\mu}\]

and

\[\boxed{\sigma^2}.\]

For a vector state,

\[\boxed{ p(\mathbf x) = \frac{ 1 }{ (2\pi)^{n/2} |\mathbf P|^{1/2} } \exp \left[ -\frac12 (\mathbf x-\boldsymbol\mu)^T \mathbf P^{-1} (\mathbf x-\boldsymbol\mu) \right] }.\]

Therefore a multivariate Gaussian is represented compactly as

\[\boxed{ \mathbf x \sim \mathcal N( \boldsymbol\mu,\mathbf P ) }.\]

This compact representation is one major reason Gaussian estimators are so useful.

4.2.5.19 Why Gaussian Assumptions Are So Common

Gaussian distributions have several attractive properties.

They are completely described by a mean and covariance.

Linear transformations of Gaussian random variables remain Gaussian.

Products of Gaussian densities lead to Gaussian forms after normalization.

Many independent small disturbances approximately produce Gaussian behaviour through central-limit effects.

Therefore, under linear-Gaussian assumptions, recursive Bayesian estimation can be performed by propagating only:

\[\boxed{\hat{\mathbf x}}\]

and

\[\boxed{\mathbf P}.\]

This leads directly to the Kalman filter.

4.2.5.20 But Real Spacecraft Errors Are Not Always Gaussian

The Gaussian assumption is extremely useful, but it is not universally valid.

Non-Gaussian behaviour may arise from:

• sensor outliers,
• multipath,
• vision mismatches,
• ambiguous feature association,
• false target detection,
• sensor saturation,
• eclipse transitions,
• intermittent measurements,
• nonlinear geometry,
• multimodal pose ambiguity,
• actuator faults,
• large initial uncertainty.

For example, a vision system may produce two plausible relative-pose solutions.

Then the probability distribution may contain two peaks.

Such a distribution is multimodal and cannot be represented faithfully by one Gaussian mean and covariance.

This motivates more advanced approaches such as particle filters and Gaussian-mixture methods.

4.2.5.21 State Transition Probability

For a dynamic spacecraft, the state evolves over time.

Instead of representing the dynamics only as

\[\mathbf x_k=f(\mathbf x_{k-1},\mathbf u_{k-1}),\]

Bayesian estimation represents uncertain state evolution through a transition density:

\[\boxed{ p( \mathbf x_k | \mathbf x_{k-1}, \mathbf u_{k-1} ) }.\]

This describes the probability of the new state given the previous state and applied control.

If the model contains process noise,

\[\mathbf x_k = f( \mathbf x_{k-1}, \mathbf u_{k-1} ) + \mathbf w_{k-1},\]

then uncertainty in \mathbf w_{k-1} creates uncertainty in the propagated state.

4.2.5.22 Measurement Probability Model

Similarly, the measurement equation

\[\mathbf z_k = h(\mathbf x_k) + \mathbf v_k\]

corresponds to the likelihood

\[\boxed{ p(\mathbf z_k|\mathbf x_k) }.\]

This distribution answers:

If the spacecraft state were \mathbf x_k, what measurements could the sensor reasonably produce?

Every sensor therefore contributes not only a measurement function

\[h(\mathbf x)\]

but also an uncertainty model.

For spacecraft sensor fusion, these two pieces are inseparable:

\[\boxed{ \text{Sensor Model} = \text{Measurement Physics} + \text{Measurement Uncertainty}. }\]

4.2.5.23 Recursive Bayesian Estimation

Spacecraft navigation is sequential.

Measurements arrive continuously:

\[\mathbf z_1, \mathbf z_2, \mathbf z_3, \ldots\]

The estimator should not recompute the entire mission history from scratch every time a new measurement arrives.

Instead, it recursively updates the probability distribution.

Let

\[\mathbf z_{1:k} = \{\mathbf z_1,\ldots,\mathbf z_k\}.\]

The desired posterior is

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) }.\]

Recursive Bayesian estimation consists of two fundamental steps:

\[\boxed{\text{Prediction}}\]

and

\[\boxed{\text{Measurement Update}}.\]

This prediction–correction cycle is the probabilistic version of the observer structure introduced previously.

4.2.5.24 Bayesian Prediction Step

Suppose the previous posterior is

\[p( \mathbf x_{k-1} | \mathbf z_{1:k-1} ).\]

The spacecraft dynamics propagate this distribution forward.

The predicted distribution is

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k-1} ) = \int p( \mathbf x_k | \mathbf x_{k-1} ) p( \mathbf x_{k-1} | \mathbf z_{1:k-1} ) d\mathbf x_{k-1} }.\]

This is the Chapman–Kolmogorov prediction equation.

It combines:

\[\boxed{\text{Previous State Uncertainty}}\]

with

\[\boxed{\text{Dynamic Model Uncertainty}}.\]

The result is the prior distribution at time k.

Conceptually:

\[\boxed{\text{Previous Posterior}} \rightarrow \boxed{\text{Dynamics}} \rightarrow \boxed{\text{Predicted Prior}}.\]

4.2.5.25 Why Uncertainty Usually Grows During Prediction

Suppose no measurements are received for some time.

The estimator continues propagating the state using the model.

Because the model is imperfect and disturbances are uncertain, the possible state region normally expands.

Therefore:

\[\boxed{ \text{Propagation Without Measurement} \rightarrow \text{Increasing State Uncertainty} }.\]

In covariance-based estimators this appears as growth of

\[\mathbf P.\]

This explains an important navigation principle:

A state estimate can continue to exist during sensor dropout, but confidence in that estimate should generally decrease.

4.2.5.26 Bayesian Measurement Update

When measurement

\[\mathbf z_k\]

arrives, Bayes' theorem is applied:

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) = \frac{ p( \mathbf z_k | \mathbf x_k ) p( \mathbf x_k | \mathbf z_{1:k-1} ) }{ p( \mathbf z_k | \mathbf z_{1:k-1} ) } }.\]

The predicted distribution

\[p( \mathbf x_k | \mathbf z_{1:k-1} )\]

is the prior.

The term

\[p(\mathbf z_k|\mathbf x_k)\]

is the likelihood.

The result

\[p( \mathbf x_k | \mathbf z_{1:k} )\]

is the posterior.

Therefore:

\[\boxed{\text{Prior}} + \boxed{\text{New Measurement}} \rightarrow \boxed{\text{Posterior}}.\]

4.2.5.27 Complete Recursive Bayesian Cycle

The entire estimator can now be represented as

\[\boxed{ p( \mathbf x_{k-1} | \mathbf z_{1:k-1} ) }\]

\[\downarrow\]

\[\boxed{\text{Dynamic Prediction}}\]

\[\downarrow\]

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k-1} ) }\]

\[\downarrow\]

\[\boxed{ \text{New Measurement }\mathbf z_k }\]

\[\downarrow\]

\[\boxed{\text{Bayesian Update}}\]

\[\downarrow\]

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) }.\]

Then the posterior becomes the starting point for the next prediction.

Thus:

\[\boxed{ \text{Posterior}_{k-1} \rightarrow \text{Prior}_k \rightarrow \text{Posterior}_k \rightarrow \text{Prior}_{k+1} \rightarrow\cdots }\]

This is the heart of recursive state estimation.

4.2.5.28 Markov Assumption

Recursive Bayesian filtering usually relies on the Markov assumption:

\[\boxed{ p( \mathbf x_k | \mathbf x_{0:k-1} ) = p( \mathbf x_k | \mathbf x_{k-1} ) }.\]

This means:

Given the current state, the next state does not require the entire past state history.

Similarly, a common measurement assumption is

\[\boxed{ p( \mathbf z_k | \mathbf x_{0:k}, \mathbf z_{1:k-1} ) = p( \mathbf z_k | \mathbf x_k ) }.\]

Thus the current measurement depends on the current state once that state is known.

These assumptions allow recursive estimation to remain computationally manageable.

4.2.5.29 Process Noise

The spacecraft process model is commonly written as

\[\boxed{ \mathbf x_{k+1} = f( \mathbf x_k, \mathbf u_k ) + \mathbf w_k }.\]

The process noise

\[\mathbf w_k\]

represents uncertainty not captured perfectly by the deterministic model.

Possible sources include:

• uncertain disturbance forces,
• uncertain disturbance torques,
• aerodynamic uncertainty,
• solar radiation pressure,
• gravity-model errors,
• actuator uncertainty,
• thruster mismatch,
• unmodelled dynamics,
• slowly varying parameters,
• random-walk sensor biases.

For Gaussian estimators,

\[\boxed{ \mathbf w_k \sim \mathcal N( \mathbf 0,\mathbf Q_k ) }\]

where

\[\boxed{\mathbf Q_k}\]

is the process-noise covariance.

4.2.5.30 Measurement Noise

The sensor model is commonly written

\[\boxed{ \mathbf z_k = h(\mathbf x_k) + \mathbf v_k }.\]

For Gaussian measurement noise,

\[\boxed{ \mathbf v_k \sim \mathcal N( \mathbf 0,\mathbf R_k ) }\]

where

\[\boxed{\mathbf R_k}\]

is the measurement-noise covariance.

The matrix \mathbf R_k describes how uncertain the sensor measurement is.

For multiple measurement channels,

\[\mathbf R = \begin{bmatrix} \sigma_1^2 & \cdots\\ \vdots & \ddots \end{bmatrix}.\]

If measurement errors are correlated, off-diagonal terms may be non-zero.

4.2.5.31 Meaning of Q and R

This distinction is extremely important.

\[\boxed{ \mathbf Q = \text{Uncertainty in State Evolution / Process Model} }\]

while

\[\boxed{ \mathbf R = \text{Uncertainty in Sensor Measurements} }.\]

They are not interchangeable.

Large \mathbf Q means:

We have relatively low confidence in the state propagation model.

Large \mathbf R means:

We have relatively low confidence in the measurement.

This later determines how a Kalman filter balances prediction against measurement.

4.2.5.32 Independence Assumptions

Basic filtering derivations commonly assume

\[E[\mathbf w_k]=0,\]

\[E[\mathbf v_k]=0,\]

and often

\[E[ \mathbf w_k\mathbf v_j^T ] = 0.\]

Process and measurement noise are then treated as uncorrelated.

However, real systems may violate these assumptions.

For example, preprocessing the same IMU signal into several quantities can introduce correlated errors.

Shared clocks, common environmental disturbances, common reference models, or shared processing chains may also introduce correlation.

Ignoring significant correlations can make the estimator overconfident.

4.2.5.33 Information Content of a Measurement

Not every measurement provides equal information about every state.

Suppose a range sensor measures

\[\boxed{ \rho = \| \mathbf r_{rel} \| }.\]

This provides information about distance but not necessarily enough information to determine the complete three-dimensional relative-position vector from one measurement.

A camera line-of-sight measurement may provide angular information but weak direct range information.

Therefore:

\[\boxed{ \text{Measurement Availability} \neq \text{Complete State Observability}. }\]

Bayesian estimation quantifies uncertainty, but it does not eliminate fundamental observability limitations.

This connects Bayesian estimation back to the observability concepts discussed in observer-based fusion.

4.2.5.34 Measurement Geometry Matters

Suppose a camera provides bearing measurements:

\[\alpha,\beta.\]

The uncertainty in Cartesian position resulting from these angular measurements depends strongly on target distance and geometry.

At long range, a small angular uncertainty can correspond to a large lateral position uncertainty.

Similarly, relative range alone may leave cross-range position poorly constrained.

Therefore:

\[\boxed{ \text{Sensor Accuracy} + \text{Measurement Geometry} \rightarrow \text{State Information}. }\]

This becomes especially important in RPO navigation.

4.2.5.35 Multi-Sensor Bayesian Fusion

Suppose several conditionally independent sensors provide measurements:

\[\mathbf z_1,\mathbf z_2,\ldots,\mathbf z_m.\]

Then, under the conditional-independence assumption,

\[\boxed{ p( \mathbf z_1,\ldots,\mathbf z_m | \mathbf x ) = \prod_{i=1}^{m} p( \mathbf z_i|\mathbf x ) }.\]

The posterior becomes

\[\boxed{ p( \mathbf x | \mathbf z_1,\ldots,\mathbf z_m ) \propto p(\mathbf x) \prod_{i=1}^{m} p( \mathbf z_i|\mathbf x ) }.\]

This is the probabilistic foundation of multi-sensor fusion.

For example:

\[\boxed{\text{Gyro}} + \boxed{\text{Star Tracker}} + \boxed{\text{Sun Sensor}} + \boxed{\text{Magnetometer}}\]

can jointly constrain spacecraft attitude states.

Similarly,

\[\boxed{\text{GNSS}} + \boxed{\text{Camera}} + \boxed{\text{LiDAR}}\]

can jointly constrain relative-navigation states.

4.2.5.36 Why Sensor Independence Must Not Be Assumed Blindly

The previous product-of-likelihoods expression assumes that measurements are conditionally independent given the state.

This assumption may fail.

For example, two navigation products may both originate from the same raw GNSS measurements.

Treating them as independent would count the same information twice.

This can cause the estimator to become artificially confident.

Therefore:

\[\boxed{ \text{Correlated Information} + \text{Incorrect Independence Assumption} \rightarrow \text{Overconfidence} }.\]

This becomes important later in federated and decentralized sensor fusion.

4.2.5.37 Sequential Measurement Updates

If several independent measurements arrive, Bayesian updates can often be performed sequentially.

Start with

\[p(\mathbf x).\]

Process sensor 1:

\[p(\mathbf x|\mathbf z_1).\]

Then use this posterior as the prior for sensor 2:

\[p(\mathbf x|\mathbf z_1,\mathbf z_2).\]

Then sensor 3:

\[p(\mathbf x|\mathbf z_1,\mathbf z_2,\mathbf z_3).\]

Conceptually:

\[\boxed{\text{Prior}} \rightarrow \boxed{\text{Sensor 1}} \rightarrow \boxed{\text{Sensor 2}} \rightarrow \boxed{\text{Sensor 3}} \rightarrow \boxed{\text{Posterior}}.\]

This is useful for asynchronous spacecraft sensors.

4.2.5.38 Multi-Rate and Asynchronous Measurements

Spacecraft sensors do not normally update simultaneously.

For example:

\[f_{gyro} \gg f_{star}\]

and an RPO system may have different update rates for:

• GNSS,
• camera,
• LiDAR,
• IMU,
• star tracker.

Bayesian recursive estimation naturally supports this.

The estimator continuously predicts:

\[\boxed{ p(\mathbf x_k|\mathbf z_{1:k-1}) }\]

and performs a measurement update whenever a valid sensor measurement becomes available.

Therefore:

\[\boxed{ \text{Prediction at Navigation Rate} + \text{Updates at Sensor Arrival Times} }.\]

4.2.5.39 Sensor Dropout in Bayesian Estimation

If no measurement arrives, there is no Bayesian correction.

The estimator simply propagates:

\[\boxed{ \text{Posterior} \rightarrow \text{Prediction} \rightarrow \text{New Prior}. }\]

Because process uncertainty continues to accumulate,

\[\boxed{ \text{Sensor Dropout} \rightarrow \text{Growing State Uncertainty} }.\]

This is one major advantage of probabilistic estimation: the estimator can explicitly represent its decreasing confidence.

4.2.5.40 Outliers and Bayesian Interpretation

Suppose a sensor measurement is extremely inconsistent with the predicted state.

Then the likelihood

\[p(\mathbf z_k|\mathbf x_k)\]

may be very small around the predicted state.

This indicates a conflict between prior information and the new measurement.

Possible causes include:

• genuine unexpected spacecraft motion,
• model error,
• sensor failure,
• outlier,
• incorrect target association,
• timing error.

A robust navigation system should therefore not automatically accept every measurement.

This leads later to:

• innovation gating,
• residual testing,
• Mahalanobis distance,
• normalized innovation squared,
• robust likelihood models,
• fault detection and isolation.

4.2.5.41 Maximum Likelihood Estimation

Suppose we have a measurement \mathbf z and seek the state that makes that measurement most likely.

Then the maximum likelihood estimate is

\[\boxed{ \hat{\mathbf x}_{ML} = \arg\max_{\mathbf x} p(\mathbf z|\mathbf x) }.\]

Maximum likelihood uses the measurement likelihood but does not directly include a prior distribution.

Therefore it differs from Bayesian posterior estimation.

4.2.5.42 Maximum A Posteriori Estimation

Bayesian estimation can instead select the state that maximizes the posterior:

\[\boxed{ \hat{\mathbf x}_{MAP} = \arg\max_{\mathbf x} p(\mathbf x|\mathbf z) }.\]

Using Bayes' theorem,

\[\boxed{ \hat{\mathbf x}_{MAP} = \arg\max_{\mathbf x} p(\mathbf z|\mathbf x) p(\mathbf x) }.\]

Thus MAP estimation uses both:

\[\boxed{\text{Measurement Information}}\]

and

\[\boxed{\text{Prior Information}}.\]

If the prior is uniform over the relevant region, MAP and maximum-likelihood estimates may coincide.

4.2.5.43 Posterior Mean vs MAP

The posterior contains more information than a single estimate.

Different point estimates can be extracted from it.

The posterior mean is

\[\boxed{ \hat{\mathbf x}_{MMSE} = E[ \mathbf x|\mathbf z ] }\]

and minimizes mean-square estimation error under standard conditions.

The MAP estimate is

\[\boxed{ \hat{\mathbf x}_{MAP} = \arg\max p(\mathbf x|\mathbf z) }.\]

For a symmetric unimodal Gaussian distribution,

\[\boxed{ \text{Mean} = \text{Mode} }\]

so these estimates coincide.

For asymmetric or multimodal distributions, they may be very different.

4.2.5.44 Uncertainty Propagation Through Linear Dynamics

Suppose

\[\mathbf x_k \sim \mathcal N( \hat{\mathbf x}_k, \mathbf P_k )\]

and the dynamics are linear:

\[\mathbf x_{k+1} = \mathbf F_k\mathbf x_k + \mathbf G_k\mathbf u_k + \mathbf w_k\]

with

\[\mathbf w_k \sim \mathcal N( 0,\mathbf Q_k ).\]

Then the predicted mean is

\[\boxed{ \hat{\mathbf x}_{k+1}^{-} = \mathbf F_k\hat{\mathbf x}_k + \mathbf G_k\mathbf u_k }\]

and the covariance propagates as

\[\boxed{ \mathbf P_{k+1}^{-} = \mathbf F_k \mathbf P_k \mathbf F_k^T + \mathbf Q_k }.\]

This equation will become one of the central equations of the Kalman filter.

4.2.5.45 Physical Meaning of Covariance Propagation

The term

\[\mathbf F_k \mathbf P_k \mathbf F_k^T\]

propagates existing state uncertainty through the spacecraft dynamics.

The term

\[\mathbf Q_k\]

adds uncertainty created by imperfect process modelling.

Therefore:

\[\boxed{ \text{Predicted Uncertainty} = \text{Propagated Previous Uncertainty} + \text{New Process Uncertainty} }.\]

This is the probabilistic equivalent of acknowledging that the model is not perfect.

4.2.5.46 Uncertainty Through Nonlinear Dynamics

For nonlinear dynamics,

\[\mathbf x_{k+1} = f(\mathbf x_k,\mathbf u_k) + \mathbf w_k,\]

a Gaussian input does not generally remain exactly Gaussian after transformation through f.

This creates a fundamental challenge.

Different estimators address it differently.

4.2.5.46.1 EKF

Approximates the nonlinear function locally using Jacobians.

4.2.5.46.2 UKF

Propagates carefully selected sigma points through the nonlinear function.

4.2.5.46.3 Particle Filter

Propagates many samples through the full nonlinear dynamics.

Therefore, later filters can be understood largely as different practical approximations to the same Bayesian recursion.

4.2.5.47 Bayesian Interpretation of Sensor Trust

Suppose two sensors observe the same physical quantity.

Sensor A has low uncertainty:

\[\sigma_A^2\ll\sigma_B^2.\]

Sensor A therefore has a narrower likelihood.

Sensor B has a broader likelihood.

When combined with the prior, sensor A generally produces a stronger posterior shift.

Thus:

\[\boxed{ \text{Lower Measurement Uncertainty} \rightarrow \text{More Information} \rightarrow \text{Greater Influence} }.\]

This is the probabilistic foundation of measurement weighting.

4.2.5.48 Information and Covariance

For a Gaussian state,

\[\mathbf x \sim \mathcal N( \boldsymbol\mu,\mathbf P ),\]

the inverse covariance

\[\boxed{ \mathbf Y = \mathbf P^{-1} }\]

is called the information matrix.

Conceptually,

\[\boxed{ \text{Large Covariance} \Rightarrow \text{Low Information} }\]

while

\[\boxed{ \text{Small Covariance} \Rightarrow \text{High Information}. }\]

This concept later becomes important in information filters and decentralized sensor fusion.

4.2.5.49 Bayesian Attitude Estimation

Attitude estimation presents an additional complication.

A quaternion

\[\mathbf q\]

must satisfy

\[\boxed{ \|\mathbf q\|=1. }\]

Therefore quaternion components are not four independent Euclidean variables.

A naive Gaussian distribution over the four quaternion components can be problematic.

Practical spacecraft estimators often represent uncertainty using a small three-dimensional attitude error:

\[\boxed{ \delta\boldsymbol\theta }\]

while maintaining a nominal quaternion separately.

This idea becomes fundamental in:

• MEKF,
• error-state filtering,
• manifold-aware estimation.

Thus Bayesian estimation must respect the geometry of the state being estimated.

4.2.5.50 Bayesian IMU/GNSS Fusion

Consider an IMU/GNSS navigation system.

The IMU provides high-rate information about motion.

The navigation state may contain

\[\boxed{ \mathbf x = \begin{bmatrix} \mathbf r\\ \mathbf v\\ \mathbf q\\ \mathbf b_g\\ \mathbf b_a \end{bmatrix}. }\]

IMU measurements propagate the state rapidly.

But uncertainty grows because of:

• gyro noise,
• accelerometer noise,
• gyro bias,
• accelerometer bias,
• integration drift.

GNSS periodically provides position and velocity information.

Bayesian fusion therefore behaves conceptually as

\[\boxed{\text{IMU}} \rightarrow \boxed{\text{High-Rate Prediction}}\]

and

\[\boxed{\text{GNSS}} \rightarrow \boxed{\text{Probabilistic Correction}}.\]

This is one of the classic applications of recursive Bayesian estimation.

4.2.5.51 Bayesian RPO Sensor Fusion

For RPO, consider relative state

\[\boxed{ \mathbf x_{rel} = \begin{bmatrix} \mathbf r_{rel}\\ \mathbf v_{rel}\\ \mathbf q_{rel}\\ \boldsymbol\omega_{rel}\\ \cdots \end{bmatrix}. }\]

The estimator may receive measurements from:

\[\boxed{\text{GNSS}}\]

\[\boxed{\text{Camera}}\]

\[\boxed{\text{LiDAR}}\]

\[\boxed{\text{IMU}}\]

and potentially other relative-navigation sensors.

Each sensor contributes a likelihood:

\[p( \mathbf z_{GNSS} | \mathbf x_{rel} ),\]

\[p( \mathbf z_{cam} | \mathbf x_{rel} ),\]

\[p( \mathbf z_{LiDAR} | \mathbf x_{rel} ).\]

The navigation system combines these with the predicted relative-state distribution.

Thus the central RPO fusion problem becomes:

\[\boxed{ \text{Relative Dynamics} + \text{Sensor Likelihoods} \rightarrow \text{Relative-State Posterior} }.\]

4.2.5.52 Mission-Phase-Dependent Uncertainty

Sensor usefulness can change with mission phase.

At long range, GNSS may provide the strongest relative-navigation information.

As the chaser approaches the target, camera measurements may become increasingly useful.

At close range, LiDAR or pose-estimation measurements may provide high-value geometric information.

Thus:

\[\boxed{ \text{Sensor Information Content} = f( \text{Range}, \text{Geometry}, \text{Mode}, \text{Environment} ) }.\]

A sophisticated estimator may therefore use measurement covariances or sensor-selection logic that changes with mission phase.

4.2.5.53 Prior Information During Sensor Handover

Suppose navigation changes from GNSS-dominated estimation to vision-dominated estimation.

The vision estimator does not need to begin with no knowledge.

The existing state distribution can serve as its prior:

\[\boxed{ p( \mathbf x_{rel} | \text{GNSS history} ) }.\]

The new vision measurement then updates that prior.

Therefore:

\[\boxed{ \text{Previous Sensor Information} \rightarrow \text{Prior} \rightarrow \text{New Sensor Update}. }\]

This is one reason probabilistic state estimation is particularly useful for multi-phase spacecraft navigation.

4.2.5.54 Probability of Capture Conditions

Suppose a capture requirement is expressed as

\[\|\mathbf r_{rel}\| < r_{capture}\]

and

\[\|\mathbf v_{rel}\| < v_{capture}.\]

A deterministic estimator might only check

\[\|\hat{\mathbf r}_{rel}\| < r_{capture}.\]

But if uncertainty is large, the true spacecraft state may still violate the capture boundary.

A probabilistic system can conceptually evaluate quantities such as

\[\boxed{ P( \|\mathbf r_{rel}\|

and

\[\boxed{ P( \|\mathbf v_{rel}\|

This illustrates why uncertainty is operationally important rather than merely mathematical.

The same concept applies to collision avoidance and keep-out-zone constraints.

4.2.5.55 Confidence Regions

For a Gaussian estimate, covariance can be used to construct confidence regions.

In one dimension, familiar intervals are associated with multiples of standard deviation.

In multiple dimensions, confidence regions are defined using

\[\boxed{ (\mathbf x-\hat{\mathbf x})^T \mathbf P^{-1} (\mathbf x-\hat{\mathbf x}) \le c }.\]

The appropriate c depends on:

• state dimension,
• desired probability level,
• assumed distribution.

This distinction is important: a multidimensional "3-sigma region" should not be interpreted exactly like a one-dimensional ±3σ interval.

4.2.5.56 Estimator Consistency

An estimator should not only be accurate.

Its reported uncertainty should also agree with its actual errors.

Suppose actual estimation errors are frequently much larger than predicted by

\[\mathbf P.\]

Then the estimator is overconfident.

If \mathbf P is consistently much larger than actual errors, the estimator is overly conservative.

A well-behaved estimator should be reasonably consistent:

\[\boxed{ \text{Reported Uncertainty} \approx \text{Observed Estimation Error Statistics}. }\]

This concept becomes important in Monte Carlo validation.

4.2.5.57 Accuracy vs Precision

These terms should not be confused.

Accuracy describes closeness to the true value.

Precision describes concentration or repeatability.

A narrow probability distribution can still be centred on the wrong state because of bias.

Therefore:

\[\boxed{ \text{Small Covariance} \not\Rightarrow \text{Correct Estimate}. }\]

A filter can be extremely confident and still wrong.

This can occur because of:

• incorrect sensor bias models,
• wrong dynamics,
• unmodelled correlations,
• poor initialization,
• invalid measurements.

4.2.5.58 Prior Quality Matters

Bayesian estimation uses prior information.

A good prior can improve estimation substantially.

But an incorrect and overly confident prior can also cause problems.

Suppose the actual state lies far from the prior mean, while the prior covariance is unrealistically small.

Then valid sensor measurements may appear inconsistent with the predicted state.

The estimator may converge slowly or even reject useful measurements.

Therefore:

\[\boxed{ \text{Prior Mean} + \text{Prior Covariance} }\]

must both be initialized realistically.

This leads later to the important topic of filter initialization and convergence.

4.2.5.59 Bayesian Estimation and Sensor Specifications

Sensor datasheets often provide quantities such as:

• noise density,
• angular random walk,
• velocity random walk,
• bias instability,
• position accuracy,
• velocity accuracy,
• range accuracy,
• angular accuracy.

These quantities can help construct stochastic measurement and process models.

However, a datasheet specification cannot always be inserted directly into

\[\mathbf Q\]

or

\[\mathbf R\]

without considering:

• units,
• sampling frequency,
• bandwidth,
• continuous vs discrete noise,
• preprocessing,
• sensor operating conditions,
• correlation,
• frame transformation.

Thus:

\[\boxed{ \text{Sensor Specification} \rightarrow \text{Stochastic Model} \rightarrow \mathbf Q,\mathbf R }\]

requires careful modelling.

4.2.5.60 Bayesian Estimation and Measurement Validation

Bayesian theory assumes that the likelihood correctly represents the measurement-generating process.

If a camera falsely tracks the wrong feature, the nominal camera likelihood may no longer be valid.

Similarly, if GNSS is degraded or LiDAR reports a false return, using the normal \mathbf R can corrupt the state estimate.

Practical systems therefore place measurement validation before the Bayesian update:

\[\boxed{\text{Raw Measurement}} \rightarrow \boxed{\text{Preprocessing}} \rightarrow \boxed{\text{Validity Check}} \rightarrow \boxed{\text{Likelihood / Measurement Update}}.\]

This connects Bayesian estimation with later sections on gating, FDIR, and robust fusion.

4.2.5.61 Bayesian Filtering vs Bayesian Smoothing

Filtering estimates the current state using measurements up to the current time:

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) }.\]

Smoothing estimates a past state using measurements that arrived later:

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:N} ), \qquad N>k. }\]

Therefore:

\[\boxed{ \text{Filtering} = \text{Present Estimate from Present and Past Data} }\]

while

\[\boxed{ \text{Smoothing} = \text{Past Estimate Improved Using Future Data}. }\]

Real-time spacecraft GNC primarily requires filtering, while smoothing can be valuable for post-flight analysis, orbit reconstruction, calibration, and offline navigation improvement.

4.2.5.62 Bayesian Filtering vs Prediction

It is also useful to distinguish:

4.2.5.62.1 Filtering

\[p( \mathbf x_k | \mathbf z_{1:k} )\]

uses measurements through the current time.

4.2.5.62.2 Prediction

\[p( \mathbf x_{k+\ell} | \mathbf z_{1:k} )\]

estimates a future state.

4.2.5.62.3 Smoothing

\[p( \mathbf x_{k-\ell} | \mathbf z_{1:k} )\]

re-estimates a past state.

These are three different inference problems built on the same probabilistic framework.

4.2.5.63 From Bayesian Estimation to Kalman Filtering

The full Bayesian recursion is conceptually elegant:

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k-1} ) = \int p( \mathbf x_k | \mathbf x_{k-1} ) p( \mathbf x_{k-1} | \mathbf z_{1:k-1} ) d\mathbf x_{k-1} }\]

followed by

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) \propto p( \mathbf z_k | \mathbf x_k ) p( \mathbf x_k | \mathbf z_{1:k-1} ) }.\]

However, evaluating and storing arbitrary probability distributions can be computationally expensive.

A major simplification occurs when:

1. the dynamics are linear,
2. the measurement model is linear,
3. process noise is Gaussian,
4. measurement noise is Gaussian,
5. the initial state distribution is Gaussian.

Then the state distribution remains Gaussian.

Therefore the entire posterior can be represented by only

\[\boxed{ \hat{\mathbf x}_k }\]

and

\[\boxed{ \mathbf P_k. }\]

This special recursive Bayesian estimator is the Kalman filter.

4.2.5.64 Where the Major Filters Fit

The Bayesian framework allows the major spacecraft estimators to be placed in one conceptual hierarchy.

4.2.5.64.1 Kalman Filter

\[\boxed{ \text{Linear Dynamics} + \text{Linear Measurements} + \text{Gaussian Uncertainty} }\]

4.2.5.64.2 Extended Kalman Filter

\[\boxed{ \text{Nonlinear System} + \text{Local Linearization} }\]

4.2.5.64.3 Error-State Kalman Filter

\[\boxed{ \text{Nominal Nonlinear State} + \text{Small Error-State Estimation} }\]

4.2.5.64.4 Multiplicative EKF

\[\boxed{ \text{Quaternion Attitude} + \text{Multiplicative Attitude Error} }\]

4.2.5.64.5 Unscented Kalman Filter

\[\boxed{ \text{Nonlinear Transformation} + \text{Sigma-Point Approximation} }\]

4.2.5.64.6 Particle Filter

\[\boxed{ \text{General Probability Distribution} + \text{Weighted Samples} }\]

These methods are not unrelated algorithms.

They are different practical ways of approximating or implementing Bayesian state estimation.

4.2.5.65 Complete Bayesian Spacecraft Estimation Architecture

A complete probabilistic spacecraft navigation chain can be represented as:

\[\boxed{\text{Previous Posterior}}\]

\[\downarrow\]

\[\boxed{\text{Spacecraft Dynamic Model}}\]

together with

\[\boxed{\text{Process Uncertainty}}\]

\[\downarrow\]

\[\boxed{\text{Predicted State Distribution}}\]

\[\downarrow\]

\[\boxed{\text{Sensor Measurement}}\]

together with

\[\boxed{\text{Measurement Model}}\]

and

\[\boxed{\text{Measurement Uncertainty}}\]

\[\downarrow\]

\[\boxed{\text{Likelihood}}\]

\[\downarrow\]

\[\boxed{\text{Bayesian Update}}\]

\[\downarrow\]

\[\boxed{\text{Posterior State Distribution}}.\]

In practical flight software, additional layers are required:

\[\boxed{ \text{Time Synchronization} + \text{Measurement Validation} + \text{Sensor Health} + \text{Mode Logic} + \text{Outlier Rejection} }.\]

4.2.5.66 Observer-Based vs Bayesian Estimation

The connection between the previous section and this one is important.

An observer uses

\[\boxed{ \text{Model} + \text{Measurement Residual} + \text{Designed Gain} }\]

to produce

\[\boxed{\hat{\mathbf x}}.\]

Bayesian estimation uses

\[\boxed{ \text{Prior Distribution} + \text{Likelihood} }\]

to produce

\[\boxed{ \text{Posterior Distribution} }.\]

Thus the observer asks primarily:

How should measurement error correct my estimated state?

Bayesian estimation asks:

How should the probability distribution describing my knowledge of the state change after receiving this measurement?

The Kalman filter will connect these two views beautifully: it looks structurally like an observer, but its correction gain is derived from uncertainty.

4.2.5.67 Key Mathematical Vocabulary

Before moving to the Kalman filter, the following terms should be clearly understood:

\[\boxed{p(\mathbf x)}\]

Prior state probability.

\[\boxed{p(\mathbf z|\mathbf x)}\]

Measurement likelihood.

\[\boxed{p(\mathbf x|\mathbf z)}\]

Posterior state probability.

\[\boxed{p(\mathbf x,\mathbf z)}\]

Joint probability.

\[\boxed{E[\mathbf x]}\]

Expected value or mean.

\[\boxed{\mathbf P}\]

State-error covariance.

\[\boxed{\mathbf Q}\]

Process-noise covariance.

\[\boxed{\mathbf R}\]

Measurement-noise covariance.

\[\boxed{ p( \mathbf x_k | \mathbf x_{k-1} ) }\]

State-transition probability.

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k-1} ) }\]

Predicted state distribution or prior.

\[\boxed{ p( \mathbf x_k | \mathbf z_{1:k} ) }\]

Updated state distribution or posterior.

These quantities form the mathematical language of recursive Bayesian estimation.

4.2.5.68 Key Takeaways

Bayesian estimation provides the probabilistic foundation of spacecraft sensor fusion.

The central equation is

\[\boxed{ p(\mathbf x|\mathbf z) = \frac{ p(\mathbf z|\mathbf x)p(\mathbf x) }{ p(\mathbf z) } }.\]

The fundamental interpretation is

\[\boxed{ \text{Prior} + \text{Measurement Information} \rightarrow \text{Posterior} }.\]

The state is represented not only by an estimate but by uncertainty:

\[\boxed{ \hat{\mathbf x},\mathbf P }.\]

The dynamic model propagates both state knowledge and uncertainty.

Process uncertainty is represented by

\[\boxed{\mathbf Q}\]

while measurement uncertainty is represented by

\[\boxed{\mathbf R}.\]

Recursive Bayesian estimation consists of:

\[\boxed{\text{Prediction}} \rightarrow \boxed{\text{Measurement Update}} \rightarrow \boxed{\text{Prediction}} \rightarrow\cdots\]

Sensor dropout normally increases uncertainty.

Reliable measurements normally reduce uncertainty.

Measurements must provide observable information about the desired states.

Gaussian distributions allow uncertainty to be represented compactly using mean and covariance.

Nonlinear and non-Gaussian problems require approximations or more advanced probability representations.

Bayesian estimation provides the common conceptual foundation behind:

\[\boxed{ \text{KF} \rightarrow \text{EKF} \rightarrow \text{ESKF} \rightarrow \text{MEKF} \rightarrow \text{UKF} \rightarrow \text{Particle Filters} }.\]

Most importantly:

\[\boxed{ \text{A Good Navigation System Must Estimate Both State and Confidence in That State} }.\]

Next: 4.2.6 Kalman Filter Fundamentals

Bayesian estimation gives us the general recursion

\[\boxed{ \text{Prior Distribution} + \text{Measurement Likelihood} \rightarrow \text{Posterior Distribution}. }\]

But propagating arbitrary probability distributions can be computationally difficult.

If we assume a linear dynamic model, linear measurement model, and Gaussian uncertainties, the complete probability distribution can be represented by only

\[\boxed{\hat{\mathbf x}}\]

and

\[\boxed{\mathbf P}.\]

This produces one of the most important algorithms in spacecraft GNC:

\[\boxed{\text{The Kalman Filter}}.\]

The next section should therefore develop the Kalman filter from the Bayesian foundation rather than simply presenting its equations. It should derive and physically explain state prediction, covariance prediction, innovation, innovation covariance, Kalman gain, measurement correction, posterior covariance, Q/R interpretation, initialization, convergence, consistency, and spacecraft sensor-fusion examples.